Dilute magnetic semiconductor and half metal behaviors in 3d transition-metal doped black and blue phosphorenes: a first-principles study
Weiyang Yu, Zhili Zhu, Chun-Yao Niu, Chong Li, Jun-Hyung Cho, Yu Jia
aa r X i v : . [ c ond - m a t . m t r l - s c i ] S e p Dilute magnetic semiconductor and half-metal behaviors in 3 d transition-metal dopedblack and blue phosphorenes: a first-principles study Weiyang Yu,
1, 2
Zhili Zhu, Chun-Yao Niu, Chong Li, Jun-Hyung Cho,
3, 1, ∗ and Yu Jia † International Laboratory for Quantum Functional Materials of Henan,and School of Physics and Engineering, Zhengzhou University, Zhengzhou, 450001, China School of Physics and Chemistry, Henan Polytechnic University, Jiaozuo, 454000, China Department of Physics and Research Institute for Natural Sciences,Hanyang University, 17 Haengdang-Dong, Seongdong-Ku, Seoul 133-791, Korea (Dated: September 29, 2015)
Abstract
We present first-principles density-functional calculations for the structural, electronic, and mag-netic properties of substitutional 3 d transition metal (TM) impurities in two-dimensional black andblue phosphorenes. We find that the magnetic properties of such substitutional impurities can beunderstood in terms of a simple model based on the Hund’s rule. The TM-doped black phosphoreneswith Ti, V, Cr, Mn, Fe and Ni impurities show dilute magnetic semiconductor (DMS) propertieswhile those with Sc and Co impurities show nonmagnetic properties. On the other hand, the TM-doped blue phosphorenes with V, Cr, Mn and Fe impurities show DMS properties, those with Tiand Ni impurities show half-metal properties, whereas Sc and Co doped systems show nonmagneticproperties. We identify two different regimes depending on the occupation of the hybridized elec-tronic states of TM and phosphorous atoms: (i) bonding states are completely empty or filled for Sc-and Co-doped black and blue phosphorenes, leading to non-magnetic; (ii) non-bonding d states arepartially occupied for Ti-, V-, Cr-, Mn-, Fe- and Ni-doped black and blue phosphorenes, giving riseto large and localized spin moments. These results provide a new route for the potential applicationsof dilute magnetic semiconductor and half-metal in spintronic devices by employing black and bluephosphorenes. Keywords:
Dilute magnetic semiconductor, half-metal, transition metal doping, phosphorene
PACS numbers: 73.22.-f, 75.50.Pp, 75.75.+a
1. Introduction
Two-dimensional (2D) materials, graphene and sil-icene, are currently the subject of intense theoreti-cal and experimental research especially for their novelelectronic device applications.
Graphene and silicenehave demonstrated many exquisite phenomena originat-ing from the characteristic conical dispersion and chiralbehavior of their valence and conduction bands aroundthe Fermi level.
Generally speaking, the nanostruc-tures of graphene and silicene such as nanoribbons,nanotubes and their interconnections have opened newroutes for experimental and theoretical studies in thefield of nanoelectronics. Very recently, black phospho-rene, a single layer of black phosphorus (BP) was success-fully fabricated through exfoliation from the bulk blackphosphorus, and therefore becomes, besides grapheneand silicene, another stable elemental 2D material. Theblack phosphorene presents some advantages superior toother previously studied 2D semiconductors because ofits intriguing electronic properties, thereby drawing enor-mous interest from the society of materials science. Recently, Li et al reported that black phosphorenecould be applied to the channel of the field effect tran-sistor (FET) device that has a high carrier mobility of ∼ cm /V . s and an on/off ratio of ∼ at room tem-perature. As the allotrope of black phosphorene, bluephosphorene has the same stability as black phosphoreneat room temperature, and its band gap is larger thanblack phosphorene . These good electronic properties of black and blue phosphorene nanosheets can be use-ful for the development of future nanoelectronic devices,spintronics, and related applications. For the design of practical electronic devices, defectsand impurities have been employed to tune the electrical,optical, and other properties. Over the last decades theresulting of dilute magnetic semiconductors (DMS) andhalf-metals have achieved important developments, both in fundamental aspects and prospective technolog-ical applications. Indeed, it was possible to understandthe underlying mechanisms of interaction between dilutemagnetic impurities allowing ferromagnetic semiconduc-tors at room temperature.
