it exhibits extraordinary electrical and mechanical properties [ 1 - 5 ] , and has inspired an array of tantalizing potential applications ( e.g. , transparent flexible displays and biochemical sensor arrays ) [ 6 - 10 ] .
the random graphene morphology can lead to unstable performance of graphene devices as the corrugating physics of graphene is closely tied to its electronic properties .
future success of graphene - based applications hinges upon precise control of the graphene morphology over large areas , a significant challenge largely unexplored so far .
recent experiments show that , however , the morphology of graphene can be regulated by the surface of an underlying substrate [ 15 - 19 ] . in this paper , we quantitatively determine the regulated graphene morphology on substrates with various engineered surface patterns , using energy minimization .
the results reveal the snap - through instability of graphene on substrates , a promising mechanism to enable functional components for graphene devices .
recent experiments show that monolayer and few - layer graphene can partially follow the rough surface of the underlying substrates [ 15 - 19 ] .
the resulting graphene morphology is regulated , rather than the intrinsic random corrugations in freestanding graphene .
the substrate - regulated graphene morphology results from the interplay between the interfacial bonding energy and the strain energy of the graphene - substrate system , which can be explained as follows . when graphene is fabricated on a substrate surface via mechanical exfoliation   or
substrate interfacial bonding energy is usually weak ( e.g. , van der waals interaction ) . as the graphene corrugates to follow the substrate surface , the graphene 
substrate interaction energy decreases due to the nature of van der waals interaction ; on the other hand , the strain energy in the system increases due to the intrinsic bending rigidity of graphene . at the equilibrium graphene morphology on the substrate ,
the above energetic consideration can be used to quantitatively determine the regulated graphene morphology on a rough substrate surface .
furthermore , with a systematic understanding of the governing mechanisms of substrate - regulated graphene morphology , we envision a promising strategy to precisely pattern graphene into desired morphology on engineered substrate surfaces . in this paper
, we illustrate this strategy by determining the regulated graphene morphology on two types of engineered substrate surfaces : herringbone corrugations and checkerboard corrugations ( fig .
these substrate surface features can be fabricated via approaches combining lithography   and strain engineering .
the graphene  substrate interaction energy can be determined by summing up all van der waals forces between the graphene carbon atoms and the substrate atoms .
the van der waals force between a graphene  substrate atomic pair of distance r can be characterized by a lennard 
jones pair potential , vlj(r ) = 4( / r  /r ) , whereis the equilibrium distance of the atomic pair and  is the bonding energy at the equilibrium distance .
the number of atoms over an area d s on the graphene and a volume d vs in the substrate are c d s and s d vs , respectively , where c is the homogenized carbon atom area density of graphene that is related to the equilibrium carbon  carbon bond length l byand s is the molecular density of substrate that can be derived from the molecular mass and mass density of substrate .
the interaction energy , denoted by eint , between a graphene of area s and a substrate of volume vs is then given by since lennard  jones potential decays rapidly beyond equilibrium atomic pair distance , eint can be estimated by adding up the van der waals forces between each graphene carbon atom and the substrate portion within a cut - off distance from this carbon atom .
if the cut - off distance is large enough , such an estimate of interaction energy converges to the theoretical value of eint . in all simulations reported in this paper , a cut - off distance of 3 nm
was used and shown to lead to variations in the estimated value of eint less than 1% .
we have developed a monte carlo numerical scheme to compute the multiple integrals in eq .
n random locations are generated in the substrate portion within the cut - off distance from this carbon atom .
the interaction energy between this carbon atom and the substrate is estimated by where rij is the distance between the ith graphene carbon atom and the jth random substrate location .
equation 2 is evaluated at m equally spaced locations over the graphene of area s. the graphene  substrate interaction energy over this area can then be estimated by as n and m become large enough , eq .
