The thermodynamic and kinetic properties of hydrogen dimers on graphene
Liang Feng Huang, Mei Yan Ni, Yong Gang li, Wang Huai Zhou, Xiao Hong Zheng, Ling Ju Guo, Zhi Zeng
aa r X i v : . [ c ond - m a t . m t r l - s c i ] N ov The thermodynamic and kinetic properties of hydrogen dimers on graphene
Liang Feng Huang, Mei Yan Ni,
1, 2
Yong Gang Li, WangHuai Zhou, Xiao Hong Zheng, Ling Ju Guo, and Zhi Zeng ∗ Key Laboratory of Materials Physics, Institute of Solid State Physics,Chinese Academy of Sciences, Hefei 230031, China School of Electronic Science and Applied Physics,Hefei University of Technology, Hefei 230009, China
Abstract
The thermodynamic and kinetic properties of hydrogen adatoms on graphene are important to the mate-rials and devices based on hydrogenated graphene. Hydrogen dimers on graphene with coverages varyingfrom 0.040 to 0.111 ML (1.0 ML = . × cm − ) were considered in this report. The thermodynamic andkinetic properties of H, D and T dimers were studied by ab initio simulations. The vibrational zero-pointenergy corrections were found to be not negligible in kinetics, varying from 0.038 (0.028, 0.017) to 0.257(0.187, 0.157) eV for H (D, T) dimers. The isotope e ff ect exhibits as that the kinetic mobility of a hydro-gen dimer decreases with increasing the hydrogen mass. The simulated thermal desorption spectra withthe heating rate α = . / s were quite close to experimental measurements. The e ff ect of the interactionbetween hydrogen dimers on their thermodynamic and kinetic properties were analyzed in detail. PACS numbers: 68.65.Pq, 67.63.-r, 68.43.Bc . INTRODUCTION The thermodynamic and kinetic properties of hydrogen (isotopes) on graphene can help us un-derstand the interaction between hydrogen and graphitic materials in outer space and fusiondevices.
Such properties also determine the realizability of the graphene(graphite)-based hy-drogen storage and the graphene-based electronic devices. And the hydrogenated graphene(graphite) has been one focus of scientific researches in recent years.Thermal desorption (TD) spectroscopy, which is also called temperature-programmed desorp-tion (TPD) spectroscopy, has been used in experiments to study the kinetic properties of hydrogenon the graphite surface.
T. Zecho et al. found that the saturation coverages ( Θ sat ) of Hand D on graphite surface are about 0.3 and 0.4 ML, respectively. The TD spectra of H and D(heating rate α = . / s) both exhibit two-peak shapes when the coverage Θ ≤ . × Θ sat , andthe increase in Θ just results in some minor modification of the two-peak shapes. L. Horkekær etal. found that there are mainly two kinds of hydrogen dimers responsible for the two-peak shapesof the TD spectra at low hydrogen coverage, which are the ortho-dimer and para-dimer. In thetwo-peak shaped TD spectra, the two desorption peaks are at 445 (490) K and 560 (580) K for H(D) adatoms, respectively, where the positions of the desorption peaks increase with the hy-drogen mass. F. Dumont et al. have simulated the TD spectra of hydrogen dimers on graphene byusing the kinetic Monte Carlo method, and reproduced the two-peak shaped TD spectra. And E.Gavardi et al. also have simulated the two-peak shaped TD spectra by the same method but takenhydrogen trimers and tetramers into consideration. These experimental and theoretical resultsindicate that the hydrogen ortho-dimer and para-dimer should be still the most important dimersresponsible for the near-two-peak shapes of the TD spectra at high coverages. This is consistentwith that the configurations of the ortho-dimer and para-dimer always tend to be preserved inhydrogen trimers and tetramers.