For prospective applica-tions, the integration between 2D semiconductors andmagnetic data storage enables the development of two-dimensional spintronics devices such as spin valve, spin-based transistors, non-volatile magnetoresistive mem-ories and even magnetically enhanced optoelectronicsdevices. Meanwhile, in spite of the success of 2D materials suchas graphene, silicene, transition metal dichalcogenides(TMDCs) and black phosphorene , there hasbeen rare study on the dilute magnetic characters ofdoped 2D black phosphorene except the work of Hashmi et al and Sui et al , while half-metal properties indoped blue phosphorene have remained unexplored sofar. From a technical point of view, 2D semiconductorshave other superior factors that can be exploited in mag-netic or spintronic devices. First, the carrier concentra-tion can be externally controlled by voltage gating. Sec-ondly, there is room to improve the control of the impu-rity concentration, for example, by employing adatoms asimpurities with concentrations above the solubility limit.In practice, studies of magnetic semiconductor nanos-tructures with lower dimensionalities, including semicon-ductor nanocrystals and nanowires doped withtransition metals (TM), demonstrated that the confine-ment effect and the improved control of magnetic dopantscan be used to increase the Curie temperature. In this work, we focus on substitutional 3 d TM im-purities (from Sc to Ni) in black and blue phosphorenesto investigate their dilute magnetic characters and half-metal properties. Using first-principles density functionaltheory (DFT) calculations, we study the structural, elec-tronic, and magnetic properties of substitutional 3 d TMimpurities in black and blue phosphorenes. One of ourkey results is that the electronic and magnetic propertiesof these substitutional impurities can be understood bya simple model based on the hybridization between theTM d orbitals and the defect (i.e., phosphorous vacancy)levels. This model together with the calculated bandstructure provides an explanation for the non-trivial be-haviors of the binding energy and the spin moments forall the systems considered. Concisely, we distinguish twodifferent regimes that depend on the electron filling ofTM-phosphorous hybridized levels: (i) completely unoc-cupied (occupied) bonding states for Sc (Co) lead to non-magnetic; (ii) partially occupied non-bonding d shell forTi, V, Cr, Mn, Fe and Ni give rise to large and localizedspin moments.This paper is organized as follows. After a brief de-scription of the computational details in section 2, wepresent the geometry structures, binding energies andmagnetic properties of all the substitutional TM impuri-ties studied in section 3. We also present the general ideasbehind our model of the metal-phosphorus hybridizationin the considered systems. In section 4, the electronicstructure of the unreconstructed D h phosphorus vacancyin pristine phosphorene, along with the electronic struc-ture of the different groups of impurities are presented.Finally, we give a summary with some general conclu-sions.
2. Computational Details
The present DFT calculations were performed usingthe Vienna ab initio
Simulation Package (VASP) codewith a plane-wave basis set.
Projector augmentedwave (PAW) potentials were used to describe the coreelectrons and the generalized gradient approximation(GGA) of Perdew, Burke and Ernzernhof (PBE) wasadopted for exchange-correlation energy. To examine thereliability of the PBE method on the magnetic proper-ties of the black and blue phosphorenes containing 3 d TM impurity atoms, we also considered the effect of theon-site Coulomb interaction U on the magnetic prop-erty within the PBE + U method. A kinetic energycutoff of the plane-wave basis set was used to be 500eV and for the structural optimization, convergence of
FIG. 1: (a) Top and side views of a diamond-like 2 × . Hellmann-Feynman residual forces less than 0.01 eV/˚Aper atom was achieved. Because the convergence withrespect to the number of k -points was especially criti-cal to obtain accurate results for the spin moment inthe systems studied, we used an adequate number of k -points for all the different supercell sizes, equivalent to9 × × sampling. The Fermi level wassmeared by the Gaussian method with a width of 0.05 FIG. 2: (a), (b) Structural parameters and binding energies (c) of the substitutional TM-doped black phosphorenes. Thecorresponding ones of the substitutional TM-doped blue phosphorenes are given in (d), (e), and (f). eV. Most of our results were obtained using 2 × × × × ∼
15 ˚A.
