invariant set is an important concept in the theory of dynamical systems and it has a wide range of applications in control .one of the reasons of the interest is due to the fact that invariant sets enable us to estimate the attraction region of a dynamical system .we consider linear continuous dynamical systems in the form and discrete dynamical systems in the form where are the state variables , are the coefficient matrices , and and indicate continuous and discrete time steps , respectively .note that equations ( [ dyna1 ] ) and ( [ dyna2 ] ) can be treated as autonomous systems or as controlled systems . in the latter case ,the coefficient matrix  can be represented in the form of , where is the open - loop state matrix , is the control matrix , and is the gain matrix . for simplicity , we use the term system to indicate dynamical system .  intuitively , a set is called an invariant set for a system , if all the trajectories of the system , which are starting in , remain in .numerous surveys on the theory and applications of invariant sets are published in the recent decades , see e.g. , blanchini .recently , several sufficient and necessary conditions , which are simply refereed to as invariance conditions , are derived to verify if a set is an invariant set for a continuous or discrete system .various convex sets with different characteristics are considered as candidates for invariant sets .invariance conditions for polyhedra are given in .ellipsoidal sets as invariant sets are analyzed in .cones as invariant sets are studied in .a novel unified approach to derive invariance conditions for polyhedra , ellipsoids , and cones is presented in .although many mathematical techniques are developed to directly solve continuous systems , in practice , one usually solves a continuous system by applying certain discretization methods .assume that a set is an invariant set for a continuous system , then it should be also an invariant set for the discrete system , which is obtained by the discretization method , i.e. , discretization should preserve the invariance . however , this is not always true for every steplength used in the discretization method , thus it will be convenient if there exists a predictable threshold for valid invariance preserving steplength . the existence of such steplength thresholds of invariance preserving on various sets is thoroughly studied in . in this paper , we consider three types of discretization methods on polyhedra and we aim to derive valid thresholds of the steplength in terms of explicit form or obtained by using efficiently computable algorithms .the popularity of polyhedra as invariant sets is due to the fact that the state and control variables are usually represented in terms of linear inequalities .for taylor approximation type discretization methods , i.e. , the coefficient matrix of the discrete system is derived from the taylor expansion of , we present an algorithm to derive a valid steplength threshold for invariance preserving . in particular , the algorithm aims to find the first positive zeros of some polynomial functions related to the system and the polyhedron . for general rational function type discretization methods ,i.e. , the coefficient matrix of the discrete system is a rational function with respect to and , we derive a valid steplength threshold for invariance preserving that can be computed by using analogous methods as for the case of taylor approximation type methods .this steplength threshold is related to the steplength threshold for the forward euler method and the radius of absolute monotonicity of the discretization method .we note that this result is similar to the one presented in , where runge - kutta methods are considered .finally , we propose an optimization model to find the largest steplength threshold for the forward euler method .we note that some results on the use of the forward euler method to analyze invariance for continuous dynamical systems can be found in ._ notation _ : for the sake of simplicity , the following notational conventions are introduced .a _ nonnegative _ matrix , denoted by , means that all entries of are nonnegative .an _ off - diagonal nonnegative _ matrix , denoted by , means that all entries , except the diagonal entries , of are nonnegative .the paper is organized as follows . in section [ sec : back ] , some fundamental concepts , theorems , and the key problems in this paper are introduced . in section [ sec : main ] , we present our main results , i.e. , deriving valid steplength thresholds for invariance preserving , for the three types of discretization methods . finally , conclusions are provided in section [ sec : con ] .we now introduce the definitions of invariant sets for continuous and discrete systems .[ def1 ] a set in is an invariant set for  * the continuous system ( [ dyna1 ] ) if  implies , for all . * the discrete system ( [ dyna2 ] ) if implies , for all .according to the definitions of invariant sets , we have that an invariant set means that the continuous ( or discrete ) trajectory of the system remains in the same set .in fact , there is an alternative perspective , see e.g. , . in that interpretation is an invariant set for ( [ dyna1 ] ) if and only if for any , and is an invariant set for ( [ dyna2 ] ) if and only if   in this paper , candidate invariant sets are restricted to convex polyhedron in .a polyhedron in can be characterized as the intersection of a finite number of half spaces .a polyhedron in is defined as where  and \in \mathbb{r}^{n\times m } ] , then the discretization method is * invariance preserving * for  ] , one needs to check every in this interval , which means that there are infinitely many values to be considered . ]this is an important property . in certain cases ,a discretization method may be invariance preserving on a set in the form of \cup[\tau_2,\tau_3] ] on . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _in this section , we present the approaches for computing a valid ( or largest ) steplength threshold such that three classes of discretization methods are invariance preserving on a polyhedron .these three classes of discretization methods are considered in the following order : taylor approximation type discretization methods , rational function type discretizatin methods , and the forward euler method .the taylor approximation type represents a family of explicit methods .the rational function type is an extended family of the taylor approximation type , which also includes some implicit methods .the relationship between these discretization methods and the forward euler method is also studied . finally ,for the forward euler method , we derive the largest steplength threshold for invariance preserving .we first consider the taylor approximation type discretization methods .note that the solution of the continuous system ( [ dyna1 ] ) is explicitly represented as , thus one can use the taylor approximation to numerically solve the continuous system .the -order taylor approximation of is given as follows : the discrete system obtained by applying the taylor approximation type discretization methods is given as where is defined by ( [ eq13 ] ) .in fact , the taylor approximation type methods form a family of discretization methods . for example, corresponds to the forward euler method , corresponds to the general runge - kutta 2nd order methods .our approach to derive steplength threshold is based on the invariance conditions presented in theorem [ poly ] .the basic ideas is that we build the relationship between these two invariance conditions of the continuous and discrete systems .in fact , conditions ( [ invcont ] ) and ( [ invdis ] ) are essentially linear feasibility problems .the unknowns in the two invariance conditions are the matrix and given by ( [ invcont ] ) and ( [ invdis ] ) , respectively .thus , the key is to find relationship between those matrices .[ lemgam ] _ _assume satisfies ( [ invcont ] ) , then there exists , such that   since we can choose , which yields the result is immediate by taking   we note that in lemma [ lemgam ] is not unique , e.g. , any value greater than a valid is also valid. we will show more about the effect of to the steplength threshold in section [ sec : ration ] , and the way to derive a larger steplength threshold based on is also presented .[ leminvar ] assume satisfies ( [ invcont ] ) , and define   * for the and given in lemma [ lemgam ] , we have where and * let where is the first positive zero of then for all , ] then we let thus we have for all  ] where is defined by ( [ valtau ] ) .thus , we have proved that the first condition in ( [ invdis ] ) is satisfied .  by recursivelyusing , for any we have then , according to ( [ p21:eq1 ] ) , and substituting ( [ p2:eq3 ] ) and ( [ eq13 ] ) , we have thus , we have proved that the second condition in ( [ invdis ] ) is satisfied .since satisfies ( [ invcont ] ) , we have . also , note that , thus we have since , we have then , according to ( [ eq3875o1 ] ) and ( [ lemma87691 ] ) , we have thus , we have proved that the third condition in ( [ invdis ] ) is satisfied .the proof is complete .lemma [ leminvar ] presents an important relationship between the two matrices and corresponding to the continuous and discrete systems invariance conditions .this relationship is explicitly represented in ( [ p2:eq3 ] ) , which is derived from the taylor approximation ( [ eq13 ] ) . according to lemma [ leminvar ] and theorem [ poly ] , we have the following theorem .[ thm123 ] assume a polyhedron be given as in ( [ poly1 ] ) is an invariant set for the continuous system ( [ dyna1 ] ) , and a taylor approximation type discretization method ( [ eq13 ] ) is applied to the continuous system ( [ dyna1 ] ) .then , the steplength threshold as given in lemma [ leminvar ] is a valid steplength threshold for invariance preserving for the given taylor approximation type discretization method ( [ eq13 ] ) on .  according to the proof of lemma [ leminvar ] , we have that a valid requires for all  ] the matrix satisfies ( [ invdis ] ) , where is defined by ( [ eq13 ] ) .a ) . according to lemma [ lemgam ] , there exists a , such that the matrix given by ( [ p2:eqd3 ] ) is represented in terms of . by substituting into ( [ p2:eqd3 ] ) , we now reformulate in terms of , i.e. , according to ( [ lem45:eqd2 ] ) , the coefficient of , for , is given as which is the same as ( [ lemma4d : eq3 ] ) .we note that is equivalent to replacing and by 1 and , respectively , in ( [ lemma4d : eq1 ] ) .then , according to ( [ lem45:eqd2 ] ) , we have for , we have implying that the right hand side of ( [ lemma453:eqd31 ] ) equals to 1 , thus ( [ lemma87dd ] ) follows immediately .the proof for part b ) is the same as the one presented for part b ) in lemma [ leminvar ] , thus we are not presenting here .we now consider the value of , i.e. , the steplength threshold . in this section, we present an algorithm to numerically compute .in particular , this algorithm aims to find the first positive zero of a polynomial function in the form of ( [ thm311:eq1 ] ) .[ lemm321 ]let be given as in ( [ thm311:eq1 ] ) .there exists a , such that for all . ] then we have note that this case only occurs when dominates .this is presented in the following lemma .[ lema : bnd ] let be given as in ( [ thm311:eq1 ] ) and .let  and .then if has no real zero in  ] we have on  ] .then we have the proof is complete .in fact , the value given in lemma [ lema : bnd ] can be considered as one of the termination criteria of the algorithm to find the first positive zero of , where is defined as ( [ thm311:eq1 ] ) .+ the sturm sequence of and the sturm theorem presented in the following definition play a key role in our algorithm . the sturm theorem aims to give the number of real zeros of a univariate polynomial function in an interval by using the property of sturm sequence on the end points of the interval .  _let be a univariate polynomial function .the * sturm sequence * , of is defined as where is the derivative of with respect to , and is the negative of the remainder on division of by   for the sake of simplicity , we introduce the following definition and notation , which are used in the statement of the sturm theorem .  for a sequence , the * number of sign changes * , denoted by , is the number of the times of the signs change ( zeros are ignored ) from to .for example , if a sequence is given as , then the signs of the sequence are . by eliminating all zeros, we have , which has 3 sign changes , i.e. ,   [ sturm ] _ _ * ( sturm theorem ) * let be a univariate polynomial function .if and then the number of distinct real zeros of in the interval  ] the number of positive zeros of at interval . ] can be computed by sturm theorem [ sturm ] .the basic idea in our algorithm is by using the bisection method to shrink the interval , which contains the first positive zero of , by 2 in each iteration .our algorithm is presented as follows .* step 0 : [ initial inputs ] * : :  set .iterate   until  =0. ] then . * step 3 : [ termination 2 ] * : :  if =1 ] and   , then   * * [ update * if =1 ] , then set    * * [ update * if =0 ] , thus we return , i.e. , the left end of the interval .  if all coefficients for in ( [ p2:eqd3 ] ) , then the algorithm is also applicable to compute a valid steplength threshold for invariance preserving for the polynomial approximation ( [ p2:eqd3 ] ) .the previous discussion is mainly about a steplength threshold for invariance preserving for a taylor approximation type discretization methods as specified in ( [ eq13 ] ) . in this section, we consider more general discretization methods , which are refereed to as the rational function type discretization methods . to be specific , these discretization methods applying to the continuous system yield the discrete system where is a rational function defined as where , and .it is clear that taylor approximation type discretization methods belong to this type .some implicit methods are also in this type , e.g. , the backward euler method , lobatto methods , etc .[ lem : rat ] _ _ let be given as in ( [ th2:eq11 ] ) , and let be a matrix .assume is nonsingular , then    in this subsection , our analysis uses the so called _ radius of absolute monotonicity _ of a function .  _let .if \}, ] this requires that the polynomial function in the numerator of is nonnegative for  ] , by the convexity of we have which completes the proof .we now consider the calculation of , where is defined as in corollary [ cor : bound ] . by the formula of , we need to compute , such that ( [ eq6 ] ) is satisfied .in fact , this can be achieved by solving the following optimization problem : since , and are known , optimization problem ( [ opt87 ] ) is a linear optimization problem .one may obtain different values of by choosing other objective functions in ( [ opt87 ] ) .the advantage by using the current objective function in ( [ opt87 ] ) is that this optimization problem yields the largest that satisfies ( [ eq6 ] ) .this is since the objective function in ( [ opt87 ] ) is thus , the value of obtained by solving the optimization problem ( [ opt87 ] ) is the largest possible value of .an alternative is presented by the following discussion .equation ( [ eq5 ] ) implies that is a feasible direction , i.e. , for sufficiently small then we can formulate the following linear optimization problem :   optimization problems ( [ opt87 ] ) and ( [ opt31 ] ) are equivalent problems , i.e. , we claim that is equal to . observing that for the first constraint in ( [ opt31 ] ) , we have i.e. , this , by letting gives the first constraint in ( [ opt87 ] ) .  according to the argument for above , we have the following theorem .  [ thm43 ]assume that the polytope defined as in ( [ eq : tope ] ) is an invariant set for the continuous system ( [ dyna1 ] ) , and the forward euler method is applied to ( [ dyna1 ] ) .then , , where is defined as in corollary [ cor : bound ] , is the largest steplength threshold for invariance preserving of the forward euler method on .  for any , and ,$ ]we have . according to corollary [ cor : bound ] and , we have .thus we have the proof is complete .many real world problems are studied by developing dynamical system models . in practice ,continuous systems are usually solved by using discretization methods . in this paper , we consider invariance preserving steplength thresholds on polyhedron , when the discrete system is obtained by using special classes of discretization methods .we particularly study three classes of discretization methods , which are : taylor approximation type , rational function type , and the forward euler method .  for the first class of discretization methods , we show that a valid steplength threshold can be obtained by finding the first positive zeros of a finite number of polynomial functions. we also present a simple and efficient algorithm to numerically compute these positive zeros .for the second class of discretization methods , a valid steplength threshold for invariance preserving is presented .this steplength threshold depends on the radius of absolute monotonicity , and can be computed by analogous method as in the first case . for the forward euler methodwe prove that the largest steplength threshold can be obtained by solving a finite number of linear optimization problems .this research is supported by a start - up grant of lehigh university and by tamop-4.2.2.a-11/1konv-2012 - 0012 : basic research for the development of hybrid and electric vehicles .the tamop project is supported by the european union and co - financed by the european regional development fund .f. blanchini , s. miani , c.e.t .drea and j.c .hennet , discussion on : ` ( a , b)- invariance conditions of polyhedral domains for continuous - time systems by c.e.t .drea and j .- c .hennet ' , _european journal of control _ , * 5 * ( 1999 ) , 82 - 86 .z. horvth , y. song and t. terlaky , invariance preserving discretization methods of dynamical systems , _ lehigh university , department of industrial and systems engineering , technical report 14t-009 _ , 2014 .

Abstract: steplength thresholds for invariance preserving of three types of discretization methods on a polyhedron are considered . for taylor approximation type discretization methods we prove that a valid steplength threshold can be obtained by finding the first positive zeros of a finite number of polynomial functions . further , a simple and efficient algorithm is proposed to numerically compute the steplength threshold . for rational function type discretization methods we derive a valid steplength threshold for invariance preserving , which can be computed by using an analogous algorithm as in the first case . the relationship between the previous two types of discretization methods and the forward euler method is studied . finally , we show that , for the forward euler method , the largest steplength threshold for invariance preserving can be computed by solving a finite number of linear optimization problems .   zoltn horvth  yunfei song and tams terlaky  ( communicated by kok lay teo )


the first data provided by corot for scuti stars made clear that the larger the precision of the observations , the larger the number of observed modes . in the case , for example , of hd174936 , a scuti target of the initial corot run , the number of frequencies extracted from the observed light curves with the traditional tools can be of the order of thousands .the same can be said from other corot scuti and doradus targets .in such a scenario , the question if all these frequencies are real or not should be a previous step to any modeling attempt . to answer this ,our aim is to check if a scuti or a doradus star has enough energy to excite such a large amount of modes simultaneously and with the observed amplitudes , for which we need to know the real energy of the modes .  up to our knowledge ,the only existing studies about the real energy of the modes , through characterisation of their amplitudes , are for stochastic pulsators ( as solar - like stars ) which were done through a description of the non - linear effects in the pulsation equations . in the case of thermodynamic pulsators ,as those driven by a mechanism , this kind of studies have not yet been attempted , since the non - adiabatic studies have been done in the linear approximation .therefore , the solutions yielded by the pulsation codes are not absolute , as they are affected by a constant indefinite factor , and only ratios of observed and theoretical quantities can be compared .we scaled the solutions given by the non - adiabatic linear pulsational code graco to absolute values using the observational amplitudes provided by space and ground - base data , as described below . in this way, we were able to compare the derived absolute values of the required pulsational energy to simultaneously excite all the observed modes , with the real energy available for exciting the modes .in our study we use one of the models best reproducing the ground - based and space observations obtained for the scuti hd174936 ( see * ? ? ?* ) , and a model at the centre of the observed hr photometric error box for the hybrid scuti - doradus star hd49434 .in order to check the consistency of our procedure , we use a representative model of the largest amplitude pulsator known , classical cepheid stars presenting variations of the order of magnitudes , and verified that it is energetically stable under its pulsations .to determine the energy needed by the system to unstabilize all of the observed modes , we have to face the problem that most of the pulsational codes , and in particular the non - adiabatic ones , use the linear approximation to solve the system of differential equations .aside from non - linear physical effects , negligible for small pulsational amplitudes , this means that every eigenfunction of an oscillation mode is known except for a multiplicative constant , i.e. , if are the solutions of the set of differential equations for a given mode , , being a constant , are also solutions . therefore , before doing any study about the global properties of the star , we have to determine the value of this constant ( ) scaling the linear solutions of each mode to the real values .  in the linear approximation ,the problem of this indefinite constant is solved by scaling all the solutions of the pulsation equations to an arbitrary normalization condition .most pulsation codes use as normalization condition    where is the radial eigenfunction .the real pulsation amplitude on the surface is given by    and following eq .[ norm ]    therefore , the multiplicative constant , scaling all the linear solutions to the real values of the pulsation , may be obtained through the oscillation amplitude of the relative radius at the surface of the star .this value can be deduced from the oscillation amplitude of the observed luminosity .we use the observed amplitude spectrum for hd174936 ( fig .[ espec ] and * ? ? ?* ) as an example to illustrate the procedure here presented .the spectrum shows a maximum oscillation amplitude of about 2 mmag , which would correspond to a low - degree mode .decreasing amplitudes are found at both sides of this highest amplitude mode .therefore , we assume that : `` the mode amplitude is given by a gaussian distribution centered in the largest amplitude mode , with the maximum amplitude of this mode , and the variance fixed by the decrease in amplitude of the modes closest to the highest one .this intrinsic amplitude is independent of the spherical degree of the mode '' .the oscillation amplitudes of the other modes , out of the highest amplitude one , would be affected by geometrical visibility , which is translated in amplitudes lower than the predicted gaussian envelope .  in this case , the distribution is centered on 377  , with a maximum amplitude of 2.12 mmag and a square root of the variance .the assumed oscillation amplitudes are also shown in fig .