However, the kinetic properties of hydrogen dimers still need more accurate simulations, espe-cially the vibrations should be taken into consideration. Because the mass of hydrogen isotopesare so small that the vibrational zero-point energy corrections may not be negligible in kinetics.The thermodynamic and kinetic properties of hydrogen dimers are expected to be sensitively de-pendent on their microscopic structures, on which systematic and accurate investigations are stilllacking.In this report, the thermodynamic and kinetic properties of H, D and T dimers on graphene2re simulated with a composite method consisting of density functional theory (DFT), densityfunctional perturbation theory (DFPT), harmonic transition state theory (hTST), and reactionrate theory. The e ff ect of the interaction between hydrogen dimers on the thermodynamic andkinetic properties of hydrogen dimers on graphene are studied. The isotope e ff ects in the kineticproperties are observed due to the inclusion of the vibrations in the simulations. The simulatedTD spectra of H and D dimers with the heating rate α = . / s are very close to experimentalmeasurements, and the TD spectra of T (radioactive) dimers are also predicted. II. METHODOLOGY
The adsorption energy of a hydrogen dimer is defined to be the energy di ff erence before andafter the adsorption of two hydrogen atoms on graphene, which is expressed as E ads = E H + E GL − E GL + H (1)where E H , E GL and E GL + H are the total electronic energies of an isolated hydrogen atom, an iso-lated graphene layer and a graphene layer with a chemisorbed hydrogen dimer on it, respectively.The frequency ( v ) of an over-barrier jump from one local minimum state (initial state) to an-other local minimum state (final state) can be calculated using the quantum-mechanically modifiedharmonic transition state theory (hTST), which is expressed as v = v ∗ qm exp ( − E ac k B T ) = k B Th N Q i = [1 − exp ( − ~ ω Ii k B T )] N − Q i = [1 − exp ( − ~ ω Si k B T )] exp ( − E ac k B T ) (2)where v ∗ qm is the quantum-modified exponential prefactor; E ac is the activation energy; N is thenumber of atoms; ω Ii and ω Si are the frequencies of the i th vibrational mode in the initial andsaddle-point states in the reaction path, respectively. The numbers 3 N and 3 N − N real vibrational modes and the saddle-point state has 3 N − E ac = ∆ E p + N − X i = ~ ω Si − N X i = ~ ω Ii = ∆ E p − ∆ F vib (0) (3)3here ∆ E p is the potential barrier in the reaction path; ∆ F vib (0) is the vibrational zero-point energycorrection. When the temperature approaches to be infinite, the classical limit of the prefactor isexpressed as v ∗ cl = π N Q i = ω Ii N − Q i = ω Si (4)This classical-limit form is also the Vineyard’s form, where all of the vibrational modes areassumed to be completely thermo-activated.In this report, a mono-layer graphene is used as a structural model, which is also a safe modelfor graphite surface because the weak Van de Waals interaction between neighboring graphene lay-ers has negligible e ff ect on the chemisorption properties of hydrogen. The vacuum betweentwo neighboring layers is 10 Å. And Θ s of 0.111, 0.063 and 0.040 ML (1.0 ML = . × cm − )are computationally achieved by chemisorbing a hydrogen dimer at the center of periodic graphenesupercells with sizes of 3 × × × × × Θ = .
056 ML). The choice of the para- and ortho-dimerfor decoration is due to that the two dimers were found to be the most stable hydrogen dimerson graphene. In experimental observations, hydrogen adatoms (including dimers) tend to clusteron graphite surface, making some spots covered by hydrogen while some not. This un-uniformdistribution of hydrogen indicates that the Θ in simulation should correspond to the e ff ective hy-drogen coverage of the hydrogen-covering spots. There are 5 configurations of hydrogen dimerson graphene considered here, which are ortho-dimer (O), meta-dimer (M), para-dimer (P), A-dimer (A) and B-dimer (B) as shown in Fig. 1, as well as the desorbed hydrogen molecule (H ).Other more extended hydrogen dimers are not considered.The structures and potential barriers are calculated using DFT and the vibrational frequenciesusing DFPT. The DFT and DFPT calculations are carried out using the Quantum Espresso codepackage, in which the ultrasoft pseudopotentials with the BLYP exchange-correlation func-tional are used. The energy cuto ff s for the wave function and charge density are 35 and 350 Ry,respectively. Uniform k-point grids are chosen to be 6 ×
6, 5 ×
5, 4 × × with an energywidth of 0.03 Ry is employed to speed the convergence of the numerical calculations. The elec-4ronic density of states (DOS) for the chemisorption states in the S5 case is calculated with a 6 × The reaction paths are described by theminimum energy paths (MEPs) between two local minimum states, which are calculated using theclimbing-image nudged elastic band method. For the calculation of the vibrational frequencies,only the Γ point at the Brillouin zone center is chosen. III. RESULTS AND DISCUSSIONA. DFT and hTST calculations
The calculated adsorption energies for the P-dimer ( E Pads ) and O-dimer ( E Oads ) on the S3, S4,S5, S6-P and S6-O graphene supercells are listed in Tab. I. In the S3, S4 and S5 cases, E ads generally increases with Θ , with the only exception that the E Oads in the S3 case is a little (9 meV)less than that in the S4 case. This indicates the attractive interaction between the neighboringdimers, which is consistent with the clustering of hydrogen adatoms on graphite surface observedin experiments.
