Novel mechanism for weak magnetization with high Curie temperature observed in H-adsorption on graphene

TL;DR

This study demonstrates that hydrogen adsorption on graphene induces ferromagnetism with a high Curie temperature by creating a $p_z$-orbital imbalance, revealing a novel mechanism for weak magnetization in nonmagnetic materials.

Contribution

The paper introduces a new model showing how hydrogen adsorption causes ferromagnetism in graphene through $p_z$-orbital imbalance, a mechanism not previously described.

Findings

Hydrogen adsorption leads to a ferromagnetic state with 1 Bohr magneton per H atom.

Estimated Curie temperature exceeds 250 K.

Magnetism arises from $p_z$-orbital imbalance between graphene sublattices.

Abstract

To elucidate the physics underling magnetism observed in nominally nonmagnetic materials with only $sp$-electrons, we built an extreme model to simulate H-adsorption (in a straight-line form) on graphene. Our first principles calculations for the model produce a ferromagnetic ground state with a magnetic moment of one Bohr magneton per H atom and an estimated Curie temperature above 250~K. The removal of the $p_z$-orbitals from sublattice B of graphene introduces $p_z$-vacancies. The $p_z$-vacancy-induced states are not created from changes in interatomic interactions but are created because of a $p_z$-orbital imbalance between two sublattices (A and B) of a conjugated $p_z$-orbital network. Therefore, there are critical requirements for the creation of these states (denoted as $p_z^{\rm imbalance}$) to avoid further imbalances and minimize the effects on the conjugated $p_z$-orbital network. The requirements on the creation of $p_z^{\rm imbalance}$ are as follows: 1) $p_z^{\rm imbalance}$ consists of $p_z$-orbitals of only the atoms in sublattice A, 2) the spatial wavefunction of $p_z^{\rm imbalance}$ is antisymmetric, and 3) in principle, $p_z^{\rm imbalance}$ extends over the entire crystal without decaying, unless other $p_z$-vacancies are crossed. Both the origin of spin polarization and the magnetic ordering of the model arise from the aforementioned requirements.

Tables

Table 1. Table 1: Total energy (eV) per supercell for FM and AFM of two H separated by units ( N 𝑁 N ). Δ ​ E = E ​ ( AFM ) − E ​ ( FM ) Δ 𝐸 𝐸 AFM 𝐸 FM \Delta E=E({\rm AFM})-E({\rm FM})

$N$	3	6	12	18	24
$E$(FM)	-889.365	-889.427	-889.419	-889.422	-889.427
$E$(AFM)	-889.338	-889.404	-889.398	-889.400	-889.405
$\Delta E$	0.027	0.022	0.022	0.022	0.022

Full text

1. $p_{z}^{\rm imbalance}$ consists of $p_{z}$-orbitals of only the atoms in sublattice A,
2. the spatial wavefunction of $p_{z}^{\rm imbalance}$ is antisymmetric, and 3) in principle, $p_{z}^{\rm imbalance}$ extends over the entire crystal without decaying, unless other $p_{z}$-vacancies are crossed. Both the origin of spin polarization and the magnetic ordering of the model arise from the aforementioned requirements.

I Introduction

Magnetism in nominally nonmagnetic materials that contain only $sp$-electrons is currently a popular research topic because of the potential practical importance of these materialsEsq13 ; Han14 ; Khe14 ; Gon19 . Of all controversial experimental observations that have been reported for these materialsMak06 ; Sep10 ; Nai12 ; Sep12 , the high Curie temperature under such weak magnetization (three or four orders of magnitude smaller than that for conventional magnets) is quite confusingElf02 ; Esq03 ; Cer09 ; Uge10 ; Gie13 ; Wan09 ; Gon16 . There is almost no means of explaining how defect-induced magnetic moments (MMs) that are localized far away from each other (where the MMs’ distance are related to the defect concentration) could be coupled in ferromagnetic (FM) ordering above room temperature. Although calculation results seem to consistently show that the $sp$-electrons are spin-polarized by defectsEsq13 ; Yaz07 ; Bou08 ; San12 ; Yaz10 ; Vol10 , no convincing description of the physics underlying this phenomenon has been put forward as yet. Exploring the origin of this phenomenon has thus become an overwhelming challenge in materials scienceGon19 ; Fis15 .