[ espec ] ( left , plus signs ) , compared with the observed amplitude spectrum .we notice that a significant number of observed amplitudes fall out of the range ~\mu]hz well comprising the canonical range of observed modes for scuti .we deliberately leave out modes under 100  as they lay far out of the instability range of scuti and no model is able to predict their excitation .the next step is to relate this intrinsic oscillation amplitude in luminosity to oscillation amplitude in relative radius .to do so , we use the definition of the apparent magnitude variation for a certain wavelength given by     where is the projected area of the photosphere , and is the variation in surface brightness defined as    with being the projected flux at a certain wavelength .given that the unknown multiplicative constant is defined by the variation of the radius , we are only interested in the intrinsic amplitude of the radial eigenfunction . therefore , although the relative variation of the projected surface area includes dependencies as the degree of the modes , the viewing angle or the limb darkening , we may adopt , for our purposes , as the radial variation of the total surface of the star .in addition , as the observed corot amplitude spectrum for hd174936 was taken in white light , we will consider as the total flux at the surface .therefore , , since the bolometric magnitude ( ) and absolute magnitude ( ) differ only in a constant .including all this in eq .[ eq : mlambda ] yields  \ ] ]  where is stefan boltzmann constant .simple calculations give :    therefore , the variations in brightness are a combination of the variations of the relative radius and the relative flux at the surface of the star , measured through relative variations of the effective temperature. the observed variations of the relative radius may be obtained as    where    the negative sign of this equation is related to the definition of the stellar magnitude and it has no physical origin. we will not take it into account in the rest of this study , since it has no influence in the results .therefore , once the variations in brightness are known , the relative radius variations can be obtained , as the variable is supplied by the non - adiabatic equations .we used the graco non - adiabatic code for this purpose , including the time dependent convection description ( tdc , * ? ? ?* ; * ? ? ?* ) , although its inclusion revealed of no significance in the results .the final relative radius variations as a function of the frequency of the modes for the two distributions are displayed also in fig .[ espec ] .once the multiplicative constant scaling all the linear solutions to their real values is known , the absolute values of the kinetic energy and the energy required to excite the observed modes can be estimated .then , they can be compared to the energy balancing the star and the energy available in the star , respectively .  to checkif the star is able to have thousands of modes excited at a time , we have to verify 1 ) if the energy flux is enough to provide the energy that all these modes need per second , and 2 ) if the total kinetic energy accumulated in them does not destroy the equilibrium structure of the star .we note that quantitative estimations of upper limits of these energies are enough to achieve the aims of this work .three stellar cases have been studied : 1 ) a representative model of the classical cepheid stellar type , adopting the largest pulsation amplitude reported ( around 2 mag ) as a falsifiability test of our main assumptions , 2 ) the scuti star hd174936 , and 3 ) the hybrid scuti - doradus star hd49434 .  for the latter two cases , we have used corot and ground - based observations . in the case of hd49434 , the publication ofthe data analysis is still in preparation and we used here only some general and preliminary results , which is , however , enough for our purpose .our first stellar case applies to the variable stellar type with the highest amplitude luminosity known to date , the classical cepheid type .this type of stars do not show a large amount of pulsational modes ; however , we are going to use it as a test of falsifiability of our main hypotheses , since for them , its high variability offsets the smaller number of pulsation modes .we built a representative equilibrium model lying in the classical cepheid instability strip .the main physical characteristics of this model are displayed in table 1 , and are similar to one of the thoeretical models studied in . the energy balance of the pulsational modesis then compared to some absolute values of the star and with some theoretical results obtained through non - linear resolutions , as described in section 2 .the variation of the relative radius of the fundamental radial mode obtained following our procedure ( ) is similar to the predicted for an equivalent non - linear model described in ( ) .the energy balance between gains and losses , for a single mode , in a complete period of oscillation is given by     this quantity is provided by graco and can also be regarded as the required energy for each mode to be driven during one oscillation cycle .the scaling of the linear solutions to their real values is done multiplying eq .9 by .as we are interested in an upper limit for this quantity , we assume that for every mode the energy balance is positive , i.e. that each mode removes energy from the star .this is only true for unstable modes , as stable modes transfer energy to the star .however , in this way , we make sure that the real amount of energy subtracted by the modes from the star , regardless of the theory used to obtain it , will be always lower than our estimated upper limit .the total energy during a cycle of oscillation is given by the sum of all the energy interchanged by all the modes in the observed frequency range .we compared this energy to the luminosity of the star , as a measure of the available radiation energy per time unit . for classical cepheid stars ,we have used the fundamental radial mode and its overtones .therefore , the sum of for each mode ( being the frequency of the mode in ) yields a total energy per time unit needed by the modes to be overstable of  ergs s .the luminosity of the star is of the order of  ergs s , that is , one order of magnitude higher .as we expected the real energy value to be even lower than the one estimated here , we conclude that this extreme case has enough energy per time unit to unstabilize all observed modes .this result is used only to legitimize our main assumptions .   for the second comparisonwe have calculated the kinetic energy of each mode as     once the kinetic energy of each mode is scaled by its variation of the relative radius ( multiplying by ) , the total sum of the kinetic energies of the selected modes gives a total kinetic energy of the order of  ergs . give the same order of magnitude for the kinetic energy of the model we are comparing with .this value has to be compared also with the total energy of the star , i.e. , the sum of the internal energy of the star , mass , rotational kinetic energy , turbulent energy , gravitational energy , etc .the estimation of all these parameters is a complex task , but the gravitational energy that holds the star together is already of the order of  ergs .therefore , we can infer that the total kinetic energy of pulsations of the radial modes for the case of the classical cepheid stars is low enough for the star to bear it maintaining the hydrostatic equilibrium .again , our assumptions are validated with this test .the scuti hd174936 was the first corot target of this stellar type ( together with hd50844 ) presenting unexpected space data .the observational spectrum and the relative variation of the radius have already been presented and explained in section 2 and depicted in fig .1 . the physical characteristics of the model used are displayed in table 1 .the total energy per oscillation cycle is given by the sum of all the energy interchanged by all the modes in the frequency range ~\mu ] , to be able to account for the thousands of observed modes .the observation of spherical degrees up to with high - precision photometric time series were predicted by .the sum of for each mode yields a total energy per time unit needed by the modes to be overstable of  ergs s , for both proposed amplitude distributions ( with and without tdc ) .the luminosity of the star is of the order of  ergs s , that is , five orders of magnitude higher .consequently , as the real energy value is even lower than the one here estimated , we conclude that the star has enough energy per time unit to unstabilize all the observed modes .on the other hand , the total sum of the kinetic energies of the selected modes gives a total kinetic energy in the range ]hz , to well cover the sct and dor range , and degrees in the range have been used .we obtain that the total energy per time unit needed by the modes to be overstable is of the order of  ergs s ( only using tdc , more suitable for dor targets ) .the luminosity of the star is of the order of  ergs s , that is , four orders of magnitude higher .we conclude again that the star has enough energy per time unit to unstabilize all the possible modes .the total sum of the kinetic energies of the modes selected are in the range $ ] ergs , depending on the amplitude distribution .the gravitational energy that holds the star together is of the order  ergs .again , we conclude that the total kinetic energy of pulsation of the thousands of modes is low enough for the star to bear it maintaining its hydrostatic equilibrium .using a non - adiabatic linear pulsational code , we have related the observed variation in brightness of classical cepheid stars , a scuti , and an hybrid sct- dor star with the intrinsic variation of their relative radius .this has allowed us to scale theoretical quantities from this linear code of oscillations to real values of the star , and to make the first comparison to date , up to our knowledge , of absolute pulsational quantities with absolute parameters of non - stochastic pulsating stars .the subsequent analysis of the total kinetic energy of the thousands of observed modes , and of the required energy to excite them , shows that all of them have enough energy to sustain their oscillation .our study shows that the results yielded by the analyses of the light curves with the classical tools ( standard sine wave fitting ) dot not come into conflict with what it is physically possible .

Abstract: the recent analyses of the light curves provided by corot have revealed pulsation spectra of unprecedented richness and precision , in particular , thousands of pulsating modes , and a clear distribution of amplitudes with frequency . in the community , some scientists have started doubting about the validity of the classical tools to analyze these very accurate light curves .   this work provides the asteroseismic community with answers to this question showing that ( 1 ) it is physically possible for a star to excite at a time and with the observed amplitudes such a large number of modes ; and ( 2 ) that the kinetic energy accumulated in all those modes does not destroy the equilibrium of the star . consequently , mathematical tools presently applied in the analyses of light curves can a priori be trusted . this conclusion is even more important now , when a large amount of space data coming from kepler are currently being analyzed .   the power spectrum of different stellar cases , and the non - adiabatic code graco have been used to estimate the upper limit of the energy per second required to excite all the observed modes , and their total kinetic energy . a necessary previous step for this study is to infer the relative radial pulsational amplitude from the observed photometric amplitude , scaling our linear pulsational solutions to absolute values . the derived upper limits for the required pulsational energy were compared with 1 ) the luminosity of the star ; and 2 ) the gravitational energy . we obtained that both upper energy limits are orders of magnitude smaller .