3. Structural, energetic, and magnetic proper-ties of TM doped in black and blue phosphorenes
In this section we provide our results for the geome-tries, binding energies, and spin moments of substitu-tional TMs in black and blue phosphorenes.
The typical structure of the systems studied in thispaper is presented in Fig. 1. Fig. 1(a) shows thediamond-like 2 × a =3.310˚A, a =4.589˚A,and a ′ = a ′ =( a + a ) / =5.658˚A, and the angle between the basis vectors a ′ and a ′ is 71.89 ◦ . These valuesare consistent with experiment and other theoreti-cal calculations . Fig. 1(b) displays the optimizedstructure of the blue phosphorene with lattice constants a ′′ = a ′′ =3.330˚A, and their angle θ ′ =60 ◦ , which are ingood agreement with previous DFT calculations .We begin to study a pure black phosphorene with amonovacancy. Fig. 1(c) shows the spin density of blackphosphorene with a monovacancy. Similar to the resultsreported by Ma et al , the phosphorus atoms around thevacancy undergo a Jahn-Teller distortion, and two of thephosphorus atoms close to the vacancy site move towardseach other to form a P-P distance of 1.832˚A, which is0.408˚A smaller than that of the intrinsic phosphorene.The ground state of the system has a magnetic momentof 1.00 µ B /unit cell, most of which is concentrated at thetwo P atoms with the unsaturated bonds, as seen in Fig.1(c).The structural parameters and energetic properties ofthe substitutional TMs in black and blue phosphorenesare shown in Fig. 2. For TM-doped black phosphorene,the bond angles θ and θ monotonically increase fromSc to Ni [see Fig. 2(b)]. Meanwhile, the bond lengths d and d decrease for Sc-, Ti-, and V-doped systems andincrease for Cr-doped system, and then decrease again forMn-, Fe-, and Co-doped systems, and then increase for FIG. 3: (a) Spin moments of the isolated TMs and their substitutions in black and blue phosphorenes as a function of thenumber of valence electrons (Slater-Pauling-type plot). Schematic diagram of spin moment in doped black phosphorene (b)and blue phosphorene (c) in terms of Hund’s rule, respectively.TABLE I: Electronic charges of each atomic species in the TM-doped black and blue phosphorenes, obtained using Badercharge analysis. The positive (negative) sign represents the gained (lost) electrons.Atoms Sc Ti V Cr Mn Fe Co NiTM -1.54 -1.23 -0.97 -0.87 -0.61 -0.37 -0.19 -0.16black-P Nearest-P +0.97 +0.78 +0.59 +0.59 +0.33 +0.12 0.00 +0.07TM -1.55 -1.27 -1.05 -0.88 -0.57 -0.40 -0.21 -0.20blue-P Nearest-P +0.32 +0.27 +0.26 +0.20 +0.12 +0.07 0.00 +0.02
Ni-doped system [see Fig. 2(a)]. These behaviors of thebond lengths and bond angles reflect the size of the TMatoms. As for TM-doped blue phosphorene, the bondlength d decreases from Sc to Mn and then increasesfrom Mn to Ni [see Fig. 2(d)], while the band angle θ shows an oscillating behavior [see Fig. 2(e)].Figure 2(c) shows the calculated binding energies ( E b )of the TM-doped black phosphorenes, where E b is de-fined as -( E total - E phosphorene - E atom ). Here, E total is theenergy of the whole configuration, E phosphorene is the en-ergy of the phosphorene with a vacancy and E atom rep-resents the energy of an isolated dopant atom. We finda continuous increase of the binding energy from Sc toCr, and then decrease from Mn to Ni, and the bind-ing energies for the considered TMs are in the range of0.375-5.466 eV. Interestingly, Cr-doped system has themaximum binding energy. This peculiar behavior is re-lated with the interplay between the energy down-shiftand the compression of the 3 d shell of the TM as theatomic number increases. Although this explanation willbe more clear when the metal-phosphorus hybridizationlevels are discussed below, we note that the behavior ofthe binding energies of the substitutional 3 d TM arisesfrom two competing effects:(i) From Sc to Cr, the decrease of d and d reflectsan increase in the bonding strength between the TM andphosphorous atoms, and (ii) From Mn to Ni, as the 3 d shell is occupied, its hybridization with the phosphorousvacancy states is weakened to decrease the binding en-ergy.It is noticeable that this trend of the energetics for theTM-doped black phosphorenes is very similar to that forthe TM-doped blue phosphorenes [see Fig. 2(f)]. The spin moments of substitutional TMs in black andblue phosphorenes are displayed in Fig. 3(a), togetherwith those of the isolated TM atoms. We find that thespin moments of the isolated TM atoms are 1, 3, 5, 6, 5,4, 3, and 1 µ B from Sc to Ni, respectively. On the otherhand, the TM-substituted black phosphorenes have thezero magnetic moment for Sc and Co, but 1, 2, 3, 2, 1,and 1 µ B for Ti, V, Cr, Mn, Fe, and Ni, respectively,which are the same as the corresponding cases in bluephosphorene. For both of the TM-substituted black andblue phosphorenes, we analyze the charge transfer usingBader charge (see Table I). We find that for both cases,the TM atoms lose electron charges while the nearestphosphorous atoms gain electron charges. It is notablethat the magnitudes of gained and lost charges decreaseas the atomic number increases.Interestingly, as shown in Fig. 3(a), the total spinmoments have the integer values of 0, 1, 2, 3, 2, 1, 0, and1 µ B for Sc-, Ti-, V-, Cr-, Mn-, Fe-, Co-, and Ni-dopedblack and blue phosphorenes, respectively.According to a recent first-principles study of substi-tutional TM impurities in graphene, the spin momentsare calculated to be 0, 0, 1, 2, 3, 2, 1, and 1 µ B for Sc-, Ti-, V-, Cr-, Mn-, Fe-, Co-, and Ni-doped graphene systems,respectively. These values are well compared with 0, 1,2, 3, 2, 1, 0, and 1 µ B for Sc-, Ti-, V-, Cr-, Mn-, Fe-, Co-,and Ni-doped black and blue phosphorenes, respectively.It is interesting to note that the spin moment of each TMimpurity (except Sc and Ni) in graphene is smaller by 1 µ B compared to the corresponding one in black and bluephosphorenes. This may be attributed to the differentbonding natures of graphene and phosphorene: i.e., sp bonding in graphene and sp bonding in phosphorene.Since one valence electron of TM impurities in graphene TABLE II: Spin moments in the TM impurity ( S M ) and the nearest phosphorus neighbors ( S P and S P ) for different substi-tutional TMs in black and blue phosphorenes, together with the spin moments of the isolated TM atoms ( S iso − atom ). S tot isthe total spin moment of the doped black and blue phosphorenes. The band gaps ( E g ) of TM-doped black (blue) phosphorenesare also given. The values in parentheses are the PBE + U band gaps.Doped-atom S M ( µ B ) S P ( µ B ) S P ( µ B ) S tot ( µ B ) S iso − atom ( µ B ) E g (eV)Sc 0.97 (1.36)Ti 0.986 -0.014 -0.017 1.00 3.00 0.36 (0.90)V 1.977 -0.001 -0.002 2.00 5.00 0.07 (0.09)Cr 3.082 -0.076 -0.085 3.00 6.00 0.72 (0.93)black-P Mn 2.207 -0.053 -0.060 2.00 5.00 0.39 (0.82)Fe 1.097 -0.036 -0.019 1.00 4.00 0.27 (0.90)Co 0.61 (1.08)Ni 0.953 0.064 -0.007 1.00 1.00 0.09 (0.46)Sc 1.35 (1.59)Ti 0.992 -0.020 -0.020 1.00 0 (0.73)V 2.032 -0.055 -0.055 2.00 0.15 (0.47)Cr 3.147 -0.083 -0.083 3.00 0.91 (1.63)blue-P Mn 1.954 -0.032 -0.020 2.00 0.12 (0.73)Fe 1.247 -0.044 -0.044 1.00 0.35 (0.91)Co 0.69 (1.12)Ni 0.967 0.039 0.039 1.00 0 (0) participates in π bonding with neighboring C atoms, thespin moment is likely to decrease by 1 µ B . To understandthis deeply, we draw the schematic diagram of spin mo-ment according to Hund’s rule in Fig. 3(b). We here notethat the valence electron configurations of Sc, Ti, V, Cr,Mn, Fe, Co and Ni are 3 d s , 3 d s , 3 d s , 3 d s ,3 d s , 3 d s , 3 d s , 3 d s , respectively. Briefly, wecan distinguish the several regimes depending on the fill-ing of electronic levels:(i) Sc-doped black phosphorene have the empty Sc-phosphorous bonding levels, leading to a zero spin mo-ment.(ii) Co-doped black phosphorene have fully occupiedCo-phosphorous bonding levels, leading to a zero spinmoment.