Two serious difficulties arise in the theoretical study of these materials. First, the origin of $sp$-electron spin polarization remains unexplained because Heisenberg established 90 years ago that the principle quantum number of electrons that contribute to magnetism must be greater than or equal to threeHei26 ; Hei28 . Second and more critical is that no current theories have convincingly explained how long-range localized MMs (on defect centers) can be coupled in FM ordering at such a high Curie temperature. There must be an unrecognized magnetic mechanism at work.

In our previous papersXu19a ; Xu19b , we showed that the spatial wavefunctions of the electronic states that contribute to magnetism are antisymmetric, thereby dispelling the uncertainties regarding $sp$-electron spin polarization. To address the question of magnetic coupling, we proposed in our previous paper on magnetism in graphene with vacanciesXu19a that the $p_{z}$-orbital imbalance that is induced by vacancy would result in magnetic ordering in the material. However, it is computationally intensive to simulate graphene with a vacancy concentration that is sufficiently similar to that for which magnetism can be experimentally observedUge10 . Hence, we did not perform first principles calculations to compare the total energy difference between the FM and antiferromagnetic (AFM) ordering but only performed an analysis that led to the abovementioned conclusions.

In a two-dimensional (2D) system, a point defect leads to a resonance that decays with $r$ (the distance from the defect)Xu19a ; Per06 ; San10 ; Pal12 ; Sun17 , whereas according to perturbation theoryPol80 , a defect that is periodically arranged infinitely throughout the entire 2D system can create a resonance without a decay. In this paper, we build an extreme model with H-adsorption (in a straight-line form) on graphene to demonstrate robustly that the FM (with a high Curie temperature) originates from a $p_{z}$-orbital imbalance between two sublattices of graphene. Our aim is not to determine whether such an extreme model could be experimentally realized; rather, we wish to use the extreme model to better understand the novel magnetic mechanismXu19a that is at work in these materials.

II Calculation methods

Our results were obtained using first principles calculations that were implemented using the VASP packagevasp with the same calculation setup as in our previous papersXu19a ; Xu19b , that is, the wavefunctions were expanded in a plane-wave basis set with an energy cutoff of 500 eV. The interaction between the atoms and electrons was described by the projector augmented plane-wave methodPAW . The generalized gradient approximationGGA-PBE for exchange-correlation effects was used. The H-adsorption on graphene was simulated using a supercell in a 48$\times$1-sized cell of the original graphene. The k-points were sampled in the 2D Brillouin zone on 4$\times$48 meshes for the total energy calculations. Vacuum thickness was maintained larger than 20 Å. All of the atoms were relaxed until the Hellmann-Feynman forces on the atoms were smaller than 0.02 eV/Å. This calculation setup was found to be sufficiently accurate for the purposes of our study. For example, the use of this calculation setup resulted in a calculated lattice constant for graphene of 1.42 Å, which is in good agreement with the experimental valueNet09 .

III Results and discussion

III.1 one H adsorption on graphene

Bonded crystals with only $sp$-electrons do not exhibit magnetism, as concluded by Heisenberg theoryHei26 ; Hei28 . However, magnetism that is induced by vacancies or nonmetal adatoms (H, F, etc.) on graphene and graphite has been experimentally observed above room temperature and predicted by calculationsEsq13 ; Kuz13 ; Han14 ; Fis15 . However, a convincing theory that explains these phenomena is still requiredHan14 ; Yaz10 ; Fen17 ; Naf17 ; Kuz13 ; Fis15 ; Kat12 ; Sin13 . Previously, we have explainedXu19a ; Xu19b why Hund’s rule does not hold when interpreting calculation results based on the singlet-electron approximation and the Bloch theorem in band theory.

In a previous studyXu19a , we proposed that the $p_{z}^{*}$ -nonbonding states (which is called the zero-mode in the literaturePer06 ; Per08 ) play a critical role in both $sp$-electron spin polarization and magnetic coupling in graphene with vacancies. As mentioned in the Introduction, in the present study, an extreme model was built to simulate H-adsorption (in a straight-line form) on graphene, as shown in Fig. 1 (a). Instead, of introducing a C-vacancy by removing a C atom in graphene, we introduced a $p_{z}$-vacancy in graphene via the H-saturation of a $p_{z}$-orbital of C below H (denoted as CH). This model enabled us to avoid cutting off graphene (if the vacancies form in a straight line) while excluding antibonding states (that are induced by the interactions among the three dangling bonds that were left by the vacancy). The model enabled us to focus only on the $p_{z}$-vacancy-induced states. Considering our computational resources, we adopted a supercell of a 48$\times$1-sized graphene to observe the long-range nature of the resonance states and examine the coupling of MMs in the model.