many important problems in study of complex systems require consideration of dense and nonlinear interaction between different functional modules .hybrid system is a framework first proposed by the engineering community to model systems that exhibit both continuous and discrete state changes , and has , in recent years , found increasingly wide applications in many practical fields in connection to both complex man - made systems or physical systems found in nature .however , most of the models proposed in literature so far are deterministic , and are not suitable to model inherent randomness .new data from high - throughput biology is enabling more ambitious , complete and validated simulation biological models and a lot of attention has recently been paid to the development of the computational methods for the _ in silico _ investigation of complex biological processes on a cellular level . to be valuable to biological and biomedical research computational methodsmust be scaled to account for to large number of interacting component of different nature and complex pathways . as it was already noted by many researchesstochastic effects play an important role in the functionality of such systems and approximation of molecular behavior by continuous models fails to reproduce many interesting biological phenomena .the deterministic approach can be particularly misleading if the molecular population of some critical reactant species becomes so small that microscopic fluctuations can conspire with reaction channel feedback loops to produce macroscopic effects .recent work shows that this can happen with dramatic consequences in the genetic / enzymatic reactions that go on inside a living cell or non - equilibrium combustion reactions .correct methods for performing stochastic simulation of the discrete event simulation are usually based on the kinetic monte carlo methods(kmc ) , stochastic simulation algorithm(ssa) .kmc / ssa methods are exact i.e. they are designed to account for _ every _ possible discrete reaction event and become notoriously inefficient for the scenarios when stochastic effects are to be ignored and model can be abstracted as system of conventional ordinary differential equations ( odes ) and ambitions goals which require the intensive use of kinetic monte carlo methods will also require immense computational resources .that became a main motivation for the development of the hybrid simulation methods which will allow adequate levels of description for different parts of complex system in hand .recently several attempts were made to bridge the gap between different levels of discretion and in particular reduce excessive information content delivered by kmc / ssa methods on a temporal scale . to overcome excessive level of resolution on the temporal scale i.e. to perform the _ temporal coarse - graining _ -leaping methodswere developed .leaping methods allow to improve the efficiency of the standard kmc / ssa while maintaining acceptable losses in accuracy by approximating the time - inhomogeneous poisson process counting the reaction events via the gaussian process / diffusion approximation .the key idea here is to take a larger time steps and allow for more reactions to take place in that step , but under the proviso that propensity functions do not change too much in that interval .when it comes to practical usage of the -leaping method a robust strategy is necessary for deciding when during the coarse of the simulation when to step exactly using kmc / ssa and when to leap approximately with diffusion methods .when leaping , consideration of the parameter values and error - reduction scheme is also necessary .this problem becomes of practical importance because it is often desirable to allow some chemical species to be treated as continuous random variables and some to be treated discretely .this is particularly true for the case of the transcriptional regulation by transcription factors ( tfs ) . in this casethere may be as few as one dna / tf binding site and messenger rna ( mrna ) abundance may be as small as tens of molecules while there may be hundreds or even thousand molecules of regulatory proteins per cell .in addition to that , biochemical networks , as probably every complex system , show remarkably interesting dynamics due to the presence of different time scales describing the evolution of discrete and continuous components .the technical difficulty with implementing the hybrid schemes is that standard kmc / gillespie approach requires constant transition rates between reaction events of the discrete species. this may not be the case if some of chemical species are evolving continuously in time or with different time scales .several attempts have been made to illustrate the relevance and feasibility of hybrid algorithm , especially for the case of highly developed separation of time - scales .rao and arkin consider the case of two groups of species evolving with two very different time scales and present stochastic averaging procedure , constructing effective rates of the `` slow '' reaction in the presence of the `` fast '' species , termed stochastic quasi steady state approximation ( qssa ) .  in their work kiehl et .al . are developing an approach to integrate ode / sde schemes with kinetic monte carlo schemes . synchronization algorithm to produce the numerical scheme which bridges together asynchronous in nature kmc / ssa algorithms and synchronous ode integration schemes .one should probably agree that following the paths presented in the literature up to date will give solutions which are very close to the exact dynamics in the limit of a very small time step , largely different time scales , etc .but verity of different implementation approaches presented in the literature to this point indicates the fact that unifying solid mathematical description has to be thought to justify the development of the hybrid simulation schemes applicable for modeling stochastic phenomena in chemical reaction networks .not many stochastic algorithms presented in the literature so far have gone through the rigorous error control analysis or even search for the fundamental justification of the presented methods .  in this publicationwe aim to provide not only the framework for the for the hybrid simulation method but also we outline it s rigorous mathematical justification including error analysis .outline of this paper is following . in section [ notation::sec ]we briefly outline some basic notation and assumption used in this article . in the next section [ stoch::sec ]our goal will be to give precise mathematical formulation of the stochastic process corresponding to the hybrid representation of the reaction network .we model the stochastic dynamics of the reaction network as a dynamics of the composite pair , governed by the system of the jump - diffusion sdes .this section will be followed immediately by simple example and view on error analysis .article is ended with discussion and outline for the future work .0.5 cm consider the reaction network which represent the dynamics of well - stirred molecular species which inter - react through ( ) channels of chemical reactions .the current state of the system is completely specified by the -dimensional vector consisting of non - negative integer numbers , where each number defined to be the current total number of molecules of type in the system .formally reaction network can be expressed as a set of the transition rules :  [ massaction ]   transition rates are the probabilities that reaction event corresponding to the reaction will take place in infinitesimal time interval somewhere inside the reaction volume .vectors are the stoichiometric al changes of number of molecules of types due to the reaction .as one can see , each reaction channel is specified by two quantities : transition rates and stoichiometric changes in molecular composition of the system due to each individual event in reaction channel .  in eqn .( [ massaction ] ) are transition probabilities for the reaction channel for any particular configurations of the interacting molecules on the right hand side of ( [ massaction ] ) .the functions are defined to be the number of distinct reaction combinations of the reacting molecules available in the state . the overall transition rate can be expressed as : for the mass - action kinetics transition rates are found to be a polynomial functions of variables .for the simple reaction system : statistical weights are : .we refer reader to the sources in and for more details .0.5 cm our interest in the next step will be to consider different reactions in the system in a different manner .we partition the set of molecular species into two groups : `` large population group '' ( with appropriate quantities labeled with `` '' ) and discrete group ( labeled with `` '' ) .species from the first group will be represented by the set of variables taking values on : and present in large copy numbers : the reminder of the species represent the discrete components of the network : partition of the species into the two separate groups correspond to the partition of the stoichiometric vectors into two sets : which correspond to the changes of species in sets and due to the reaction event in the channel .based on this partition of species we can introduce three groups of reactions :  first group of reactions which we shall call , consist of reactions with large combinatorial weights , i.e. which satisfy the condition .we state that this condition is sufficient for the use of the diffusion approximation ( see appendix sec .[ appendix::diffusion ] )  second group , consists of reactions which do not satisfy the constraint but change the species of the group , i.e. those reactions for which   finally the third group , which we call consists of reactions for which   note that , those subset of the reactions are not necessary independent , i.e . without the loss of generality we shall assume that reactions from the set ( set of `` jump '' reactions ) are formed by reactions in eqn .( [ massaction ] ) with indices ,the rest , are assumed to satisfy the condition of having large combinatorial weight .0.5 cm  chemical master equation is usually employed to describe the stochastic dynamics of reaction networks of the type ( [ massaction ] ) .it is notoriously hard to solve analytically even for the simple systems and , as we mentioned before , one usually has to resort to so called kinetic monte carlo methods to obtain numerical solution . in this articlewe shall abandon the description based on the chemical master equation(cme ) and resort to the explicit treatment of the stochastic trajectories of the process as a solution of the system of stochastic differential equations ( sdes ) providing the recepie for the construction of numerical schemes .poisson counter process is a simple but important process and below we extend its utility by combining with ideas from differential equations .we briefly describe this approach to pave our way to specific applications .  to describe the structure of the stochastic jump processes let us introduce in the rather traditional way a stochastic basis supplied with the filtration large enough to include -dimensional unit poisson processes .we recall that state variables are defined on this probability space . on this space transition rates eqn .( [ rate ] ) define the set of time non - homogeneous counting processes :    notation is used to underline the existence of jumps in the process : .the above equation states that differentials of processes , depend upon local intensities .notation is used to underline the existence of jumps in the process : .counting process can be also represented as a time changed unit - rate poisson processes  :   in turn , counting processes represent the stochastic dynamics of the markov process through the `` mass - balance '' equation : solution of eqn .( [ sdepoisson ] ) is a constant on the interval where all are constants and jumps by whenever the reaction event takes place in the reaction channel at the moment of time : using the partitioning which was introduced earlier , stochastic dynamics of the state vector can be presented as following system of sdes :    where integer vectors and represent the change of the components and due to the reaction event in the channel . in general , propensity functions depend on both and .large combinatorial weights of some reactions allow us to follow the _ diffusion approximation _ , i.e. by using the weak convergence methods we express differentials of the time - inhomogeneous poisson processes in eqn .