3 converges to the theoretical value of eint . in all simulations in this paper , n = 10 , m = 400 .
the strain energy in the graphene  substrate system results from the corrugating deformation of the graphene and the interaction - induced deformation of the substrate .
when an ultrathin monolayer graphene partially conforms to a rigid substrate ( e.g. , sio2 ) , the substrate deformation due to the weak graphene 
also , when the graphene spontaneously follows the substrate surface under weak interaction ( imagine a fabric naturally conforming to a rough surface ) and is not subject to any mechanical constraints ( e.g. , pinning ) , the in - plane stretching of the graphene is also expected to be negligible . under the above assumptions , the strain energy in the graphene  substrate system can be reasonably estimated by the graphene strain energy due to out - of - plane bending , denoted by eg .
effect of the above assumptions on results is to be further elaborated later in this paper . denoting the out - of - plane displacement of the graphene by wg(xy )
, the graphene strain energy over an area s can be given by where d and  are the bending rigidity and the poisson s ratio of graphene , respectively .
the out - of - plane herringbone corrugations of the substrate surface ( fig . 1a ) and the out - of - plane corrugations of the graphene regulated by such a substrate surface are described by respectively , where as and ag are the amplitudes of the substrate surface corrugations and the graphene corrugations , respectively ; for both the graphene and the substrate , x is the wavelength of the out - of - plane corrugations , y and ay are the wavelength and the amplitude of in - plane jogs , respectively ; and h is the distance between the middle planes of the graphene and the substrate surface . given the symmetry of the herringbone pattern , we only need to consider a graphene segment over an area of x/2 by y/2 , and its interaction with the substrate . by substituting eq .
4 , the strain energy of such a graphene segment is given by as shown in eq .
6 , for a given substrate surface corrugation ( i.e. , as , ay , x , and y ) , eg increases monotonically as ag increases . on the other hand , the graphene 
substrate interaction energy , eint , minimizes at finite values of ag and h , due to the nature of van der waals interaction . as a result
, there exists a minimum of ( eg + eint ) where ag and h reach their equilibrium values .
the energy minimization was carried out by running a customized code on a high performance computation cluster . in all computations , d = 1.41 ev , l = 0.142 nm , s = 2.20  10/m, = 0.353 nm and as = 0.5 nm , which are representative of a graphene - on - sio2 structure .
various values of x,y , and ay were used to study the effects of interfacial bonding energy and substrate surface roughness on the regulated graphene morphology .
figure 2a plots the normalized amplitude of the regulated graphene corrugation , ag / as , as a function of d/ for various x .
thus , various x define a family of substrate surfaces with self - similar in - plane herringbone patterns and the same out - of - plane amplitude ( i.e. , as ) . for a given substrate surface pattern ,
if the interfacial bonding energy is strong ( i.e. , small d/ ) , ag tends to as . in other words ,
in contrast , if the interfacial bonding is weak ( i.e. , large d/ ) , ag approaches zero . that is , the graphene is nearly flat and does not conform to the substrate surface ( fig . 2c ) .
interestingly , there exists a threshold value of d/ , below and above which a sharp transition occurs between the above two distinct states of the graphene morphology .
. the threshold value of d/ increases as x increases . for a given interfacial bonding energy , ag increases as x increases .
that is , graphene tends to conform more to a substrate surface with smaller out - of - plane waviness .
aag / as on substrates with herringbone surface corrugations as a function of d/ for various x . at a threshold value of d/ ,
the graphene morphology snaps between two distinct states : b closely conforming to the substrate surface and c remaining nearly flat on the substrate surface figure 3 shows the effect of in - plane waviness of the substrate surface on graphene morphology .
here x = 6 nm and ay = y/4 . for a given substrate surface pattern ,
if the interfacial bonding energy is strong ( i.e. , small d/ ) , ag tends to as . for a given interfacial bonding energy , ag increases as y/x increases
that is , graphene tends to conform more to a substrate surface with smaller in - plane waviness . in particular ,
when y/x is large ( e.g. , 100 ) , the predicted graphene corrugation amplitude converges to that of graphene regulated by straight substrate surface grooves with the same x and as .
figure 3b further plots ag / as as a function of d/ for various ay with fixed x and y
. similar effect of in - plane waviness of the substrate surface on graphene morphology emerges from fig .
3b . moreover , the snap - through instability of graphene , similar to that illustrated in fig .
the snap - through instability of graphene morphology , similar to that shown in fig .