The C($2s^{2}2p^{2}$) atom in graphene prefers to hybridize in one $p_{z}$ and three $sp^{2}$ orbitals (where the $z$-axis is perpendicular to the graphene plane)Net09 . The $sp^{2}$ orbitals form $\sigma$-bonds as a backbone of the honeycomb lattice that is composed of two sublattices that are denoted by A and B. Each of the atoms in sublattice A has three neighboring atoms in sublattice B and vice versa. The $p_{z}$ and three $sp^{2}$ orbitals in graphene are orthogonal and thus do not interact with each other. Therefore, the $p_{z}$-orbitals ($p_{z}$-electrons) over the two sublattices under C3V symmetry form a conjugated $p_{z}$-orbital ($p_{z}$-electrons) network in a coherent manner, which is the critical factor for producing magnetism in graphene.

Fig. 1 (a) shows the atomic configuration of the supercell for the adsorption of one H atom onto the 48$\times$1-sized graphene with unit vectors, ${\bf a}_{1}=48(\sqrt{3/2},-1/2)a$ and ${\bf a}_{1}=48(\sqrt{3/2},1/2)a$. CH has three carbon atoms as its nearest neighbors. However, only two of the three carbon atoms could lie in the supercell because of the period along ${\bf a}_{2}$. The optimized atomic structure shows that the adsorption of H caused CH to lift 0.68 Å above the graphene plane, whereas the two nearest neighbors of the H atom were at 0.34 and 0.26 Å above the graphene plane. The distances between CH and its two nearest neighbors were 1.48 and 1.52 Å. For comparison, the corresponding value in a perfect graphene is 1.42 ÅNet09 . Clearly, CH was transformed from $sp^{2}$ hybridization in graphene to $sp^{3}$ hybridization after H-adsorption, and its $p_{z}$-orbital (dangling bond) was saturated as expected, introducing to a $p_{z}$-vacancy, that is, there was a $p_{z}$-orbital imbalance between the two sublattices. Except for the aforementioned changes, there were no other changes in the C-C bond lengths that were larger or smaller than 0.04 Å relative to those in a perfect graphene.

To determine the origin of the magnetism, the band structures for the ground state of the 48$\times$1-sized graphene with and without H-adsorption are shown in Figs. 2 (a) and (b), respectively. Comparing the two band structures, there are two features after H-adsorption on the 48$\times$1-sized graphene. First, the degeneration at the Dirac-point of graphene, as shown in Fig. 2 (b), was lifted after H-adsorption, creating a gap of approximately 0.3 eV. Second, the red (solid and dashed) bands in Fig. 2 (a) exhibit spin-splitting from 0.1 to 0.9 eV along 2D-BZ. The graphene with a $p_{z}$-vacancy was thus semiconducting with a gap of 0.1 eV. The majority (solid) red band was fully occupied, and the minority (dashed) red band was empty, contributing 1 $\mu_{\rm B}$ per H to the MM. Fig. 2 (b) clearly shows the folded structure of the original graphene band that resulted from a 48-folded 2D-BZ.

In previous studiesYaz07 ; Bou08 ; San12 ; Pal12 ; Gon16 , the H atom was considered to be a point-defect adsorption on graphene, and the induced 1 $\mu_{\rm B}$ per H was ascribed to Hund’s rule because the induced electron states appeared to be quite localized. However, as discussed in our previous papersXu19a ; Xu19b , caution is required when using first principles calculations that are based on band theory because all of the electrons in band theory extend over the entire crystal as per the Bloch theorem, even if the bands appear to be flatBloch .

To examine the nature of the majority red band, we calculated the charge distribution of the majority red band at the K-point, which is shown in Figs. 2 (c) and (d). At first glance, the electrons were distributed on each of the atoms in sublattice A with almost the same isosurfaces, indicating that the red state extended over the entire supercell without decaying, as expected, which motivated us to study straight-line defects in graphene instead of point defects. In fact, the electron distribution on only sublattice A indicated an important feature: the spatial wavefunction of the state should be antisymmetric. That is, the phases of the wavefunction components on the two neighboring atoms belonging to sublattice A should be antisymmetric. Otherwise, electrons would accumulate on the atom (sublattice B) between the two neighboring atoms (sublattice A), resulting in an extra imbalance. Therefore, it is impossible. Note that the length of the 48$\times$1-sized supercell was already at 120 Å. In principle, a consequence of the antisymmetric wavefunction is that the state should extend infinitely. Otherwise, terminating the wavefunction at one point will create an additional imbalance without any wavefunction at the opposite side of the broken point.