( [ sdepoisson ] ) corresponding the group as a gaussian processes with the drifts : where are brownian motions independent of each other and .this approximation of the stochastic dynamics allows us to make a simplification in which we can consider the components as defined on a hybrid space ; takes values on the continuum state space , , while still takes values on discrete state space .this brings us to the following formulation of the stochastic dynamics of the pair   [ jump_diffusion ]   where we explicitly demonstrated the dependence of the differential of the counting processes on local intensities .note that according to the presence of the jump component term in equation ( [ jump_diffusionx ] ) and is a jump - diffusion process in .processes with state - dependent intensities of jump are notoriously hard to model .one complications which makes the study of the sdes of the type eqn .( [ jump_diffusion ] ) hard is the nature of the noise source driving the dynamics .in contrast to very well studied wiener or poisson driven sdes is that intensity of the noise source is stochastic in nature .we shall take a different representation of the noise source .namely we use the technique of the poisson random measures .our discussion of the random measures will be brief and rather informal ; for mathematical background see monograph by jacod and shiryaev or cox and isham .poisson random measure defines a sequence of points and marks with the simple interpretation that the mark arrives at time ;  take values on .marks take values in a general space , which in the context of the problem considered here can be taken as a subset of the euclidian space , .deterministic intensity of the poisson process is with being a measure on .arrival times follow a poisson process with deterministic intensity and marks a i.i.d distributed in the interval .  important procedure which can be used to simulate the intensity dependent process eqn .( [ massaction ] ) is based on introduction of the partition on the marked space . at every point of the state space construct the set of disjoint intervals based on transition rate of `` jump '' reactions , i.e. reactions from the subset , i.e. :  [ intervals ]   where index runs over reactions which either do not satisfy condition of for which vector is non - zero i.e. they change discrete components .thus in general length of the interval is .now define a set of functions : suppose now that the space supports a poisson random measure in with a compensator and a set of wiener processes .then time non - homogenous counting process can be represented as : and valued process can be viewed as a solution of coupled ito - skorohod sdes driven by the poisson random measures :  [ random_measure ] [ jump_diffusion_poisson ]   function in eqn . ( [ random_measure ] )transforms arrived marks into magnitude of the jump of due to appropriate reaction channel .  for the practical insight on the dynamics described above, we will introduce _ reference poisson process_.  since the length of each interval is and bounded function for every then it is follows that the length of the interval ( we denote it ) is bounded function of .therefore , it has a maximum at some point : denote the maximum length of the joint intervals in the marked space .now consider a standard poisson process with intensity .let denote the jump times of this process ( sampled , perhaps , as the partial sums of the exponentially distributed random variables with mean ) .let be the marked space and be the sequence of i.i.d .distributed random variables with uniform distribution on , independent on . in this casewe can represent a poisson random measure with intensity is related to the marked point process , constituting the sample path of the process .namely for each :  ,a)=\sum_{n\geq 1}1 _ { \tau^{\lambda}_n\leq t } 1_{z_n \in a } , \\\mathbb{e}(n_{\lambda}((0,t],a))=\lambda_{max}t\mathbb{p}(z_n\in a)= t\mu(a)\end{gathered}\ ] ]  this representation ( sometimes referred to as thinning of the poisson measure ) is very useful in practical numerical applications . in the next sectionwe will use the reference process to construct the discretization scheme taking ( [ intervals ] ) for various .in this section we outline basic principles for the construction of the numerical scheme for the solution of the system given by eqn .( [ jump_diffusion_poisson ] ) . to begin with, we should introduce the appropriate discretization of the interval  ] , perhaps the usual equidistant , discretization of the time interval , i.e. .suppose now that  ] and jumps of the reference poisson process [ poisson_master ] .it is important to note that jump - times of the poisson measure eqn .( [ poisson_master ] ) can be modeled without discretization error and with little prior information ; one has only make sure that transition rates from the groups are bounded , i.e. are less then chosen intensity of the reference poisson measure .armed with the time discretization scheme assembled from equidistant discretization and jump - times of the reference process we introduce the following simple jump - diffusion numerical scheme :    which recursively determines the values of the discretized processes and at points , starting from values .increments of the winer processes are zero - mean gaussian random variables : . according to the ( [ euler_diff_x ] ) , between the jumps dynamics of the component is purely diffusive . integral in eqn . ( [ euler_jumps_sigma ] )is evaluated at the single point which belongs to the uniform distribution ) ] , generated on the step satisfies the condition :   depending on the relative ratio between characteristic time - scales of diffusion and discrete species different numerical schemes must be considered to achieve substantial speed - up of the simulation .here we consider only the case of fast diffusion modes and slow discrete modes . in this caseevery interval should be partitioned onto smaller intervals of the discrete grid on which numerical scheme ( [ euler_diff_x ] ) for diffusion approximation is used .opposite case of the fast switching in discrete component will be considered elsewhere .we now wish do demonstrate application of the above numerical scheme . in this sectionwe consider the simple example , which nevertheless , is interesting enough for the purposes of the demonstration of the technique discussed in this paper .consider the following model of the reaction network :    with continuous and discrete components and respectively . the functions and vectors for this systemare :    we have used the following set of kinetic parameters: , , and leads to the bursts of the transcription of protein with proteins per transcription event . ] .analysis of the propensity functions ( ) shows the following partition of the original reaction set : i.e. reactions correspond to the diffusion modes .we also constraint ourselves in this example considering the case : propagation of the diffusion mode was performed via euler - maruyama scheme , with set of different time - steps : .we performed monte carlo runs with both standard kmc / ssa and hybrid scheme outlined above to obtain the distributions of the number of molecular species at time - slice .figures([fig::histogramx1],[fig::histogramx2 ] ) show that distributions of the species , obtained with the hybrid scheme is almost indistinguishable from the results of the obtained by following complete kmc / ssa - based approach .we also report an observed speed - up in terms of the ratio of the cpu times , fig.([fig::speed_up ] ) as a function of time step used in numerical integration of diffusion modes . 0.5 cmadvances in field of molecular biology outlined new directions of research in stochastic chemical kinetics .we now need to develop the software tools and mathematical approaches to integrate models from micro - scales to macro - scales in a seamless fashion .such multi - scale models are essential if we are to produce quantitative , predictive models of complex biological behaviors .an important step on this path is implementation of hybrid simulation methods , capable to account for heterogeneity of properties of interacting components .we have outlined rigorous framework for development of hybrid simulation schemes based on the path sample representation of reaction dynamics rather then on cme based approach . for the best of our knowledgethis type of analysis has not being performed before .it was not the primal goal of this publication to justify the partitioning of the species / reactions into the different groups .one should probably have some prior knowledge when partitioning the species into different types .choice of the reactions ( `` diffusive '' or `` jump '' ) is based on functional law of large numbers and error generated by this step can probably be controlled rigorously with some insights on this subject outlined in section [ appendix::diffusion ] .  numerical solution for the propagation of the diffusion modeswere outlined via simple constant step explicit euler scheme .typical explicit or implicit scheme based on stochastic taylor approximation , might run into problems , since they do not conserve the non - negativity of the numerical solution , i.e. i.e. it can not guarantee almost surely .we point out here that use of balanced - implicit stochastic schemes ( bim ) , look promising when it comes to the construction of the numerical solution with the property of `` almost sure '' positivity .bim scheme known to have the same order of convergence as the euler - maruyama stochastic scheme , namely , the error for approximations of the individual trajectories and for moments .adaptive time - step control systems also look promising as a direction for the research aimed for achieving the simulation speed up .author would like to acknowledge valuable discussions and useful comments from prof . c. myers , prof .d. steinsaltz , k. ericksson and prof .

Abstract: the existing literature on stochastic simulation of chemical reaction networks has a tendency to move as quickly as possible to the abstract formulation of the stochastic dynamics in terms of probabilities based on the concept of the chemical master equation ( cme ) , largely ignoring sample path representation . in this publication we discuss both theoretical basis and numerical approach for the problems in this area using sample path methods as a crucial part of the process . relying as it does on a representation of the underlying stochastic processes as a weak solution of a system of stochastic differential equations driven by poisson random measures this approach brings to bear a heretofore ignored but quite effective problem solving methodology .   we first present a simple and intuitive way of partitioning species and reactions of the interaction network into different groups . we then discuss how original stochastic dynamics with state dependent intensities of transitions can be reformulated in terms of jump - diffusion stochastic differential equations driven by both wiener noise sources and poisson random measures . finally , we show that this approach facilitates the construction of hybrid simulation techniques , an important step in the creation of efficient techniques for modeling multi - scale stochastic dynamics of the reaction networks . numerical methods related to sampling events from poisson random measures are demonstrated on simple intuitive examples . error control analysis of the finite differences scheme is also presented .   current version : march 20,2005  stochastic algorithms , chemical reaction networks , ito - skorohod stochastic differential equations , poisson random measure , + hybrid systems  91b74 , 60h35 , 60h10 ,