2 , is evident in both cases the snap - through instability of graphene on a substrate surface can be explained as follows .
figure 4 plots the normalized total system energy as a function of ag / as for various d/. here x = 9 nm , y = 2 x and ay = y/4 .
if the interfacial bonding energy is weaker ( d/ = 575 ) than a threshold value , the total energy profile reaches its minimum at a small graphene corrugation amplitude ag / as = 0.14 .
if the interfacial bonding energy ( d/ = 750 ) is stronger than the threshold value , the total energy profile reaches its minimum at a large graphene corrugation amplitude ag / as = 0.93 . at the threshold value of d/ = 650
, the total energy profile assumes a double - well shape , whose two minima ( ag / as = 0.20 and 0.91 ) correspond to the two distinct states of the graphene morphology on the substrate surface . the normalized total system energy as a function of ag / as for various d/. at a threshold value of d/ = 650 ,
the total system energy minimizes at two points , corresponding to the two distinct states of graphene morphology .
here x = 9 nm , y = 2 x and ay = y/4 in the case of graphene regulated by a substrate surface with checkerboard pattern ( fig .
1b ) , the substrate surface corrugations and the regulated graphene corrugations are described by respectively , where  is the wavelength of the out - of - plane corrugations for both the graphene and the substrate surface .
the numerical strategy similar to that aforementioned was implemented to determine the equilibrium amplitude of the regulated graphene morphology .
figure 5 plots ag / as regulated by the checkerboard substrate surface as a function of d/ for various . for a given substrate surface roughness , ag / as decreases as d/ increases . for a given interfacial bonding energy , ag / as increases as  increases . on a substrate surface with checkerboard corrugations
, graphene exhibits the snap - through instability as well , which also results from the double - well shape of the system energy profile at the threshold value of d/ , similar to that shown in fig .
4 . the threshold value of d/ at the graphene snap - through instability increases as  increases . ag / as on substrates with checkerboard surface corrugations as a function of d/ for various . the insets illustrate the two distinct states of graphene morphology at the snap - through instability in this paper we focus on graphene morphology spontaneously regulated by substrate surfaces via weak interaction . when a graphene / substrate structure is subject to external loading
, the graphene strain energy due to stretching and the substrate strain energy may also need to be considered . in this sense , the present model overestimates the graphene corrugation amplitude .
also the graphene / substrate interaction can be enhanced by the possible chemical bondings or pinnings at the interface . in this sense ,
in summary , we investigate the graphene morphology regulated by substrates with herringbone and checkerboard surface corrugations . depending on interfacial bonding energy and substrate surface roughness ,
the graphene morphology exhibits a sharp transition between two distinct states : ( 1 ) closely conforming to the substrate surface and ( 2 ) remaining nearly flat on the substrate surface .
the quantitative results suggest a promising strategy to control the graphene morphology through substrate regulation . while it is difficult to directly manipulate freestanding graphene , it is feasible to pattern the substrate surface via lithography   and strain engineering .
the regulated graphene morphology on such engineered substrate surfaces may lead to new pathways to control the graphene electronic properties , introducing desirable properties such as band - gap , or p / n junction behavior . in particular , the results shown in this paper ( e.g. , figs .
235 ) reveal a wide range of design tunability of the graphene snap - through instability on substrates through substrate surface patterning and interfacial adhesion tailoring , which offers abundant unexplored potential toward the design of functional graphene device components ( e.g. , nano - switches , nano - resonators ) .
this work is supported by the minta - martin foundation , a umd general research board summer research award to t. l. , and nsf cmmi 0856540 .
 we determine the graphene morphology regulated by substrates with herringbone and checkerboard surface corrugations . as the graphene  
 substrate interfacial bonding energy and the substrate surface roughness vary , the graphene morphology snaps between two distinct states : ( 1 ) closely conforming to the substrate and ( 2 ) remaining nearly flat on the substrate . 
 since the graphene morphology is strongly tied to the electronic properties of graphene , such a snap - through instability of graphene morphology can lead to desirable graphene electronic properties that could potentially enable graphene - based functional electronic components ( e.g. nano - switches ) . 
Introduction
Computational Model
Results and Discussion
Concluding Remarks
Acknowledgment