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Low entropy in graphene through the Co-C system

Uniform, mostly single-layer graphene with enhanced stability is demonstrated over Co film. The polycrystalline Co film deposited on a SiO2/Si substrate gives a continuous graphene layer that is easily transferred without the aid of any polymeric support, but preserving the material quality, as evidenced by Raman analysis. Great stability to the damaging action of the laser beam, as compared to the Cu-grown material is also observed. The better structural and electrical properties of the material are interpreted in terms of thermodynamics of the cooling-down process. It is suggested that the reduction in entropy, due annihilation of vacancies caused by C atoms precipitating during cooling, directly depends on the activation energy of C solubility into Co, which is considerably high, due to Co magnetic ordering at the process temperature. Our work expands the possibility of synthesizing single-layer graphene keeping into account the thermodynamics of various C-metal systems.


Introduction
Chemical Vapor Deposition (CVD) on catalyzing metals promises to be a simple, reliable and relatively cheap technique for producing large areas of good quality graphene, to be employed in the future electronics generation. amount of carbon can be dissolved in Cu.

Continuously imaging the C monolayer coverage on Cu using Low--Energy Electron
Microscopy confirmed no C precipitation or island growth during cooling [10], suggesting that the process is confined to the surface [11].
In contrast, Co and Ni can dissolve much more C atoms. The graphene growth comes also from the precipitation during the cool--down of the process and "polygraphene" was detected in most cases [11], as a consequence of an excess of precipitation of C atoms. For Co and Ni, the solubility and precipitation process must be controlled to some extent with the annealing, isothermal growing and cooling rates.
This means that cooling is an additional way to reduce the entropy of the system. In fact, entropy related to vacancies in Cu--grown graphene can be reduced during cooling through their diffusion, solely. It can be minimized if, e.g., two vacancies collapse into a di--vacancy (the number of loose atoms reduce from 6 to 4) or by their motion towards the grain boundaries, which act as vacancy sinks. In the case of Co and Ni, on the other hand, an extra, effective mechanism for vacancy reduction is provided by the C atoms precipitating from the substrate.
The aim of this work is to elucidate the effect of the supply of C atoms during cooling on the reduction of the entropy of the growing graphene layer. For the purpose, we will discuss in detail the properties of graphene grown onto Co substrates and discuss the effect of the activation energy of C solubility. It will be shown that this parameter plays an important role in dictating the rate of C precipitation, with important benefits on the entropy of the resulting graphene film.

Experimental
Graphene was synthesized via CVD, using CH4 as a precursor gas in a vacuum chamber of a Rapid Thermal Annealing apparatus [12]. For the purpose, 500 nm thick, Co polycrystalline thin films were deposited by RF Sputtering onto Si substrates with thermal SiO2 (500 nm thick). The Co deposition parameters were fixed at 100 W RF power and 10 --2 mbar Ar + pressure, leading to a deposition rate around 3 Å/s. The Co/SiO2 catalyst was sonicated in acetone and ethanol and then inserted in the reactor chamber. After removal of the residual Co oxide in H2 atmosphere for 5 min, 10 sccm of CH4 and 20 sccm of H2 were delivered for 5 min at a constant temperature of 1000 °C and pressure of 6.7 mbar. Cooling down was carried out in two steps: from 1000°C to 600°C in Ar and H2 atmosphere at a rate of 3.5°C/s, and from 600°C to room temperature at the same rate with a flow of N2 (200 sccm ) .
Taking advantage from its high structural integrity, the resulting graphene was then transferred onto a Si substrate with 300 nm SiO2, without the aid of any supporting layer.
Removal of the Co film was achieved by sample dipping in a 1 M FeCl3 solution. After repeated rinsing in H2O, the graphene sheet was picked up with the Si/SiO2 target substrate and then dried.
Differently from Cu, Co does not emit any photoluminescence when exposed to green light. This allows for accurate Raman comparisons onto the growth substrate and the destination one. A typical one, reported in Fig. 1, indicates that the graphene layer excellently withstands the transfer step, despite of the absence of any supporting material.
In order to measure the sheet resistance of the graphene sample, Cr/Au (20 and 75 nm thick, respectively) strip contacts were deposited in a E--Gun Evaporator and defined by a mechanical mask, thus avoiding any lithography technique to get rid of the contamination by PMMA and related chemicals. The metallic strips, 150 µm wide, 766 µm spaced, define three graphene areas of interest of about 1,32 mm 2 (see Fig.  2) The electrical measurements were performed in a Janis Cryostat [13], at room temperature and 2 10 --2 mbar pressure.
Micro--Raman analysis has been has been performed in a homemade system described elsewhere [2]. All acquisitions of Raman spectra were performed with a laser (λ = 532 nm, wavelength) spot diameter of 1µm and power of P ≈ 2mW. The spectra where collected at power density Ψ≈6.4×10 4 W/cm 2 and acquisition time tacq = 60 s.