epitaxial graphene growth on sic  is a scalable fabrication method of high quality graphene for post - silicon electronics and opto - electronics  . among novel devices ,the growth technique , together with sic wafer preparation , stands at the beginning of whole manufacturing process .it is therefore a key to understand conditions under which reproducible high quality graphene can be reached  .the growth mechanisms has been studied both theoretically and experimentally .first - principles calculations has been done to study graphene buffer layer formation on sic(0001 ) , diffusion of carbon on sic  , stability and reactivity of atomic steps of sic in the initial graphene growth stage  .experimental studies have been done to understand mechanisms of epitaxial graphene growth on sic(000 )  , sic(0001 )  , non - polar sic surfaces  and to elucidate role of carbon diffusion  and silicon sublimation  .it has been shown that graphene with reduced pit density can be grown on nominally flat sic substrates  and graphene quality can be further improved when grown in high argon pressure  . on the other hand ,the carrier mobility has been observed to decrease with increasing argon pressure when time and temeperature are kept constant  .other strategies to improve graphene quality involve thermal decomposition of deposited polymer adsorbate which acts as a carbon source  .the sic step bunching , as another issue reducing graphene mobility on sic , has been solved by amorphous carbon step pinning  .the thermodynamics of stable phases that governs the onset of graphene formation  , oxidation  and other chemical reactions  has been discussed , too . however , there is a little or no experimental evidence of graphene growth conditions and composition of a residual gas inside a graphene furnace  .these studies are restricted to gas phase dynamics during chemical vapour deposition ( cvd ) graphene growth  and to carbon nanotubes growth  . here, we determine experimentally the residual gas content in a furnace for epitaxial graphene growth on sic .the measurements are performed in - situ by residual gas analyzer during whole graphene growth .we show that the initial graphene growth is accompanied by chemical reaction of residual water and carbon .the results of norimatsu  indicate that control of the number of graphene layers require precise control of the first stage of sic decomposition .we discuss that hydrogen , one of the products of the reaction of residual water and carbon , might be one reason for sic step bunching during initial graphene growth stage .we compare the chemistry of graphene growth in high vacuum and in low argon pressure .the comparison of both growth techniques demonstrates that argon reduces the rate of water and carbon reaction .therefore amount of hydrogen is smaller and graphene quality is higher .we discuss also the role of argon in terms of growth temperature , time and gas purity .the graphene quality is analyzed by raman spectroscopy mapping and by hall effect measurements .we compare growth in vacuum and high pressure argon in terms of charge density , strain , graphene grain size and their spatial variation .the wafers of 6h - sic were purchased from ii - vi inc . and diced into 3.5.5 mm rectangles .the samples are ( 500 )  m thick , semi - insulating high resistivity ( cm ) vanadium doped sic .graphene is grown on the silicon face sic(0001 ) with a wafer miss - cut .the wafers were chemically - mechanically polished and the surface roughness is about 0.5 nm .diced samples were rinsed and sonicated in both aceton and iso - propylalcohol for 10 minutes .the semi - insulating samples were heated in a graphite crucible by radio - frequency ( rf ) induction heating at 250 - 270 khz .the graphite used for fabrication of the crucible is iso - statically pressed ( isp ) graphite and glassy ( vitreous ) graphite .the isp graphite has density 1850 kg / m and electrical resistivity 11 cm .the glassy graphite , purchased from final advanced materials , has been used as a reference to study effect of crucible pre - baking .the high epitaxial graphene quality has been reached by confinement controlled sublimation ( ccs ) technique  .the processing environment is either high vacuum  mbar or high purity argon ( 6n purity gas vessel ) .the argon is further purified to the level of 9n purity by gas purifiers to remove residual oxygen and water .the stainless steel gas tubes have been outgassed at temperature about 200 for 2 hours prior to graphene growth to avoid contamination by residual water and impurities from the stainless steel surface .the temperature is monitored by either type c thermocouple or by two - color pyrometer working at 0.95 and 1.05  m wavelengths .the pyrometer allows to measure temperature with spatial resolution of about 0.2 mm .the thermocouple type c is electrically isolated by alumina shielding .the graphite crucible is placed inside a hot wall reactor made of 4 mm thick and 250 mm long quartz tube .the quartz tube is opened from both sides to allow effective gas circulation .the growth environment has been studied by residual gas analyzer ( prisma plus qmg 220 ) .the cf-40 flanges are used for all connections except of the adapter between the furnace and the turbo - molecular pump and quartz tube feedthrough .the quartz feedthrough is made of two o - rings on both sides to allow leak - tight connection up to high vacuum  mbar and yet allowing easy access to the reaction chamber .the standard furnace design for epitaxial graphene growth uses either thermocouple or pyrometer to monitor growth temperature . we have used both methods and compared them . the thermocouple is a cheaper method and it measures temperature inside the graphite crucible .it requires electrical isolation ( made of alumina for high temperature purposes up to 2000 ) complicating furnace design and introducing other elements beside carbon and silicon ( sic sample and c based crucible ) .we have found that the electrical shield made of alumina ( polycrystalic al ) is sputtered and deposited on a relatively colder quartz tube during cool down process .energy - dispersive x - ray spectroscopy ( eds ) has been performed to determine chemical composition of the sputtered material , as shown in fig .[ figeds ] for three different positions along the inner part of quartz tube .we have found presence of aluminium and al mixture .their relative ratio changes along the quartz tube .as we have used thermocouple type c , which is composed of rhenium and tungsten , we have looked carefully for the traces of these metals as well .the negative result of our search has led us to conclusion that the sputtered alumina from the thermocouple electrical shield is a source of furnace contamination , very often seen in high temperature graphene furnaces .we expect alumina traces can also contaminate sic wafer and graphene .hence , in order to keep high graphene quality , we propose to use only remote temperature sensing by pyrometer .we also note it is essential to use two - color ratio pyrometer for the reason of possible contamination of furnace quartz windows .the contamination leads to decreased light intensity and to the lower effective temperature determined by one - color pyrometer .c for 30 minutes .the pre - baking at 800 for 10 minutes and at 1000 for 10 minutes has been performed prior to graphene growth .the growth atmosphere has been studied for the case of high vacuum at  mbar using graphite crucible ( solid lines ) , low pressure 9n purity argon at  mbar ( dotted lines ) and at high vacuum  mbar using unbaked glassy graphite crucible ( dashed lines ) .the base pressure was  mbar prior to rf heating in all cases.,width=264 ]we study here the effect of graphite crucible gas adsorption . the commonly used isp graphite and glassy graphite cruciblesare compared .the major issue of isp graphite is its high porosity and permeability to gases .the high permeability leads to undesired contamination by air when samples are exchanged or graphite crucible is stored at ambient conditions for longer time ( order of hours and longer ) .a glassy graphite is impermeable to gasses hence it is more convenient to keep the furnace environment clean of air residual gasses .the residual gas stored in isp graphite is removed within the first two pre - bake graphene growth stages .the pressure usually grows from  mbar to  mbar or more if crucible is not well baked .the well baked crucible exhibits small rise of the residual gas pressure by less than one order of magnitude , typically  mbar to  mbar .contrary to isp graphite , glassy graphite is impermeable to gasses and the pre - baking removes surface absorbants only .the glassy carbon crucible is less susceptible to long exposure to air , however , its pre - baking at temperature much higher than the growth temperature is also necessary , as is shown in fig .[ figrga ] .the well - baked graphite crucible exhibits comparable strength of peaks at m / z=12 ( carbon ) and 14 ( nitrogen ) , which is also signature of leak - tight vacuum chamber . the unbaked glassy graphite exhibits more than 2 stronger nitrogen peak which is supposed to come from glassy graphite surface .we use peak at m / z=14 instead of m / z=28 because the latter one overlaps with carbon monoxide .mbar using graphite crucible ( a ) , low pressure 9n purity argon at  mbar ( b ) and at high vacuum  mbar using unbaked glassy graphite crucible ( c ) .the three growth stages are marked as following : ( i. ) prebake at 800 for 10 minutes , ( ii . ) prebake at 1000 for 10 minutes , ( iii . ) graphene growth at 1550 for 30 minutes .the temporal evolution is shown for major growth environment components ; nitrogen / carbon monoxide ( black solid line ) , carbon dioxide ( red solid line ) , argon ( green solid line ) , oxygen ( blue solid line ) , water ( magenta solid line ) , nitrogen monoxide ( black dashed line ) , hydrogen ( red dashed line ) , -oh fragment ( green dashed line ) , -ch fragment ( magenta dashed line).,width=340 ]  c in high vacuum at 1 - 2  mbar ( a , d ) and total pressure and temperature in shown in part ( f ) .the m / z ion current is compared with low argon pressure growth at 1.1 mbar ( 9n purity ) in part ( b , c , e ) and corrensponding pressure and temperature time evolution is shown in part ( g).,width=302 ]  mbar ,  mbar ,  mbar and  mbar in diagram ., width=264 ]the general growth process consist of three stages .the pre - baking stage ( i. ) , depicted in fig .[ figmcd ] , is used for crucible outgassing at 800 for 10 minutes . the second pre - baking stage ( ii . ) at 1000 - 1100 for 10 minutesis used to decompose native oxide layer on sic wafer . the graphene growth stage ( iii . )is performed at temperatures 1400 - 2000 at times in the range of minutes to hours .[ figmcd ] depicts relative content of growth ambient in all three growth stages for the case of growth in high vacuum ( a ) under continuous vacuum pumping by a turbo - molecular pump , low pressure argon ( b ) under continuous pumping and small argon leak and growth in high vacuum using glassy graphite crucible ( c ) . the graphene growth stage ( iii . )has been kept at 30 minutes to show long - time evolution of the growth ambient .a change of relative content is caused by different pumping rate ( e.g. slow hydrogen pumping ) , outgassing from vacuum chamber walls ( mainly water ) and chemical reactions .we have identified that oxidizing reactions appear at every temperature rise ( from step ( i. ) to ( ii . ) and from step ( ii . ) to ( iii . ) ) resulting in reduced both relative and absolute amount of oxygen .these reactions are however negligible for very small content of oxygen in the chamber .the major identified reaction is a reaction of two major components in the furnace , carbon and water .we discuss this process in detail here .the epitaxial graphene can be grown fast ( 2 - 5 minutes ) at higher temperatures or slowly ( 20 - 90 minutes ) at lower temperatures .we have found that 1430 for 20 minutes is equivalent to 1500 for 5 minutes by amount of grown graphene .it is however not equivalent in growth conditions concerning the growth environment .we show difference between the two regimes .the main issue is a content of water and carbon monoxide , as shown in fig .[ figrgadetail ] at the beginning of graphene growth .a water content is about 10 - 15% for timesshorter than 5 minutes , but it grows then steadily and it completely dominates the growth atmosphere at times longer than 30 minutes .a detail of growth ambient in high vacuum in fig .