Magnetic response of triangular graphone

We consider triangular graphone structure – a semi-hydrogenated layer of graphene with hydrogen atoms boned to one of its sub-lattices only. The response function of graphone to an external magnetic field is evaluated in general using the Heisenberg Hamiltonian model of the structure with exchange energy coefficient J 0 of unknown value (negative due to ferromagnetic behaviour of triangular graphone). The spin wave approach in the limit of near zero temperature is used for the description of the magnetisation. A specific case when the graphone is exposed to a magnetic pulse with a given carrier frequency is examined in greater depth. To obtain the magnetization response, integration over both the frequency space and momentum space is necessary. Due to inapplicability of the isotropic approximation for the given geometry of graphone, integration over momentum space is performed numerically. The calculations show that the resonance of the system occurs at frequencies which correspond to the upper limit of the spin wave energy band and the saddle points of the energy surface. Using these results, further experimental investigation based on THz or far-66 infrared spectroscopy can be performed, which can determine the as-yet-unknown exchange energy coefficient J 0 . The coefficient can in turn provide an estimate of a temperature range for which the spin wave approach utilised in our investigation is valid.


Introduction
Over the last decades, graphene has gained increasing attention due to its two-dimensional planar structure, band crossing at the Dirac points, two sub-lattice semi metallic conduction and hence its possible use in nanoelectronics [1][2][3][4][5]. Graphene however does not exhibit strong magnetic properties, but only weak antiferromagnetic order [6,7]. Several methods such as functionalization of graphene by chemisorptions of atomic hydrogen have been proposed to tune electronic properties and induce magnetic behaviour [8][9][10][11]. For this purpose however, only partial hydrogenation proves useful as fully hydrogenated graphene (graphane) does not have a strong magnetic response either [12,13].
In our investigation, we focus on triangular graphone structure (see Fig. 1, [14]). Triangular graphone is a semi-hydrogenated graphene sheet with all hydrogen atoms bonded only to carbons in one sublattice of graphene (key characteristic of the trigonal structure) with all hydrogen atoms on one side [12]. When we refer to graphone, we mean the triangular form, unless stated otherwise. Semi-hydrogenation breaks the delocalized π-bonding present in graphene. As a consequence the p-electrons in the un-hydrogenated carbon atoms are unpaired and localised, allowing for isotropic exchange interactions. According to the works of Zhou et al, computer simulations based on spin-polarized density functional theory suggest that these interactions result in ferromagnetic properties of graphone [12]. Moreover it also possesses an energy gap, making it an attractive material for use in nano-electronics [7,[12][13][14][15].
Possible uses of graphone largely depend on its thermal stability as uncontrollable changes to its structure by hydrogen hopping to neighbouring carbon atoms may significantly alter its conducting and magnetic properties [14]. Several authors including Podlivaev and Openov report high instability of triangular graphone (with characteristic time of disordering of the structure shorter than 1 ns at liquid-nitrogen temperatures) [13,16], and its evolution via hydrogen hopping into more stable rectangular graphone (see Fig.2, [14]) which exhibits anti-ferromagnetic behaviour [14]. The study of Hemmatiyan et al. however suggests a method of stabilising triangular graphone using hexagonal boron-nitride (h-BN) [7]. Substrate creates dipole moments for each nitrogen site that break the equivalency of two carbon atoms in two different graphene sub-lattices, hence suppressing the hydrogen migration.

System response function
Throughout our analysis, we will be using Planck units and hence ħ normalizes to 1. The triangular graphone layer is assumed to be effectively a 2D sheet oriented in the x-y plane. We consider the limit of 0 K temperature.
In the presence of localised spins S of the pelectrons of un-hydrogenated carbons at lattice sites i and j (i ≠ j), the tight-binding free hopping Hamiltonian of the structure † † 0 , which allows for electrons to hop (quantum tunnel) between sites i and j on the same graphene sub-lattice (different sub-lattice jump not possible due to C-H bonds) is effectively represented by the spin exchange Heisenberg Hamiltonian In the equations above, † i a ( i a ) creates (annihilates) an electron on site i, and t' is the next nearestneighbour hopping energy (value not well known, calculations show value 0.1 eV t   ) [1,17]. Because of the isotropy of the triangular graphone structure J ij is constant J 0 , which is negative due to ferromagnetic behaviour [4,12].
Using spin raising and lowering operators and Holstein-Primakoff transformation [18] (assuming s is the total spin on the site) ).
Here we have neglected terms of order β 4 and higher.
The operators β i † and β i are spin deviation creation and annihilation operators, respectively. They satisfy the commutation relations: † , , Response of the material (change of magnetisation ) to the field given by where  indicates that one should average over a grand canonical ensemble and () t
We consider the field H causing a time dependent Zeeman perturbation Hamiltonian  We consider a pulse with carrier frequency Ω lasting for duration of 2τ acting on the graphone system in the x-direction: In frequency space, this signal can be expressed as 0 2 sin(( ) ) .
) ( Hence the response of the system in x-direction to the signal (2.13) is given by (2.7) (2.16) and ε → 0 is a small parameter shifting the poles of response function away from the integration path which includes the x-axis.
Integration over frequency domain can be performed analytically using the residue theorem and Jordan's lemma. During the duration of the signal (t < τ), the response becomes

Dispersion relation
The un-hydrogenated carbon in triangular graphone has Z = 6 next nearest-neighbours which are in the vertices of the hexagon units (see Fig. 1). Direct evaluation of (2.11) gives The energy surface contour plot and its crosssections are shown in Fig. 3,a and 3,b, respectively; the hexagonal shape bounds the 1st Brillouin zone. The energy surface has three types of critical points.
The first is a global minimum at the origin, where 0 k   . The energy is proportional to k 2 around the centre, and the dispersion relation is almost isotropic here. Nevertheless, the integrals for magnetisation response or internal energy of spin waves gas diverge making the isotropic approximation inapplicable. The Brillouin zone perimeter includes 6 maxima where . We will show below that these characteristic points are responsible for the resonance behaviour of the spin waves in graphone.