Although the red state in Fig. 2 (a) was an H-induced state, its wavefunction was not localized but extended over the entire supercell and consisted of $p_{z}$-orbitals of only the atoms in sublattice A. Note that the atoms in sublattice A belonged to the second nearest neighborsNet09 . Therefore, the state was not caused by a change in the localized atomic interaction due to H-adsorption, but was created as a response of the conjugated $p_{z}$-electron network to a $p_{z}$-electron imbalance between the two sublattices.

In our previous paperXu19a , we used ”nonbonding state ($p_{z}^{*}$)” to describe the involved state following molecular orbital theory from quantum chemistry. However, imbalance is more essential than nonbonding in the inherent nature of this state. In the absence of a suitable name, we temporarily refer to this state as an imbalanced state (denoted as $p_{z}^{\rm imbalance}$) for the purposes of discussion in the following section, to distinguish this state from the lone pair dangling bond state, which is also referred to as a nonbonding state in the literature.

As the induced state was caused by a $p_{z}$-electron imbalance, the response of the conjugated $p_{z}$-electron network to the imbalance should not create any further imbalance and should minimize the effect on the conjugated $p_{z}$-electron network. When one $p_{z}$-electron in sublattice A loses its pairing in sublattice B, the imbalance could be rectified by filling the induced state with one $p_{z}$-electron in sublattice A. However, if one $p_{z}$-electron was provided by one C atom in sublattice A (denoted by CA) near the $p_{z}$-vacancy, this response would be both electrostatically unstable and break its pairing with the atom in sublattice B near CA. Thus, unpairing would continue indefinitely, alternating on the two sublattices. Therefore, to minimize the effect of the response on the conjugated $p_{z}$-electron network, each atom in sublattice A approximately provided $1/N_{\rm A}$ electrons to form the $p_{z}^{\rm imbalance}$-imbalanced state, where $N_{\rm A}$ is the number of atoms in sublattice A in the supercell.

Therefore, rectifying the imbalance involved the following requisites: 1) $p_{z}^{\rm imbalance}$ consisted of $p_{z}$-orbitals of only the atoms in sublattice A, 2) the $p_{z}^{\rm imbalance}$ wavefunction was antisymmetric, and 3) $p_{z}^{\rm imbalance}$ extended infinitely over the entire crystal without decaying. The electron antisymmetric exchange principleQM states that the spin wavefunction of $p_{z}^{\rm imbalance}$ should be symmetric. This is the origin of the MMs caused by the $sp$-electrons in this material. That is, the MMs originated from the response of the conjugated $p_{z}$-electron network to the $p_{z}$-electron imbalance between two sublattices.

Counting electrons within the supercell, only one electron filled $p_{z}^{\rm imbalance}$ and was distributed over all of the 48 atoms of sublattice A of the entire supercell, which corresponded to a contribution of 1 $\mu_{\rm B}$. From the isosurfaces that are shown in Figs. 2 (b) and (c), we concluded that the charge distribution on these atoms was approximately the same. That is, there was a contribution of 1/48 $\mu_{\rm B}$ from each atom of sublattice A in the supercell. Note that these MMs were not atomic MMs isolated on atoms but were inherently parallel on the atoms because the MMs were the collective contribution from one electronic state. This result explains why such weak MMs with only 1 $\mu_{\rm B}$ and in parallel alignment could be distributed so widely over the entire supercell.

Here, we should explain the implications of an antisymmetric wavefunction for the supercell, as is shown in Fig. 1 (a). There was a phase difference of $e^{-i2\pi/3}$ between neighboring atoms in sublattice A. The angle between ${\bf a}_{2}$ and ${\bf a}_{1}$ was $2\pi/3$. Each arrow in Fig. 2 (d) represents a phase factor of $e^{-i2\pi/3}$ between the two respective atoms. Thus, an antisymmetric wavefunction means that the sum of the $p_{z}$-orbital components of the three atoms surrounding any atom in sublattice B was zero (within an accuracy of $10^{-6}$), thereby guaranteeing that there were no orbital components for $p_{z}^{\rm imbalance}$ on sublattice B.