This is because one chemisorbed dimer can cause the carbon lattice aroundwrinkled, and partly destroy the sp orbital hybridization of the carbon atoms nearby, which canincrease the a ffi nity of those carbon atoms to bond with another hydrogen dimer. However, the E Pads in the S6-P(O) case is larger than those both in the S4 and S5 cases, and the E Oads in the S6-P(O)case is larger than those in all the S3, S4 and S5 cases, although the Θ of the S6-P(O) case is lowerthan those of the S3 and S4 cases. This is due to the e ff ect of the periodic boundary conditions(PBC) we take for the DFT calculations here. In the S3, S4 and S5 cases, two interacting dimersare the periodic images of each other. While in the S6-P and S6-O cases, a dimer can interact withthe decoration dimer at the corner of the same supercell. Thus, in the S3, S4 and S5 cases, thePBC will partly suppress the wrinkling of the carbon lattice caused by the adsorption of a dimer,while in the S6-P and S6-O cases, a dimer feels more wrinkling caused by the decoration dimerin the same supercell. Thus, this PBC e ff ect on the thermodynamic stability of hydrogen dimerson graphene also approves that the wrinkling of the carbon lattice will result in the attractiveinteraction between dimers on graphene.The calculated potential barriers ( ∆ E p ) for various transitions of the hydrogen dimers on theS3, S4, S5, S6-P and S6-O supercells are listed in Tab. II. It can be seen that ∆ E p is dependenton both Θ and the structural configuration, the same as E Pads and E Oads described in the previous5aragraph. In all cases, the optimal reaction path for an O-dimer to be desorbed is O-M-P-H , andthat for the P-dimer is P-H , which is consistent with Hornekær’s results. Other transitions outof the O-M-P-H reaction path should overcome higher potential barriers, e.g. ∆ E p (M-B) is 0.189eV larger than ∆ E p (M-P) in the S6-P case (much larger in other four cases). This magnitude ofdi ff erence will result in that the possibility of the M-P transition is about 3 orders larger than thatof the M-B transition at 300 K (estimated by Equ. 2). Thus, the M-B transition could be safelyneglected when compared with the M-P transition. In addition, according to the calculations byHornekær and ˇSljivanˇcanin, the evaporation of hydrogen monomers from dimers also needsovercome much higher potential barriers. Therefore, only the transitions in the optimal path O-M-P-H are considered in the simulation of the TD spectra in the following. For an isolated dimeron graphene, there may be more than one paths that are equivalent for one kind of transition,e.g. there are 4 equivalent paths for the O-M transition. The number of equivalent paths for atransition is defined as the degeneracy of the reaction path ( g path ). The values of the g path s for thetransitions of an isolated dimer on graphene are listed in Tab. II, where the g path s for the adsorbingtransitions (H -M, H -P and H -O) are absent, because they are hardly and unnecessarily definedin the simulations here. However, when Θ is high enough, the interaction between dimers willtend to make two di ff erent paths of the same kind transition have di ff erent ∆ E p s, namely the g path is reduced. Among the transitions in the optimal path, the P-H and O-M transitions arethe most important ones (the rate-limiting steps) for the TD spectra, which will be shown in thefollowing. The ∆ E p (O-M) and ∆ E p (P-H ) generally increase with Θ in the S3, S4 and S5 cases,which indicates that it is harder for hydrogen dimers to be desorbed from graphene at higher Θ s.This is the same as the e ff ect of Θ on the thermodynamic stability of hydrogen dimers on graphene,as discussed in the previous paragraph. However, the ∆ E p (O-M) in the S6-O case and the ∆ E p (P-H ) in the S6-O(P) case are larger than those in the S3, S4 and S5 cases. This is because the PBCe ff ect tend to make