(iii) Ti-, V-, and Cr-doped black phosphorene have par-tially occupied TM-phosphorous bonding levels with thespin moments of 1.00, 2.00, and 3.00 µ B , respectively.(iv) Mn-, Fe-, and Ni-doped black phosphorene havepartially occupied non-bonding 3 d levels with the spinmoments of 2.00, 1.00, and 1.00 µ B .It is notable that the spin moments of TM-doped bluephosphorene [see Fig. 3(c)] are the same as those of blackphosphorene because the energy states of s and d in theoutermost orbital of TM atoms and phosphorus atom arevery close to each other.To explore the underlying mechanism of the magneticmoments in TM-doped black and blue phosphorenes, theMulliken population analysis was performed to list theresults in Table II. We find that the spin moments of theTM impurities ( S M ) have a dominant contribution forthe nearest phosphorus neighbors ( S P and S P ). Thecalculated spin moments of TM impurities for Ti, V, Cr,Mn, Fe and Ni in doped black phosphorene are S M =0.986, 1.977, 3.082, 2.207, 1.097, and 0.953 µ B , respec-tively, close to the above-discussed Hund’s analysis. Sim- FIG. 4: (color online). Band structure and density of statesof the undoped defective black phosphorene. The red linesrepresent majority spin band, while the black lines representminority spin band. The energy zero represents the Fermilevel. ilarly, the spin moments of TM impurities for Ti, V, Cr,Mn, Fe and Ni in doped blue phosphorenes are S M =0.992, 2.032, 3.147, 1.954, 1.247 and, 0.967 µ B , respec-tively.
4. Analysis of the electronic structures
We first examine the electronic structure of a singlephosphorous vacancy in black phosphorene. As substitu-tional impurities in black phosphorene, most of the TMatoms studied here exhibit a symmetrical configurationof C h . For this reason, it is particularly instructive toanalyze their electronic structures with the hybridizationbetween the atomic levels of the TM atoms and thoseassociated with a relaxed C h symmetrical phosphorus FIG. 5: (color online). Band structures of Sc-, Ti-, V-, Cr-, Mn-, Fe-, Co- and Ni-doped black (a) and blue (b) phosphorenes,respectively. The red (black) lines represent the majority (minority) spin band. The energy zero represents the Fermi level.FIG. 6: (color online). Band structures of Ti-, Ni-dopedblue phosphorenes obtained using the PBE + U calculationwith U = 5.5, 6.5 eV, respectively. The red (black) linesrepresent the majority (minority) spin band. The energy zerorepresents the Fermi level. vacancy. As shown in Fig. 4, the C h vacancy showsa considerable spin polarization of 1.00 µ B , indicating adilute magnetic property.To further shed light on the underlying mechanismof magnetic properties in the TM-doped black and bluephosphorene structures, we plot the spin-polarized bandstructures of TM-doped black and blue phosphorenes inFig. 5(a) and (b), respectively. Interestingly, the ma-jority and minority spin bands for Ti-, V-, Cr-, Mn-,Fe-, and Ni-doped black phosphorene show semiconduct-ing characters [see Fig. 5(a)], indicating dilute magneticproperties. On the other hand, Sc- and Co-doped black phosphorene exhibit zero spin moment, whereas Ti- andNi-doped blue phosphorene show half-metal characters[see Fig. 5(b)]. Note that V-, Cr-, Mn-, and Fe-dopedblue phosphorenes exhibit dilute semiconducting charac-ters, while Sc- and Co-doped blue phosphorenes have zerospin moment.In general, substitutional TM impurities in black andblue phosphorenes exhibit very similar behaviors in theirenergetic and magnetic properties. This result indi-cates that the structural differences of black- and blue-phosphorene lattices are insensitive to determine the en-ergetic and magnetic properties of TM-doped black andblue phosphorenes, as shown in Fig. 2, 3, and 5.It is interesting to examine the effect of the on-siteCoulomb interaction U on the magnetic properties ofthe substitutional 3 d TM impurities in black and bluephosphorenes. We perform the PBE + U calculationsfor all the considered systems, where the values of U =4.0, 5.5, 3.3, 3.5, 3.5, 4.3, 3.3, and 6.5 eV are chosen forSc-, Ti-, V-, Cr-, Mn-, Fe-, Co-, and Ni-doped systems,respectively. The calculated PBE + U band gaps( Eg ) of Sc-, Ti-, V-, Cr-, Mn-, Fe-, Co-, and Ni-dopedblack and blue phosphorenes are listed in Table II. Wefind that Eg of the magnetic semiconductor obtained us-ing PBE + U increases by ∼