Fig. 4. The 1st Brillouin zone (solid line) and rectangular integration area (dashed line)
The transversal magnetisation given by (2.16) is evaluated numerically. Integration with respect to wave-vector is provided within the 1st Brillouin zone (Fig. 4). The applied method uses a uniform Cartesian grid in the rectangle and checks whether current wavevector k is within the hexagon: where k  is the grid step in momentum space -equal for both x and y directions, ,, ( , , ) x i y j q kt fk is the integrand in (2.16), and t q is the discretised time. Calculated magnetisation is normalized to the maximum of its absolute value. The grid has 100 nodes along k x -axis. Test calculations with 50 and 200 nodes show that the integration is converged.
Fourier spectrum of magnetisation signal is evaluated from the previously calculated data. It is realized by approximation of continuous Fourier transform by numerical integration with respect to the time: where t  is the time discretisation step. The algorithms are executed in FORTRAN-90 language with OpenMP parallelization over independent variable t and ω, respectively.

Magnetisation response
The magnetisation response to a transversal pulse of a duration τ is evaluated for different Ω. The pulse excites oscillations of magnetisation, which decay after it ends. Characteristic decay time strongly depends on carrier frequency. There are three main signal types (based on carrier frequency) which can be considered.
The first occurs when Ω < 4ω 0 /3 (we refer this as a "low-frequency" pulse). The dynamics of the positive envelope of M x for Ω = ω 0 is shown in Fig. 5,a, and its Fourier spectrum is visualised in Fig. 5,b. The magnetisation signal is normalized to its maximum value. This interval does not contain any characteristic frequency of spin waves. Therefore the magnetisation decays fast with characteristic time near 30ω 0 -1 . The Fourier spectrum shows oscillations in the whole interval of possible spin wave frequencies with sharp peak at 3ω 0 /2. These oscillations are induced by the edges of the pulse.
The second type of dynamics occurs when Ω > 3ω 0 /2 (referred to as "high-frequency"). Here, the magnetisation oscillates weakly with frequency Ω during the pulse, and large-amplitude signal is caused by the falling edge of the pulse (Fig. 6,a). After the pulse, the magnetisation oscillates with frequency close to the dispersion relation maximum 0 3 / 2 k   , therefore two peaks occur in spectrum (Fig. 6,b). The characteristic decay time is the same as in the previous case.
The pulse with carrier frequency in the interval 4ω 0 /3 < Ω < 3ω 0 /2 ("near resonance" pulse) may produce a resonance excitation of spin waves when Ω is close to the frequency corresponding to the critical points of dispersion relation (see Fig. 3). In this case, the transversal magnetisation decays slowly during the time 1 0 150) (100    (Fig. 7, a). At this frequency, the Fourier spectrum contains only one dominant peak (Fig. 7, b).
Carrier frequency between the critical points (Ω = 7ω 0 /5) produces relatively weak response during the pulse and the beating with the period is close to 30ω 0 -1 after the pulse; the beat pattern amplitude also decays over a long period of time. The spectrum of such signal also contains one bright peak. Its fine structure cannot be resolved in Fig. 7, b. К. Б. Циберкин, М. Гажи

Conclusion
The results demonstrate a possible way for experimental measurement of the exchange energy J 0 be-tween un-hydrogenated carbons in graphone as it determines the characteristic frequencies in the magnetisation spectra (see (2.16) and (3.2)).
There are few known papers evaluating J 0 using density functional theory [20,21]. In these studies, the  [20] to -33 meV when the spin-spin interaction is fullyscreened by Coulomb repulsion of the conduction electrons, and to -98 meV if there is no screening [21]. The first estimate corresponds to frequency around 1.5·10 12 Hz; the two latter values give the frequency in interval from 4.8·10 13 Hz to 1.4·10 14 Hz. The characteristic decay time for "near resonance" pulse is of the order 10 -10 s in the former case, and 10 -12 s in the latter. This suggests that THz or far-infrared spectroscopy could be used to measure the exchange energy parameter. Our results show the resonance will be at frequencies which correspond to the upper limit of spin wave energy band (Ω = 3ω 0 /2) and the saddle points of energy surface (Ω = 4ω 0 /3).
The exchange value also determines the applicability of the spin waves approach for the description of magnetisation. If the coefficient is of the order 1 meV, the spin waves will be suppressed by thermal fluctuations at temperatures greater than 10 K. Larger values of the exchange energy extend the domain of spin wave existence to 380 K with interaction screening and to 1100 K without it. These temperatures are much larger than the thermal limit of graphone stability estimated in literature [13].
We are grateful for the financial support of the scholarship provided by the British Petroleum and the University of Oxford's Career Service Office which enabled us to get these scientific results. K. Tsiberkin is also thankful to Russian Foundation for Basic Research, Grant N. 17-42-590271.