the dimers on a S6-P(O) supercell more stable than those on the S3, S4 and S5supercells. Thus, it costs more energy for a dimer in the S6-P(O) case to escape from one state totransit to another state.The adsorption of a hydrogen dimer onto two C atoms in graphene will change these C atomsfrom sp to sp hybridized. Seen from the DOS spectra for the systems of S5 + O-dimer, S5 + M-dimer and S5 + P-dimer (Fig. 2), the hybridization between hydrogen dimers and graphene makesgraphene be semiconducting. The band gap are 0.43, 0.36 and 0.38 eV for S5 + O-dimer, S5 + M-dimer and S5 + P-dimer, respectively. However, the DOS spectra for S5 + M-dimer are di ff erent6rom those for S5 + O-dimer and S5 + P-dimer. Only in the DOS spectra of S5 + M-dimer, thereare four narrow peaks around the Fermi level, two spin-up (occupied) and two spin-down (unoc-cupied). And S5 + M-dimer is magnetic with a spin moment of 2.0 µ B , while the other two arenonmagnetic. The two C atoms bonded with M-dimer are equivalent in graphene, a bipartite sys-tem, while those C atoms bonded with O(P)-dimer are inequivalent. The di ff erence in magnetismis due to this di ff erence in the hydrogen-bonded sites, which can be understood from the Lieb’stheorem for the bipartite system. The adsorption of a M-dimer leaves two unsaturated p z orbitals,which then form two quasilocal states around the M-dimer, with each state holding one electronand a spin moment of 1.0 µ B . This is the same as the adsorption of the hydrogen monomer ongraphene. The contributions of the H(1 s ) orbitals to these quasilocal states are very small.However, the two unsaturated p z orbitals in S5 + O-dimer or S5 + P-dimer bond with each other,thus, there are no quasilocal states and magnetic moment in these two systems. This bondingbetween these two p z orbitals makes the O-dimer and P-dimer be more stable than the M-dimer(Tab. II). Although the transition of a hydrogen dimer changes the magnetism of the hydrogenatedgraphene, the system is dominated by nonmagnetic states. Because the M-dimer is metastable(seen from the ∆ E p s in Tab. II), and its lifetime at 300 K is less than 1.0 s, as simulated below.The two hydrogen atoms of a dimer prefer to adsorb on two inequivalent C atoms in graphene.Likewise, from the energetic calculations for more extended dimers and trimers on graphene, aconfiguration is also much more stable when two nearest-neighboring hydrogen atoms are bondedwith inequivalent C atoms in graphene.The calculated ∆ F vib (0)s for the transitions of H, D and T dimers in the optimal path O-M-P-H are listed in Tab. III, where the results for the S3 and S5 cases are compared. The di ff erencesbetween these two groups are very small, especially for the ∆ F vib (0)s of the two rate-limitingtransitions, O-M and P-H (within 6 meV). This implies that the vibrational properties of thehydrogen dimers on graphene are mainly determined by the localized vibrational modes, whichis the same as those of the hydrogen monomer on graphene. It also can be seen that ∆ F vib (0)decreases with increasing the hydrogen mass. This isotope e ff ect is also the same as that of thehydrogen monomer on graphene in Ref. 29, where the relationship between the isotope e ff ectsin phonon spectra and ∆ F vib (0) was analyzed in detail. The ∆ F vib (0)s for the di ff using transitions(O-M, M-O, M-P, P-M) are less than those for the desorbing transition (P-H ), which is also thesame as that of the hydrogen monomer on graphene. The values of the ∆ F vib (0)s for H (D, T)dimer vary from 0.038 (0.028, 0.017) to 0.257 (0.187, 0.157) eV, which are large enough that can7ot be omitted in kinetics.The small di ff erence between the vibrational properties of hydrogen dimers in the S3 and S5cases has also been found by comparing the prefactors for the O-M and P-H transitions of H, Dand T dimers in these two cases (Fig. 3). In the plotted temperature