[ figrgadetail ] ( a , d , f ) shows sharp pressure rise accompanied by sharp rise of ion current at m / z=28 by 8 a. similarly , the ion current increases at masses m / z=44 by 0.6 a and m / z=12 by 0.25 a. a significant rise of hydrogen is also present by amount of 6 a. the water ion current is unchanged within first 60 s of growth stage ( iii . ) .the step rise of ion current is expected instead due to the step rise of temperature and consequent step rise of mean velocity of ions .the lack of such ion step rise of water can be attributed to the reduction of total amount of water in the chamber .the rise of m / z=28 by 8 a is attributed to co. the rise of co is confirmed by rise of ion current at m / z=12 by 0.25 a , which is 3% ( expected co contribution is 5% ) of the major ion current rise at m / z=28 .the minor ion current peak at m / z=14 by 0.05 a ( 0.6% of m / z=28 , expected co contribution is 1% ) is also attributed to co. we conclude that there is no absolute change of nitrogen ( m / z=28 , 14 ) and all content changes can be explained by increasing amount of co and similar amount of hydrogen .the shift of hydrogen peak towards longer times is caused by slower pumping of hydrogen than pumping of co , water and other heavier ions .the ion current rise at mass m / z=44 by 0.6 a is due to the carbon dioxide ( co ) .therefore the dynamics of growth ambient is dominated by chemical reaction the major chemical reaction given by equation ( [ eqreaction ] ) is observed mainly on the short time scale below 30 s after each temperature step rise .the time scale between 30 - 100 s is dominated by pumping the products of eq .( [ eqreaction ] ) out of the furnace .the long time scale above 2 minutes is dominated by both pumping of the residual gas and water outgassing from the stainless steel furnace walls .this behavior has been observed both in high vacuum and low pressure argon and also in the case of glassy graphite crucible in high vacuum , as is shown in fig .[ figmcd ] .the content changes only slowly for longer times (  s ) , however , water becomes slowly dominant gas .the water source is at the surface of stainless steel furnace walls .the main issue is that it can promote reaction with carbon ( [ eqreaction ] ) even further , hence creating defects in a graphene layer .therefore it is more preferential to grow graphene at higher temperatures for shorter times .as the residual gas analysis has shown preferable shorter growth times and higher temperatures , we grow single layer graphene between 1500 - 1550 for 5 minutes in high vacuum - mbar .the 1500/5 minutes growth seems to result in a patchy graphene with about 20 - 40% of buffer layer .the single layer graphene is proven by raman spectroscopy and discussed later .there are already signatures of bilayers at higher temperature 1550/5 minutes .growth in 800 - 1050 mbar of argon is approximatelly 30 - 50 shifted towards higher temperatures with respect to growth in high vacuum .buffer grows at about 1410 - 1430 in high vacuum and at about 1450 at 800 - 1050 mbar argon pressure .the growth has been always done in the same graphite crucible with a hole 1 mm in diameter and 10 mm long .argon acting on graphene growth is twofold .first , it is equivalent to ccg method , where the lowered silicon sublimation is due to the small volume in nearly closed graphite crucible .second , argon pressure acts as an inhibitor for water outgassing , as has been discussed in previous paragraphs as the major source of residual gas and potential cause of defects formation in epitaxial graphene .we note , that high purity argon is necessary in order to obtain overall purity of growth conditions comparable to those in high vacuum at  mbar .an equivalent purity of argon gas for graphene growth at 1000 mbar is 8n ( = 1000 mbar/10 mbar ) . a lower argon purity results in higher relative content of residual gas , which leads to higher total amount of mainly water at 1000 mbar argon pressure growth ( the total amount of water is discussed with respect to the total amount of water at high vacuum ( 10 mbar ) growth ) .beside the gas purity , also the background outgassing in the furnace has to be taken into account .a typical water backround pressure in unbaked furnace is 10 mbar-10 mbar .if the background water pressure is higher , the overal argon purity in the furnace will be lower even with gas purifiers , as is described in fig .[ figpurity ] .[ figpurity ] demonstrates overall gas purity as a function of total argon pressure for two degrees of argon purity .the solid ( dashed ) lines depict total argon purity as a function of total argon pressure for inlet of 9n ( 6n ) pure argon. the total gas purity can not be higher than the inlet gas purity , therefore all curves saturate at the value of inlet gas purity 9n ( solid lines ) and 6n ( dashed lines ) . fig .[ figpurity ] also shows influence of four different background impurity pressures .the background impurity pressure is detrimental at low total argon pressures , therefore it is expected that higher quality graphene will be grown at higher argon pressures , as has been evidenced in literature  .a further reduction of water outgassing could be done by growth in argon flow .the graphene quality should be independent on the flow rate as long as the flow rate is much higher than outgassing rate .another important role of argon is reduction of water and carbon reaction by factor of three , as can be seen in fig .[ figrgadetail ] ( c , e , g ) , as compared to growth in high vacuum , fig . [ figrgadetail ] ( a , d , f ) .it can be assumed that the reaction rate will be reduced even further at higher argon pressure .mbar ( black curve ) and in 9n pure argon at 800 mbar ( red curve ) .the positions of d , g and 2d peaks are marked by labeled arrows.,width=264 ]  mbar at 1550 for 5 minutes .the analyzed raman spetra parameters are g peak position ( a ) , g peak fwhm ( b ) , 2d peak position ( c ) , 2d peak fwhm ( d ) , ratio of integrated g peak and d peak intensities ( e ) and ratio of integrated 2d peak and g peak intensities .the scale bar m is shown in the left bottom corner in each map .the statistical analysis is depicted in a form of histogram for each analyzed parameter including units and the range of values used in color scales . , width=340 ]  c for 5 minutes .the analyzed raman spetra parameters are g peak position ( a ) , g peak fwhm ( b ) , 2d peak position ( c ) , 2d peak fwhm ( d ) , ratio of integrated g peak and d peak intensities ( e ) and ratio of integrated 2d peak and g peak intensities .the scale bar m is shown in the left bottom corner in each map .the statistical analysis is depicted in a form of histogram for each analyzed parameter including units and the range of values used in color scales.,width=340 ]  mbar ( a , b ) and 800 mbar 9n - argon atmosphere ( c , d ) .the statistical distribution of strain and charge density in both types of graphene samples is shown in corresponding histograms for high vacuum grown graphene ( blue histograms ) and high pressure argon grown graphene ( red histograms ) .the scale bars are shown in the left bottom corner of each map ; 1  m for high vacuum grown graphene and 0.5  m for high pressure argon grown graphene.,width=340 ]we compare here two epitaxial graphene samples grown in high vacuum ( hv - graphene ) at 10 mbar and in 9n pure argon at 800 mbar ( ar - graphene ) .first , raman spectroscopy has been performed in order to determine strain , charge density distribution and graphene grain size .we have used confocal microscope with 100 magnification microscope objective and numerical aperture .the excitation laser wavelength was 532 nm and the laser power at the sample was 5 mw .the characteristic raman spectra of hv - graphene and ar - graphene are shown in fig .[ figramanspectra ] . nearly absent d - peak shows on better graphene quality grown in ultra pure argon atmosphere .we have analyzed the positions , full width at half maximum ( fwhm ) and integrated intensities , and of d , g and 2d peak , respectively .we have constructed spatial maps of g and 2d peak positions , g and 2d peak fwhm and spatial maps of and , as shown in fig .[ figramanvac ] and fig .[ figramanargon ] for hv - graphene and ar - graphene , respectively .the positions of g and 2d peak has been calculated as center of mass ( com ) of each peak after subtructing sic background signal .the larger grain size of ar - graphene can be seen from the fine structure of all spatial maps .we have used these spatial maps further to determine mechanical strain  given by measured position of 2d peak , position of 2d peak at zero strain  and grneisen parameter for 2d peak  .the effects of charge density on the position of 2d peak are negligible  .the charge density is determined from g peak position and strain , ,\ ] ] where is grneisen parameter for g peak ,  is g peak position for unstrained ( ) and neutral ( n=0 ) graphene . parameter =10/1  is approximately a proportionality factor between g peak position and graphene charge density , as determined theoretically  and confirmed experimentally  .graphene grain size can be determined from the ratio  of integrated g peak intensity ( ) and d peak intensity ( ) , giving for excitation wavelength 532 nm .number of graphene layers has been determined by several authors  from the ratio of integreated 2d peak and integrated g peak intensity .this ratio for single layer graphene , for bilayer and it is for more than two graphene layers .we note this relation is the most discutable and it is probably valid only in the limit of few graphene layers .it can not be used e.g. for number of graphene layer determination of multilayer graphene on c - face of sic .our graphene samples have been grown about 50 above the lowest graphitization temperature at 5 minutes and graphene has been always grown on si - face of sic , therefore we are in the limit of 1 to 2 layers and this method can be used to determine number of graphene layers .[ figramanvac ] ( f ) and fig .[ figramanargon ] ( f ) show spatial map of the ratio including statistical analysis in a form of histogram .both samples consist of mostly single layer .a 2d - peak fwhm is ( 35 )  in both samples .the narrow 2d peak is also a fingreprint of a single layer graphene  and fwhm values about 30  has been correlated with charge mobility of about 10 000  at dirac point  .the grain size in hv - graphene is  nm and it reaches  m in ar - graphene .strain is  % and  % in hv and ar grown graphene , respectively .the expected strain is determined from graphene ( )  and sic ( ) lattice constants at 300 k assuming sic - graphene super - cell  containing 13 graphene unit cells and mutual rotation of graphene and sic lattices by 30 .both samples show low doping level in the range  and the statistically significant amount of low density areas show on electron - hole puddles  . a small doping is also evidenced by g peak fwhm in the range of 14 - 18   .  .the charge density of electrons ( black lines ) and holes ( red lines ) is shown in part ( a ) .the hall coefficient ( black lines ) and product ( red lines ) are shown in part ( b ) .the standard deviation of the electrostatic potential is taken to be  mv ( all dashed curves).,width=188 ]we have measured hall effect to determine charge density and carrier mobility .the room temperature measurements have been performed in magnetic field up to .35 t in van der pauw configuration .a direct current ( dc ) source has been used ( ) and all combinations of current directions and voltages have been measured to determine specific resistance and hall coefficient .typical transport coefficients are shown in tab .[ tabtransport ] .we have observed twice lower specific resistance and 1.8 higher apparent carrier mobility of argon grown graphene ..transport properties of epitaxial graphene grown in high vacuum  mbar ( hv - graphene ) and 800 mbar 9n - argon atmosphere ( ar - graphene ) .the typical values are presented for specific two - dimensional resistivity ( resistance per square ) , hall coefficient , apparent carrier density and apparent carrier mobility .