III.2 Two H adsorption on graphene

A single $p_{z}$-vacancy in sublattice B created $p_{z}^{\rm imbalance}$, which was the response of the conjugated $p_{z}$-electron network to a $p_{z}$-electron imbalance between two sublattices. How did the conjugated $p_{z}$-electron network respond to two $p_{z}$-vacancies simultaneously on sublattice B? More $p_{z}$-electron in sublattice A were required to rectify the imbalance caused by increasing the $p_{z}$-vacancies in sublattice B because $p_{z}^{\rm imbalance}$ was not created by a change in localized atomic interactions but from a $p_{z}$-electron imbalance. The requirements (no additional imbalance and minimized the effect on the conjugated $p_{z}$-electron network) on $p_{z}^{\rm imbalance}$ resulted in only a recombination of the $p_{z}$-orbitals of sublattice A for this case.

The conclusion was obtained by considering two $p_{z}$-vacancies separated by different units ($N$) on sublattice B. The atomic configuration of two H atoms on one sublattice ($N$ = 6) is shown in Fig. 1 (b). The configurations for $N$ = 3, 6, 12, 18, and 24 were similar. Table I lists the total energies of FM and AFM for different $N$. From the table, FM was energetically more favorable than AFM by at least 220 meV. Because the current models calculating the transition temperature are based on the magnetic coupling (interaction parameters) between the nearest neighbors, we cannot use them for our cases, in which one MM contributed of one electronic state widely distributes as a whole moment on all atoms of sublattice A (see below). However, we would point out that the exchange energy of 220meV corresponds (1 eV = 1.16048*104 K) to a temperature higher than 2500K, in agreement with experimental observations of a high Curie temperature in proton-irradiated graphene. For all of the five cases, the MM for FM was 2 $\mu_{\rm B}$ per supercell, or 1 $\mu_{\rm B}$ per H.

The band structures of two H atoms separated by $N$ = 3, 6, 12, 18, and 24 units adsorbed on one sublattice of the 48$\times$1-sized graphene are plotted in Figs. 3 (a)-(e), respectively. The band structure of one H-adsorbed on the same model is also plotted in Fig. 3 (f) for comparison. First, it can be clearly seen that the MMs of 2 $\mu_{\rm B}$ per supercell for these systems came from the two red bands in Figs. 3 (a)-(e). Second, both the dispersion and energy level of these red bands in Figs. 3 (a)-(e) were similar to that of Fig. 3 (f), except for the small splitting of the two red bands for $N$ = 3 and $N$ = 6 near the K-point. The splitting was attributed to the charge distributions for the two red bands and is discussed below.

The charge distributions for the two solid red bands in Figs. 3 (a)-(e) at the K-point are shown in Figs. 4 (a)-(e), respectively. The top and bottom subfigures in each panel of Figs. 4 (a)-(e) correspond to two imbalanced states: the top subfigure had a lower energy than the bottom subfigure. From Figs. 4 (a)-(d), it was found that the 48$\times$1-sized supercell could be divided into two segments with the H atom ($p_{z}$-vacancy) as the boundary: short and long segments with lengths of $N$ (3, 6, 12, and 18) and $48-N$ units, respectively. The electrons that occupied each of the red bands were distributed on only one of the two segments, leaving the other segment empty, and the electrons were evenly distributed on each atom of sublattice A within the segment. These results showed that the requirement for the antisymmetry of the wavefunction was fulfilled.

The splitting of the two red bands that appeared near the K-point for the cases $N$ = 3 and $N$ = 6 was caused by the difference in the exchange energy between the short and long segments, because the exchange energy depended on the charge density of the involved atomsmagnetism . One electron was evenly distributed on each atom of sublattice A within the segment, $p_{z}^{\rm imbalance}$ distributed in the short segment had thus a lower energy than that in the long segment. The band-splitting for the cases $N$ = 12, 18, and 24 was smaller than 0.02 eV.