range, the v ∗ cl di ff erences be-tween these two cases are within 20%, and the v ∗ qm di ff erences within 15%. Since v is just linearlydependent on v ∗ qm (Equ. 2), a small di ff erence ( < v ∗ qm will not result in anysignificant deviation in the kinetic properties, e. g. the TD spectra. Consequently, the vibrationalfrequencies from the S3 case are used in the simulations of the kinetic properties of hydrogendimers in all cases, which is a quite accurate treatment and brings much convenience to the nu-merical simulations. Seen from Fig. 3, v ∗ cl decreases with increasing the hydrogen mass, while thevariation of v ∗ qm with the hydrogen mass is di ff erent. The v ∗ qm for the O-M transition is nearly in-variant with the hydrogen mass in the plotted temperature range, while that for the P-H transitionincreases with the hydrogen mass. These isotope e ff ects in v ∗ cl and v ∗ qm of the hydrogen dimer arethe same as those for the desorption and di ff usion of the hydrogen monomer on graphene, whichhas been understood from the isotope e ff ect of the spectra of the localized vibrational modes ofhydrogen and analyzed in detail in Ref. 29. The hydrogen dimers are kinetically active in thetemperature range from 350 to 650 K, above which the dimers have been completely desorbed(shown below). In this temperature range, the v ∗ qm s for the O-M transition vary from 8 to 15 × s − , and those for the P-H transition vary from 5 to 9 × s − .The calculated jump frequencies for the transitions of H, D and T dimers on the S6-P supercellare shown in Fig. 4 (a - e). The jump frequency of a transition in other cases ( v X , X = S3, S4, S5,S6-O) can be deduced from the corresponding v in the S6-P case by the equation v X = v S − P exp ( − δǫ k B T ) (5)where δǫ is the di ff erence of the ∆ E p for a transition in the X case and the S6-P case. The valuesof the ∆ E p s for all the transitions in all the cases are listed in Tab. II. The isotope e ff ect in v isthat it decreases with increasing the hydrogen mass, which is due to the isotope e ff ect in ∆ F vib (0)(Tab. III) and is the same as that of the hydrogen monomer on graphene . In the temperaturerange from 350 to 650 K, the jump frequencies for the M-O and M-P transitions are at least4 orders larger than those for the O-M, P-M and P-H transitions. Because v is exponentiallydependent on E ac or ∆ E p (Equ. 2 and 4) and the ∆ E p s for the M-O and M-P transitions aremuch less than those for the O-M, P-M and P-H transitions. The life time of the M-dimer ( τ M ∼ v M − P + v M − O ) in the S6-P case is less than 1.0 s at 300 K, and decreases exponentially with temperature.And τ M is much less in other cases due to their lower ∆ E p s for the M-P and M-O transitions(Tab. II). Thus, although the M-dimer is an important configuration in the kinetic transitionsof a hydrogen dimer on graphene, it has not been observed in experiments and theoreticalsimulations. Furthermore, v P − H transition is at lest 2 orders larger than v O − M , which indicatesthat the O-dimer is kinetically more stable than the P-dimer on graphene. B. The simulation of the TD spectra
In the optimal path of O-M-P-H , the jump frequencies for the M-O ( v M − O ) and M-P ( v M − P )transitions are at least 4 orders of magnitude larger than those for other transitions in the temper-ature range from 350 to 650 K. Thus, it is reasonable to assume that when a dimer succeeds injumping from O-dimer state to M-dimer state, it will immediately turn to be O-dimer or P-dimerstate. Following the treatment by Toyoura in Ref. 28 for the di ff usion of lithium in the interca-lated graphite, the mean time of the total jump is the sum of those of the individual jump steps, τ = τ + τ , where τ = ( g O − M v O − M ) − and τ = ( g M − P v M − P + g M − O v M − O ) − , with the g path ofeach transition path being accounted. An O-P transition is accomplished if the M-dimer finallysucceeds in jumping to be a P-dimer. Then the jump frequency for this O-P transition is given by v O − P = g M − P v M − P g M − O v M − O + g M − P v M − P × τ + τ (6) = g O − P v M − P v M − O + v M − P v O − M (7)where Equ. 7 is exact only for the isolated dimers on graphene and g O − P ( = g O − M g M − P /