[ cols="<,^,^,^,^ " , ]   we note that all samples show small negative hall coefficient in the range from -50 to -100 /t , indicating n - type doping on the order of upper 10  .such charge densities are consistent with data in ref .however , it has been realized later  that instead of assuming single type conductivity , both electrons and holes should be considered in order to correctly interpret transport data exhibiting small hall coefficient .this is so called model of electron - hole puddles  .the electron - hole puddles have been observed previously by several authors  .for this reason , we use notation of ref . and call the transport properties to be apparent charge density and apparent carrier mobility .the small hall coefficient then leads to low carrier density .the low charge density is also consistent with our raman scattering measurements , as shown in fig .[ figstrainchargemap ] .the presence of electron - hole puddles leads to very high carrier mobilities above 10 000 cm for pristine graphene at 300 k. more realistic value can be estimated by considering electrostatic potential fluctuations .electrostatic potential fluctuations smear out the fermi level and increase an intrinsic charge density .we assume normal potential distribution with standard deviation .the charge density is calculated for given fermi level by where is fermi velocity .the mean charge density caused by the potential fluctuations is calculated as a weighted sum the mean hole charge density has been calculated by considering electron - hole symmetry .the mean electron and hole densities are plotted as a function of mean fermi level in fig .[ fighallcoef ] ( a ) by solid lines for pristine graphene (  mv ) and by dashed lines for graphene influenced by electrostatic potential fluctuations with standard deviation  mv .the total electrical current ( two - dimensional , units ( a / m ) ) is a sum of electron and hole currents . where ( ) is electron ( hole ) mobility .a specific resistivity , or resistance per square , is we plot a product of the specific resistance and mobility of pristine graphene and graphene influenced by electrostatic potential fluctuations (  mv ) in fig .[ fighallcoef ] ( b ) .the is lowered for graphene influenced by electrostatic potential fluctuations because mean charge density ( and intrinsic charge density ) in graphene rises .a hall coefficient is given in two - dimensions and in the limit of  t by eq .( [ eqrh ] ) the hall coefficient is plotted in fig .[ fighallcoef ] ( b ) for pristine graphene ( solid black curve ) and for the case of electrostatic potential fluctuations  mv by dashed black curve .the hall coefficient is largely reduced for all mean fermi level positions .the value  mv is taken as an upper bound determined in other works  .a small measured hall coefficient in the range of -50 to /t suggests fermi level in the range of 5 - 10 mev for  mv , which corresponds to about 5  electron and 4  hole density .the analysis of transport properties reduces now to determine , , , . because we have determined and and estimated , we can not determine any transport properties prior to further assumptions .the electron and hole charge densities are coupled by two common parameters , mean fermi level and electrostatic potential fluctuations .we assume equal electron and hole mobilities , which reduces number of unknown parameters to three ( being equal to three measured properties ( , , ) .we use measured hall coefficient to determine mean fermi level from fig .[ fighallcoef ] ( b ) .the mean fermi level provides a value of  , which leads immediately to carrier mobility provided the specific resistance is known .we get carrier mobilities 2900 and 6000  for vacuum and argon grown graphene , respectively .the results of raman scattering and hall effect measurements lead us to conclusion that graphene grown in ultra - pure argon at 800 mbar is of higher quality than graphene grown in high vacuum .the growth in high vacuum leads to smaller graphene grain size below 1  m even if the growth conditions are well optimized by high temperature bakeout of graphite crucible , gas inlet pipes to vent the furnace and the furnace itself .the key isssue in graphene growth is a presence of water and water outgassing from stainless steel surfaces .we have observed chemical reaction of water and solid state carbon in all stages of graphene growth .this reaction is important within first one minute of each graphene growth step .it becomes important again when graphene is grown for longer times above 20 - 30 minutes .this is caused by gradual furnace warm up and consequent water outgassing from previously colder furnace parts .even though it would be thermodynamicaly more preferable to grow graphene slowly , providing high crystal quality , the water outgassing acts against long growth times . as we do not know the origin of carbon , we assume that the main source is a graphite crucible for a time scale below 60 s at the beginning of each growth step . however, as graphene grows , the carbon source can be also graphene itself .this reaction of water and carbon from partially formed graphene will be a cause of higher defect density and smaller grain size , as we have observed in raman spectra analysis .a disadvantage of short growth time is that there is no temporally stable growth atmosphere content .moreover , it differs slightly from one sample to another .the hydrogen produced in a reaction of carbon and water can also cause sic etching or it can react with carbon , increasing defect density in graphene even more .hydrogen is therefore undesired gas during graphene growth .the growth in argon has been shown to lead to more stable growth atmosphere content with small rate of carbon and water reaction and also smaller amount of hydrogen .the high argon pressure reduces both water outgassing and also mean free path of water molecules .the shorter mean free path can be used to restrict water transport towards grown graphene when growth is performed in argon flow .we also point out that it is essential to use ultra - pure argon ( 8 or 9n ) at the furnace inlet . a common 6n argon at 1000 mbar is in terms of total amount of residual gas equivalent to vacuum growth at  mbar ( if background outgassing causes lower or equal pressure ) .therefore , growth in 6n argon at 1000 mbar will lead to much worse graphene than the one grown in hv at - mbar .the argon flow rate is not supposed to improve the overall amount of residual gas because this residual gas in homogeneously spread within the furnace volume .contrary to that , the 9n argon at 1000 mbar is equivalent to hv at  mbar in terms of total amount of residual gas .these two growth conditions are initially equivalent . however, the growth conditions are more preferable for argon growth at the later stages of graphene growth for two reasons .first , water outgasses in hv and causes graphene degradation ( pressure rises to 10 mbar ) .water ougassing is strongly reduced in high pressure argon , therefore the total amount of impurities remains nearly constant .second , any possible residual gas in ultra pure argon can be removed by growth in argon flow .the argon flow should be only higher than the outgassing rate .air permeability of graphite brings constrictions on a crucible storage .the nitrogen and water is absorbed into the crucible after longer exposure to air .it is therefore suggested to store isp graphite in vacuum or inert gas .glassy graphite is impermeable to air components , however , the surface absorbants still lead to higher desorption when glassy graphite crucible is heated to growth temperatures .raman spectroscopy has proven to be a versatile tool to determine most of basic graphene properties as mechanical strain , charge density , graphene grain size , number of graphene layers or even to estimate electrostatic potential fluctuations and carrier mobility .we have measured graphene raman spectra using several microscope objetives with different magnification and numerical apertures .we point out that although relative strength of sic raman spectrum and intensity of graphene spectrum ( particularly intensity of g peak ) have been used to determine number of graphene layers by some authors  , this method can be only calibrated ( intensity ratio to the number of layers ) to a given experimental set up and it is not transferable .the reason is mainly due to the different microscope objective numerical apertures ( na ) .the smaller the apperture is the longer the rayleigh length .it causes that small na objective probes signal deeper from bulk crystal , even if it is focused properly on a surface graphene layer . as a result ,relative intensity of graphene to sic bulk spectrum will be smaller than the ratio determined from raman spectra measured with high - na objective .a similar issue comes up in a grain size determination from the ratio of d to g peak intensity . if low - na objective is used , graphene signal is weak realitively to sic signal and the d peak can seem to be negligible with respect to sic  .the small d peak is then mistakenly identified as a signature of high quality graphene .hall effect measurements have been used to quantify graphene electronic quality .the apparent carrier mobilites are almost twice higher in argon grown graphene .the analysis of transport measurements has been done also within a model of electron hole puddles .this model is in agreement with our raman scattering analysis which has indicated very low charge density below 1.5   .the carrier mobility is twice higher for argon grown graphene within the model of electron hole puddles , too .we have analyzed experimentally graphene growth environment and its influence on graphene quality .residual gas analysis has shown important role of water reaction with carbon in high vacuum growth ambient .we have shown reduced role of the residual gas in 9n pure argon and we have compared high vacuum and argon grown graphene . the gas purity is a detrimental factor for high quality graphene for electronic applications . the mechanical strain , charge density , graphene grain size and number of graphene layers has been determined from raman spectroscopy mapping .hall measurements in van der pauw configuration have provided information about charge density consistent with raman spectroscopy .a carrier mobility has been modeled within electrostatic potential fluctuations model to provide realistic model of mean fermi level dependence of hall coefficient .the estimated lower bound of carrier mobilities in lithographically unprocessed epitaxial graphene provides a new route towards high - frequency electronic devices based on epitaxial graphene on sic .financial support from the grant agency of the czech republic under contract 16 - 15763y is gratefully acknowledged .raman spectroscopy has been measured within project vavpi cz.1.05/4.1.00/16.0340 .we acknowledge technical support from k. uhlov and fruitful discussion with p. hldek and r. grill .

Abstract: in recent years , graphene growth optimization has been one of the key routes towards large - scale , high - quality graphene production . we have measured in - situ residual gas content during epitaxial graphene growth on silicon carbide ( sic ) to find detrimental factors of epitaxial graphene growth . the growth conditions in high vacuum and purified argon are compared . the grown epitaxial graphene is studied by raman scattering mapping and mechanical strain , charge density , number of graphene layers and graphene grain size are evaluated . charge density and carrier mobility has been studied by hall effect measurements in van der pauw configuration . we have identified a major role of chemical reaction of carbon and residual water . the rate of the reaction is lowered when purified argon is used . we also show , that according to time varying gas content , it is preferable to grow graphene at higher temperatures and shorter times . other sources of growth environment contamination are also discussed . the reaction of water and carbon is discussed to be one of the factors increasing number of defects in graphene . the importance of purified argon and its sufficient flow rate is concluded to be important for high - quality graphene growth as it reduces the rate of undesired chemical reactions and provides more stable and defined growth ambient .