It is shown in Figs. 4 (a)-(d) that the $p_{z}$-vacancy acted as a boundary in dividing the supercell into two segments, that is, the charge distribution for $p_{z}^{\rm imbalance}$ did not have the $p_{z}$-vacancy as its center, although $p_{z}^{\rm imbalance}$ was induced by the $p_{z}$-vacancy. This also implies that $p_{z}^{\rm imbalance}$ resulted from an imbalance, not an interaction. Most importantly, this characteristic led to the recombination of the $p_{z}$-orbitals in sublattice A, thereby forming individual segments for each of the $p_{z}^{\rm imbalance}$ states to rectify the imbalance.

However, for $N$ = 24, the electron distribution in the supercell could not be divided into two segments. Instead, the electrons of both red states were distributed over the entire supercell, as shown in the top and bottom subfigures of Fig. 4 (e). The two red bands were almost degenerate, as in the case of the single H atom that was adsorbed on the 24$\times$1-sized graphene. We performed calculations for the case $N$ = 24 with two different initial atomic configurations:

1. two H atoms were placed on the original graphene, and 2) one H atom was placed on the relaxed graphene with one adsorbed H atom. The two initial configurations produced the same ground state: the total energy and MMs were the same at the level of accuracy of the calculation, and the DOS and charge distribution were similar. Most importantly, the electrons of the two red bands beginning with both initial configurations were distributed over the entire supercell, unlike for the cases of $N$ = 3, 6, 12, and 18, for which the electrons were distributed over one segment of the supercell, leaving the other segment empty.

The analysis above led us to conclude that the response of the conjugated $p_{z}$-electron network to two $p_{z}$-vacancies on one sublattice was similar to that for one $p_{z}$-vacancy because the origin of $p_{z}^{\rm imbalance}$ was the same for one or two $p_{z}$-vacancies. Regardless of the number of $p_{z}$-vacancies that appear on one sublattice, the aforementioned requirements should always be fulfilled. Therefore, constructing the wavefunction for each of the two $p_{z}^{\rm imbalance}$ states corresponded simply to a recombination of the $p_{z}$-orbitals in sublattice A.

This conclusion is borne out by Figs. 4 (a)-(d): two red states ($p_{z}^{\rm imbalance}$), respectively, are distributed on two individual segments without overlapping with each other, as shown in the top and bottom subfigures in each of Figs. 4 (a)-(d). The two individual segments shared a common $p_{z}$-vacancy as a boundary.

The magnetic ordering of the MMs in the two segments is illustrated in Fig. 5. In the figure, the two segments are differentiated by color and black-white lobes. The up-down colors (black-gray) of the alternative lobes indicate antisymmetric wavefunctions. In principle, the wavefunction phases of the two $p_{z}^{\rm imbalance}$ states should be independent, that is, the up-down colors of the lobes in the left segment could be inverted relative to the black-gray up-down order in the right segment, as shown in Figs. 5 (a) and (b).

Let up-pink and down-green (up-black and down-gray) denote the positive phase: then, if the orbital phases on two atoms on the two sides of the $p_{z}$-vacancy (denoted by Cleft and Cright respectively) were antisymmetric, as shown in Fig. 5 (a), no electrons accumulated on the $p_{z}$-vacancy site. Otherwise, electrons could have appeared on the $p_{z}$-vacancy-site, if the orbital phases on Cleft and Cright are symmetric, as shown in Fig. 5 (b). Therefore, the condition that no additional imbalance was allowed for $p_{z}^{\rm imbalance}$ required the orbital phases on Cleft and Cright to be antisymmetric, thereby favoring parallel alignment between the two MMs (blue arrows) on Cleft and Cright. The white arrows on the other atoms in each of two segments followed the individual blue arrows, leading to all the MMs in the two segments being aligned parallel to each other because the MMs (blue and white arrows) within the same segment belonged to one electronic state, which should be parallel. By contrast, Fig. 5 (b) shows that the two boundary atoms, Cleft and Cright, had symmetric orbital phases; thus, the MMs on the two boundary atoms (blue arrows) should have been antiparallel. For the case $N$ = 24, the MM alignment could be easily understood: the charge distributions of the two $p_{z}^{\rm imbalance}$ states overlapped; thus, the MMs of one $p_{z}^{\rm imbalance}$ state could act as an MM field to induce the MMs of the other $p_{z}^{\rm imbalance}$ state to align in parallel.