2) equals4 there; g M − P v M − P g M − O v M − O + g M − P v M − P or v M − P v M − O + v M − P is the success probability of the M-P jump from a M-dimer. g O − P is regarded as the path degeneracy of the O-P transition, and Equ. 7 is used for non-isolateddimers on graphene where g O − P is treated as a variable. Likewise, the jump frequency for the P-Otransition is given by v P − O = g P − O v M − O v M − O + v M − P v P − M (8)The interaction between neighboring hydrogen dimers tends to lower the values of g O − P and g P − O from 4. Both of g O − P and g P − O are taken to be equal in the simulation of the TD spectra forconvenience. In Fig. 4 (f) and (g), the calculated v O − P s and v P − O s for H, D and T dimers areshown, with g O − P and g P − O both being taken an intermediate value of 2. v O − P and v P − O are mainly9etermined by v O − M and v P − M , respectively, with the multiplying factors in Equ. 7 and 8 justvarying from 0.5 to 1.0 in the temperature range from 350 to 650 K. v O − P and v P − O decrease withincreasing the hydrogen mass, the same isotope e ff ect as other transition frequencies in the optimalpath of O-M-P-H (Fig. 4).The ratio of O-dimer and P-dimer ( n O : n P ) at the starting moment (t =
0) was reported as0.15 , 0.2 and 0.47 , respectively. In this report, we take this ratio as 0.2 ( n O = . n P = . ff ect, the reaction of the hydrogen dimers on graphene is first-order, which is consistent withexperimental observations. The reaction-rate equations for the TD spectroscopy of hydrogendimers on graphene can be expressed as dn O ( t ) dt = − v O − P ( T ) n O ( t ) + v P − O ( T ) n P ( t ) (9) dn P ( t ) dt = − [ v P − O ( T ) + v P − H ( T )] n P ( t ) + v O − P ( T ) n O ( t ) (10) n O ( t + ∆ t ) = n O ( t ) + dn O ( t ) dt ∆ t (11) n P ( t + ∆ t ) = n P ( t ) + dn P ( t ) dt ∆ t (12)where ∆ t is a time interval. The desorption rate of hydrogen dimers, or the hydrogen-gas yield perunit time, is defined as R des ( T , t ) = v P − H ( T ) n P ( t ) (13)where T = α t and the heating rate ( α ) is set as 1.0 K / s in accordance with experiments. The simulated TD spectra for H, D and T dimers in the S6-P case are shown in Fig. 5 (a).The two-peak shaped TD spectra have been reproduced, with the desorption peaks being at 436(460, 469) and 562 (574, 581) K for H (D, T) dimers, respectively. These simulation results arevery close to experimental measurements.
The kinetic processes behind the TD spectra aredescribed in the following. As the temperature increases (T = . t ), the P-dimers are first desorbedthrough the P-H transition above 350 K. At this time, there are negligible amount of P-dimerstransiting to be O-dimers, because v P − M ( v P − O ) is at lest 2 orders less than v P − H . Afterwards,the O-dimers are reduced at higher temperatures ( >
475 K) through the O-M-P-H transition.The variations of n P ( T /α ) and n O ( T /α ) are shown in Fig. 5 (b). When a dimer succeeds intransiting from an O-dimer through a M-dimer to a P-dimer, it stays as a P-dimer for a shorttime of about 1 / v P − H seconds before being desorbed, which is reflected by the small occupation10umber of the P-dimers ( n P ( T /α )) at the temperatures around the 2 nd desorption peak (the insetin Fig. 5 (b)). The mobilities of the hydrogen dimer in di ff usion and desorption decrease withincreasing the hydrogen mass, which is due to the isotope e ff ect in v or ∆ F vib (0). In experimentalmeasurements, the positions of the two desorption peaks in the TD spectra for H (D) dimersare at 445 (490) K and 560 (580) K (the average Θ < .