Results and Discussion
SEM and AFM images of the graphene/Co system, displayed in Fig 3, show the presence of micrometer--sized holes in the Co thin film, upon which, suspended graphene can be observed. The AFM image also shows that the suspended graphene is stretched and corrugated. Even if the detailed discussion of the formation and properties of such graphene-on--nothing will be presented in a forthcoming paper, it will be however shown in section 4 that the occurrence of such a growth is consistent with the predictions of our model.
The SEM micrograph of the transferred sample in Fig. 3 shows the uniformity of the graphene sheet, with little bilayer islands, further confirmed by the Raman spectroscopy analysis whose results are displayed in Fig. 5. ID/IG and I2D/IG maps (50µm x 50µm) are shown.
These maps confirm the uniformity of the sample, its mostly monolayer character, and the low quantity of defects. A more specific and quantifiable measure of these two features, is provided by the histograms counting the number of spots with different ID/IG and I2D/IG ratios, peaking at 0.05 and 3.4 respectively.
As described in section 2, electrical measurement has been performed on a mm--sized graphene sheet, obtaining a value of the order of 200 Ω/Sq, comparable with the results reported for high quality CVD graphene [14--17]. The resistance of the Cr--Au contacts has been evaluated to be about 30 Ω. These results highlight the advantage of getting rid of any polymer, commonly employed in the standard transfer technique and in the lithography steps, since it can affect the electrical properties of graphene, e.g. by doping. Moreover, incomplete removal of the masking polymer can degrade the quality of the contacts on graphene [18].
The possibility of transferring the graphene sheets without the aid of a supporting layer deserves attention. Defects are commonly expected to originate from unsaturated bonds at grain boundaries, affecting both mechanical stability and carrier transport. In our case it seems that there are stiffer interconnections between grains, maybe due to the high crystallographic order of the layer, even if the presence of bilayer graphene at grain boundaries cannot be ruled out. In any case, the combination of the two effects allows for a polymer--free transfer method, which preserves the electrical properties of graphene.
We analyzed the evolution of the Raman spectra in order to determine the kinetics of defect formation. An example is reported in Fig.  6, obtained by focusing the laser onto three different spots of the sample, exhibiting similar ID/IG ratios in the pristine state, but different I2D/IG. The laser continuously irradiated those spots for 90 minutes and spectra were acquired at 10 min intervals.
The reported results give evidence that in Co--grown graphene, no additional defects are induced by laser irradiation, a very different behavior if compared to CVD growth on Cu foils.
It has to be pointed out that the model for the kinetics of defect formation under laser irradiation proposed in a previous work [2], predicts that fragmentation preferentially occurs at defective regions. The basic assumption of the model is that at a defective region, the energy released from the non--radiative recombination of photogenerated electron--hole pairs is sufficient to break a number of weakened bonds of loose atoms surrounding the defect. The We will demonstrate that the content of entropy of the graphene films is a direct consequence of the growth mechanism. As mentioned in section 1, graphene can grow by CVD on catalyzing metals by means of two main mechanisms, the first being the isothermal growth when the substrate is exposed to the C gaseous precursor, the second is the C precipitation upon cooling--down. In case of Cu the first mechanism dominates since only a negligible amount of additional C atoms precipitate on the metal surface [10,11], whereas, in case of high C solubility, graphene forms in both ways [19].

Theory
The theoretical treatment of the problem considers the variation of the free energy of the graphene layer during sample cooling--down. Fig. 7 displays the dependence of C solubility on the inverse T. The thermally activated behavior holds for the three metals into consideration, with the following activation energy values: ΔCo=0.791 eV, ΔNi,Cu=0.484 eV.
The peculiar value for the activation energy of C segregation in Co has been explained by Hasebe et al. [20] in terms of an additional contribution to the Gibbs' energy of the internal magnetic field. It is worth mentioning that graphene growth is normally carried out at temperature values below the Curie temperature of Co, even at high concentration of segregated C [20].
The total free energy G of a graphene layer in which the vacancies left by the incomplete, isothermal growth have to be annihilated by the successive precipitation step, can be written, following Hayes and Stoneham [21], as:

