[Paper Review] Disorder-induced rippled phases and multicriticality in free-standing graphene

David R. Saykin, V. Yu. Kachorovskii+1 more|arXiv (Cornell University)|Mar 6, 2020

Graphene research and applications49 references4 citations

TL;DR

This paper establishes a four-phase phase diagram for free-standing graphene with short-range disorder, demonstrating that strong disorder stabilizes rippled flat phases by counteracting thermal flexural fluctuations. Using a renormalization group approach in $d_c \to 2$ dimensions, it identifies a multicritical point where rippling and crumpling transitions intersect, revealing non-commutative limits between thermal and rippling fluctuations that stabilize two distinct flat phases at a single fixed point.

ABSTRACT

One of the most exciting phenomena observed in crystalline disordered membranes, including a suspended graphene, is rippling, i.e. a formation of static flexural deformations. Despite an active research, it still remains unclear whether the rippled phase exists in the thermodynamic limit, or it is destroyed by thermal fluctuations. We demonstrate that a sufficiently strong short-range disorder stabilizes ripples, whereas in the case of a weak disorder the thermal flexural fluctuations dominate in the thermodynamic limit. The phase diagram of the disordered suspended graphene contains two separatrices: the crumpling transition line dividing the flat and crumpled phases and the rippling transition line demarking the rippled and clean phases. At the intersection of the separatrices there is the unstable, multicritical point which splits up all four phases. Most remarkably, rippled and clean flat phases are described by a single stable fixed point which belongs to the rippling transition line. Coexistence of two flat phases in the single point is possible due to non-analiticity in corresponding renormalization group equations and reflects non-commutativity of limits of vanishing thermal and rippling fluctuations.

Motivation & Objective

• To resolve the long-standing question of whether rippled phases survive in the thermodynamic limit of disordered 2D membranes like suspended graphene.
• To clarify the interplay between thermal flexural fluctuations and static ripples induced by short-range curvature disorder.
• To construct a complete phase diagram for disordered suspended graphene, identifying distinct phases and their transitions.
• To investigate the non-commutativity of limits in vanishing thermal and rippling fluctuations, leading to coexistence of two flat phases.

Proposed method

• Employing a $1/d_c$ expansion in the $d_c \to 2$ limit, the authors derive renormalization group (RG) equations for bending rigidity $\varkappa$ and disorder parameter $f$.
• The RG flow is computed up to second order in $1/d_c$, incorporating self-energy corrections from both thermal fluctuations and disorder.
• The analysis includes momentum-scale-dependent renormalization of $\varkappa$ and $f$, with distinct characteristic scales $q_*$, $q_*'$, and $q_*''$ for different physical regimes.
• The RG equations are derived from perturbative corrections to the self-energy, with coefficients $\alpha_i$, $\gamma_i$, and $\tilde{\alpha}_i$, $\tilde{\gamma}_i$ computed from Feynman diagrams.
• The fixed points of the RG flow are analyzed to identify stable phases and multicritical behavior at the intersection of phase boundaries.
• Non-analytic behavior in the RG equations is used to explain the coexistence of clean and rippled flat phases at a single stable fixed point.

Experimental results

Research questions

• RQ1 Does a stable rippled phase exist in the thermodynamic limit of disordered suspended graphene?
• RQ2 How do thermal flexural fluctuations and static ripples compete to determine the phase diagram?
• RQ3 What is the nature of the multicritical point where rippling and crumpling transitions meet?
• RQ4 Why do two distinct flat phases—clean and rippled—coexist at a single fixed point despite different physical origins?
• RQ5 How does the non-commutativity of the limits $T \to 0$ and $f \to 0$ affect the phase structure?

Key findings

• The phase diagram of disordered suspended graphene contains four distinct phases: clean flat, rippled flat, clean crumpled, and rippled crumpled.
• A multicritical point exists at the intersection of the rippling and crumpling transition lines, where all four phases meet.
• The rippled and clean flat phases coexist at a single stable fixed point due to non-analyticity in the RG equations and the non-commutativity of $T \to 0$ and $f \to 0$ limits.
• For strong disorder, the rippled phase is stabilized and survives in the thermodynamic limit, while weak disorder leads to dominance of thermal fluctuations and crumpling.
• The RG flow shows that the $T=0$ rippled phase is marginally unstable, but its effects persist over a wide range of length scales in the presence of weak thermal fluctuations.
• The second-order $1/d_c$ corrections to the RG equations reveal non-trivial scaling of $\varkappa$ and $f$, with logarithmic dependence on momentum scale and coupling to disorder strength.