12 ML), while those in the simulatedspectra are 436 (460) K and 562 (574) K (the e ff ective Θ = .
056 ML). The simulated isotopicdi ff erences of the desorption peaks of H and D dimers are 24 and 12 K for the 1 st and 2 nd peaks,respectively, in comparison with the experimental measurements of 45 and 20 K, respectively.These little deviation between our theoretical simulations and experimental results maybe comefrom that a small amount of uncertainty has been introduced in experiments when estimating thedesorption-peak positions from the somewhat chaotic TD spectra of H, or from the isotope e ff ectin the e ff ective Θ of the hydrogen-covered spots on graphene in experiments. The latter possibilityis understandable since that the thermodynamic stability of the hydrogen dimer on graphene shouldbe hydrogen-mass dependent, thus, the distribution of hydrogen dimers on graphene should behydrogen-mass dependent. This is like the isotope e ff ect of Θ sat , which are about 0.3 and 0.4 MLfor H and D on graphene, respectively. The di ff erence in the e ff ective Θ for H and D dimersresults in the di ff erence in the interaction between dimers and then the di ff erence in ∆ E p . Basedon this point, the isotope e ff ect in the experimental TD spectra of hydrogen dimers consists of thecontributions from the isotope e ff ects in v and Θ . The simulations here, however, just consider theisotope e ff ect in v .In order to observe the e ff ects of Θ and the structural configuration on the TD spectra, the ∆ E p sfor the transitions of O-M, M-O, M-P, P-M and P-H in all the cases considered here (Tab. II) areused to simulate the TD spectra. The positions of the two desorption peaks in each case are listedin Tab. IV. The 1 st desorption peak shifts nearly linearly with the ∆ E p of the P-H transition by34 K / nd peak does not have a simple monotonic variation due to the complextransitions of hydrogen dimer behind but still generally increases with ∆ E p . Additionally, theinteraction between dimers also tends to lower the degeneracy of the di ff using reaction paths, g O − P and g P − O , which are taken equally here and together denoted as g path below. The position of the 1 st desorption peak is obviously not dependent on the di ff using g path . Seen from Fig. 5 (c) whereas, thepositions of the 2 nd desorption peaks for H, D and T dimers are nearly logarithmically dependent11n g path , which is approximately expressed as T ( g path ) = T (1) − × lg ( g path ) (14)where T(1) is the peak position when g path =
1. And T( g path ) increases by about 11 K with g path decreasing by a half.From the analysis above, the interaction between hydrogen dimers on graphene is coordinatedwith the e ff ective hydrogen coverage and the structural configuration. Such interaction a ff ectsthe potential barriers and the reaction-path degeneracies of the hydrogen-dimer transitions ongraphene. As a result, the jump frequencies and TD spectra vary with the hydrogen coverageand the structural configuration. However, the localized vibrational properties of hydrogen dimersare not significantly influenced by the hydrogen coverage. IV. CONCLUSIONS
The thermodynamic and kinetic properties of H, D and T dimers on graphene have been simu-lated with a composite ab initio method consisting of density functional theory, density functionalperturbation theory, harmonic transition state theory, and reaction rate theory. The e ff ects of thehydrogen coverage and the structural configuration on the thermodynamic and kinetic propertieshave been revealed by varying the coverage from 0.040 to 0.111 ML (1.0 ML = . × cm − ).It has been found that the interaction between hydrogen dimers influences the e ff ective hydrogencoverage and the configuration of hydrogen dimers deposited on graphene and a ff ect the thermo-dynamic and kinetic properties ( E ads , ∆ E p , v ( T ) and R des ( T , t )) of those dimers. The vibrationalzero-point energy correction ( ∆ F vib (0)) and the jump frequency ( v ) both decrease with increasingthe hydrogen mass. However, the isotope e ff ect in the positions of the desorption peaks in the TDspectra is inverted. In a word, the mobility of the hydrogen dimer in desorption and di ff usion de-creases with increasing the hydrogen mass. The simulated TD spectra (heating rate α = . / s)are quite close to experimental measurements, and the e ff ect of the dimer interaction has beenclarified in detail via the potential barrier ∆ E p in various cases and the reaction-path degeneracy g path . 