G = G0 + nG1 + G2 --nG--1, (1.1)
where n is the number of vacancies, G0 is the free energy of the perfect crystal, G1 the energy needed to generate one vacancy, G2 = -T · Sc (Sc being the configurational entropy of n vacancies), and G--1, the energy for a vacancy annihilation.
Imposing the equilibrium condition (δG/δn=0) yields: The configurational entropy Sc is generally calculated by means of the Boltzmann's famous formula Sc=kB ln P with P the number of possible microstates for the same macrostate.
Following Ref [21], eq. (1.2) becomes: where an additional entropy term α has been introduced, as additional degrees of freedom for the combinatorics needed to calculate the entropy. Nowadays, the molar enthalpy of formation of graphene is still unknown. Herein, it is estimated to be 513 kJ/mol from the bond enthalpy of graphene [22]. Consequently, this value is the same for both formation and annihilation of a vacancy, but with opposite signs, that have already been considered in eq to the forming graphene. The total amount of the entropy gain is obtained by multiplying ΔS times the number of precipitated atoms. This last number is dramatically lower in Cu, then, no noticeable entropy gain takes place during rapid cooling of the graphene--Cu system.

kB T ln = ΔH --TΔS
The model discussed so far considers single vacancies as the unique sources of entropy in the graphene system. The real situation is much more complicated, however the model can be extended to larger vacancies. At first, we can consider a di--vacancy as the result of the chemical reaction [21]: 1V+1V=1V2+ E2V, where E2V is the binding energy of the di--vacancy consequent to the collapse of two single vacancies. This means that: It is trivial to demonstrate that the quantity E2V (EnV in the case of a cluster with n--vacancies) adds to Δ in eq. (1.8), meaning that the gain in entropy when a cluster of n--vacancies is annihilated, is larger than in the case of n single vacancies.
Discontinuities in the underlying metal film are sources of large clusters of vacancies in the growing graphene layer. The case of grain boundaries is a well--known example. As it has been elucidated [11], the growth of graphene is impeded by grain boundaries in case of Cu (where precipitation is negligible), whereas enhanced growth of graphene is reported for the Ni and Co cases. The widely accepted explanation is the contribution to the growth from atoms precipitated from the vis--a--vis surfaces of two contiguous grains.
It has also been reported that graphene can grow on wide discontinuities of the metal (Ni, in that case) film [25]. This does not happen in the case of Cu: no Raman signatures of graphene are observed in holes opened by the dewetting phenomenon occurring in the Cu films before exposure to C precursor [26].
The occurrence of the growth of a suspended layer of graphene on holes in the underlying Co film deserves special consideration. The origin of such holes, few µm wide, is ascribable, like in case of Cu, to dewetting, which is unlikely in the case of pure Co (with melting temperature Tm=1495 °C), but can occur at C contents close to saturation (Tm ˜1300 °C [20]). Dewetting preferentially occurs at the boundary of three or more grains [27], and it is strongly inhibited by the presence of the growing graphene layer [12]. Basing on these considerations, it is likely that holes in the Co film form during the isothermal gas exposure step. In other words, the graphene growth occurs On--Nothing during the following step.
Furthermore, the occurrence of the alternative mechanism, the growth of graphene at a grain boundary, followed by hole opening, would imply the formation of a multi--layer at that boundary. Raman investigation, on the contrary, confirms that the graphene on Co holes grows as defect--free monolayer.
Some considerations are needed about such growth of Graphene--On--Nothing in view of the present model. The graphene giant vacancy corresponds to the Co hole, and is a great source of entropy of the film. Nevertheless, the system tends to reduce its entropy by precipitation of C from the Co boundaries surrounding the hole, arranging them in order to progressively reduce entropy at the perimeter of the giant vacancy cluster. A continuous graphene layer is then obtained in spite of discontinuities in the metal substrate. This underlines the crucial role played by the high activation energy of C solubility in Co in sustaining this kind of unconventional growth.

Conclusions
In summary, a comparison among the different mechanisms of graphene growth in Cu, Ni, and Co, is presented. It is shown that C atoms precipitation from the metal substrate is an efficient mechanism to reduce the total entropy of the graphene layer. A model is proposed, which shows that not only the absolute value of C solubility, but also its dependence on temperature play an important role in reducing entropy. Thanks to the high activation energy of C solubility, Co shows advantages with respect to other, more employed, catalyzing metals. This viewpoint is confirmed by the good structural and electrical properties of the graphene layers obtained in this way.