Considering that the $p_{z}^{\rm imbalance}$’s electron was evenly distributed on the atoms of sublattice A and there were $N_{\rm AL}$ and $N_{\rm AS}$ atoms of sublattice A within the long and short segments respectively, the two boundary atoms, Cleft and Cright, had thus $1/N_{\rm AL}$ and $1/N_{\rm AS}$ electrons. According to the above analysis about the orbital phases on the two boundary atoms, the coupling of the MMs in the two segments depended on the $1/N_{\rm AL}$ and $1/N_{\rm AS}$ electrons. The lower the $p_{z}$-vacancy concentration, the smaller the $1/N_{\rm AL}$ ($1/N_{\rm AS}$) electrons, and the lower the exchange energy. Compared with the case of a point defect, such as graphene with vacancies, the model presented here represents an extreme case in which a state without decay can be realized. In the point-defect case, such as a vacancy in graphene, the charge density of a defect state can fast decay to a value that is smaller than a thermal fluctuation on atoms of sublattice B. This difference between point-defects and straight-line-defects may explain the occurrence of controversial experimental observationsMak06 ; Sep10 ; Nai12 ; Sep12 .

IV Conclusions

In summary, we studied magnetism that has been observed in proton-irradiated graphene by constructing an extreme model of a supercell for an H-adsorbed (modeled as a straight line) onto graphene. We present a novel mechanism to explain the magnetic phenomena. This mechanism was first suggested in our previous papersXu19a , and it is substantially different from conventional models such as the Heisenberg model, the indirect exchange model, the superexchange model, the RKKY model, and the itinerant electron modelmagnetism .

We showed that the conjugated $p_{z}$-electron network in graphene plays a critical role in $sp$-electron spin polarization and magnetic ordering. The H-adsorption on graphene saturates a $p_{z}$-orbital of C under the H atom, creating a $p_{z}$-vacancy on the conjugated $p_{z}$-electron network, thereby inducing a $p_{z}$-electron imbalance between two sublattices. As the $p_{z}^{\rm imbalance}$ state is caused by an imbalance, no further imbalance and minimized effect on the conjugated $p_{z}$-electron network should be required for the creation of $p_{z}^{\rm imbalance}$. That is,

1. the state consists of $p_{z}$-orbitals only on the atoms in sublattice A,
2. the spatial wavefunction of the state is antisymmetric, and
3. the state extends over the entire crystal without decay unless other $p_{z}$-vacancies are crossed. The $sp$-electron spin polarization in graphene with a $p_{z}$-vacancy originates from the spatial antisymmetric wavefunction of $p_{z}^{\rm imbalance}$ as per the electron exchange antisymmetric principle.

An increase in $p_{z}$-vacancies results in the creation of more $p_{z}^{\rm imbalance}$ states because $p_{z}^{\rm imbalance}$ states are created from a $p_{z}$-orbital imbalance and not from interatomic interactions. Although the response of the conjugated $p_{z}$-electron network to the imbalance is to create more $p_{z}^{\rm imbalance}$ states, the requirement of the formation of the induced states must still be fulfilled, resulting in only the recombination of $p_{z}$-orbitals in sublattice A. If two $p_{z}$-vacancies exist, the electrons that fill up each of two $p_{z}^{\rm imbalance}$ states are not distributed over the entire crystal but in two individual segments where the $p_{z}$-vacancies serve as boundaries. The wavefunction phase of each state should be independent, although all of the $p_{z}^{\rm imbalance}$ states consist of $p_{z}$-orbitals of sublattice A in each segment that are antisymmetric. However, the requirements that no additional imbalance should be introduced and that the effect on the network should be minimized result in the orbital phases on two boundary atoms, Cleft and Cright, to be antisymmetric, leading to the presence of MMs on the two boundary atoms in FM alignment with a high Curie temperature. Therefore, both the spin polarization and magnetic ordering that is produced by our extreme model originate from the requirements for the imbalanced states.

Because $p_{z}^{\rm imbalance}$ is filled by only one electron but has a long-range extent and is distributed on a $p_{z}^{\rm imbalance}$-dominant segment, each MM on the atoms is very small, and the sum of the MMs is only one Bohr magneton. The requirement for the imbalanced states that are induced by the $p_{z}$-vacancies on one sublattice leads to the MMs of all of the $p_{z}^{\rm imbalance}$-dominant segments being in coherent FM alignment. The total MM depends on the $p_{z}$-vacancy concentration. These results explain the experimental observations of very weak magnetization with a high Curie temperature.

This work was supported by NFSC (No.61274097) and NBRPC (No. 2015CB921401).