12 cknowledgments The first author (Huang) wish to thank Liv Horkekær for helpful email exchanges. This workwas supported by the special Funds for Major State Basic Research Project of China (973) undergrant No. 2007CB925004, Knowledge Innovation Program of Chinese Academy of Sciencesunder grant No. KJCX2-YW-N35, and Director Grants of CASHIPS. Part of the calculations wereperformed in Center of Computational Science of CASHIPS and the Shanghai SupercomputerCenter. ∗ Email: [email protected] J. M. D. Coey, M. Venkatesan, C. B. Fitzgerald, A. P. Douvalis, and I. S. Sanders, Nature , 156(2002). L. 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ABLE I: The hydrogen coverage ( Θ ) and the adsorption energy of the para-dimer ( E Pads ) and ortho-dimer( E Oads ) in various cases (1.0 ML = . × cm − ).Supercell Θ (ML) E Pads (eV) E Oads (eV)S3 0.111 2.714 2.555S4 0.063 2.562 2.564S5 0.040 2.538 2.518S6-P 0.056 2.604 2.758S6-O 0.056 2.578 2.652TABLE II: The reaction-path degeneracy ( g path ) for an isolated hydrogen dimer on graphene, the potentialbarrier ( ∆ E p ) for the transitions of a hydrogen dimer on various graphene supercells. ∆ E p (eV)Path g path S3 S4 S5 S6-P S6-OO-M 4 1.679 1.699 1.664 1.690 1.713M-O 2 0.405 0.510 0.596 0.689 0.609M-P 2 0.351 0.538 0.607 0.773 0.660P-M 4 1.792 1.711 1.674 1.620 1.690P-A 2 1.821 1.637 1.602 1.605 1.636A-P 1 0.223 0.449 0.525 0.558 0.587M-B 2 0.605 0.806 0.836 0.962 0.912B-M 2 1.106 1.139 1.110 1.121 1.113O-H -O - 4.414 4.298 4.470 4.191 4.345M-H -M - 4.191 4.187 4.143 4.126 4.227P-H -P - 3.398 3.448 3.461 3.593 3.573 ABLE III: The vibrational zero-point energy correction ( ∆ F vib (0), in eV) for the transitions of H, D and Tdimers in the optimal path (O-M-P-H ) in the cases of S3 and S5.S3 S5Path H D T H D TO-M 0.141 0.107 0.093 0.135 0.102 0.088M-O 0.038 0.028 0.017 0.051 0.034 0.027M-P 0.053 0.032 0.023 0.079 0.053 0.042P-M 0.138 0.106 0.093 0.144 0.111 0.075P-H α = . / s). 1st desorption peak 2nd desorption peakSupercell H D T H D TS3 434 458 468 534 536 551S4 401 424 434 551 563 569S5 393 417 427 537 548 554S6-P 436 460 469 562 574 581S6-O 420 443 453 562 573 579 M P H A BPO S5S6-O(P) S4S3
FIG. 1: (Color online) The structures of the periodic graphene supercells with the sizes of 3 × × × × ). And the S6 graphene supercell is decorated with a para-dimer (S6-P) or an ortho-dimer(S6-O) at the corner (shaded in gray) of the supercell. The gray spheres are carbon atoms, and the bluesmaller spheres are hydrogen atoms. E-E F (eV) -10010 -10010 Total H(1s) spin upspin down
OrthoMeta
Para D O S FIG. 2: (Color online) The DOS spectra of the S5 graphene supercell with an O-dimer, M-dimer and P-dimer. The vertical line at the Fermi level guides the eyes. H D T P r e f a c t o r ( s - ) Temperature (K) (a) v * cl v * qm O-M P r e f a c t o r ( s - ) Temperature (K) v * cl v * qm (b) P-H FIG. 3: (Color online) The variation of the v ∗ qm and v ∗ cl with respect to temperature in the S3 and S5 casesfor (a) O-M and (b) P-H transitions of H, D and T dimers. The prefactors for the S3 case are plotted withthicker lines, and those for the S5 case with thinner lines. .5 2.0 2.5 3.010 -9 -7 -5 -3 -1 H D T ( s - ) -1 )O-M(a) (b) ( s - ) -1 )M-O (c) ( s - ) -1 )M-P -10 -8 -6 -4 -2 (d) ( s - ) -1 )P-M -7 -5 -3 -1 (e) ( s - ) -1 )P-H -12 -10 -8 -6 -4 -2 (f) ( s - ) -1 )O-P -10 -8 -6 -4 -2 (g) ( s - ) -1 )P-O FIG. 4: (Color online) The variation of v with respect to the inverse of temperature for the (a) O-M, (b)M-O, (c) M-P, (d) P-M, (e) P-H , (f) O-P and (g) P-O transitions of H, D and T dimers. The corresponding v ∗ qm s are used as the prefactors.
50 400 450 500 550 600 6500.000.060.120.18
H D T nd peak1 st peak R de s ( s - ) Temperature (K) (a)
350 400 450 500 550 600 650012345
450 500 550 600 65010 -6 -5 -4 -3 -2 -1 (b) n O n P N o . o f d i m e r s ( a r b . un i t ) Temperature (K -1 ) n P (c) T e m pe r a t u r e ( K ) g path FIG. 5: (Color online) (a) The TD spectra of H, D and T dimers ( α = . / s) with the starting dimerratio n O (0) : n P (0) = n P (T / α ) and n O (T / α ). The inset shows the details of thevariation of n P (T / α ) (in the logarithmic scale) from 450 K to 650 K. (c) The variations of the positions ofthe 2 nd desorption peak in the TD spectra for H, D and T dimers with respect to the degeneracy ( g path ) ofthe di ff using reaction path . g path is plotted in the logarithmic scale.is plotted in the logarithmic scale.