30% compared to the PBEresults. However, it is noticeable that the spin momentdoes not change depending on the PBE and PBE + U methods. Interestingly, we find that the PBE + U bandstructure of Ti-doped blue phosphorene shows a magneticsemiconductor property with a gap opening (see Fig. 6),different from the half-metallic character predicted byPBE. This indicates that U in Ti-doped blue phospho-rene splits the narrow half-filled bands crossing the Fermilevel into lower and upper Hubbard bands. On the otherhand, the half-metallic character of Ni-doped blue phos- FIG. 7: (color online). Spin polarized total (upper panel) and partial (lower panel) density of states of Sc-, Ti-, V-, Cr-, Mn-,Fe-, Co- and Ni-doped black (a) and blue (b) phosphorenes, respectively. The energy zero represents the Fermi level. phorene predicted by PBE is preserved in the PBE + U band structure (see Fig. 6), because the bands crossingthe Fermi level have relatively larger band widths com-pared to those in Ti-doped blue phosphorene [see Fig.5(b)].A general picture of the dilute magnetic and half-metalfeatures of the TM-doped black and blue phosphorenescan be seen from the analysis of the spin-polarized totaland partial DOS, as shown in Fig. 7. As for Sc- and Co-doped black phosphorenes, the total DOS of the major-ity and minority states are completely compensated witheach other, yielding zero spin moment [see Fig. 7(a)].It is found that for Ti-, V-, Cr-, Mn-, Fe- and Ni-dopedblack phosphorenes, the total DOS of the majority andminority states are not compensated below E F and showa gap opening, indicating dilute magnetic semiconduct-ing properties. We note that the DOS of Sc- and Co-doped blue phosphorene show nonmagnetic properties;those of Ti- and Ni-doped blue phosphorene show half-metal properties; V-, Cr-, Mn-, Fe-doped blue phospho-renes show dilute magnetic semiconductor characters [seeFig. 7(b)]. From the analysis of the spin-polarized totaland partial DOS, it is seen that the magnetic momentsare well localized at the TM atom site, and the d xy and d x − y orbitals are dominant for the contribution to thepartial DOS.
5. Conclusions
We have performed a first-principles DFT calculationfor the structural, energetic, and magnetic properties ofa series of substitutional 3 d TM impurities in black and blue phosphorenes. We provided a simple model based onHund’s rule for understanding the calculated electronicand magnetic properties of the considered systems, wherethe dilute-semiconducting and half-metal features, spinmoment, and binding energy are varied depending onthe atomic number of the TM atoms. The spin-polarizedband structures and DOS calculations show that for blackphosphorene, the Ti-, V-, Cr-, Mn-, Fe- and Ni-dopedsystems have dilute magnetic semiconductor properties,while Sc- and Co-doped systems have no magnetism. Forblue phosphorene, the Ti- and Ni-doped systems showhalf-metal properties, while V-, Cr-, Mn- and Fe-dopedsystems show dilute magnetic semiconductor characters,Sc- and Co-doped systems show non-magnetism.Since substitutional impurities of 3 d TM atoms inblack and blue phosphorenes exhibit some intriguing elec-tronic and magnetic properties, such doped systems canprovide an interesting route to tune various functions forspin electronic devices based on black and blue phos-phorenes. This functional ability together with the highstability of substitutional impurities can open a route tofabricate ordered arrays of these impurities at predefinedlocations, which would allow the experimental tests of thetheoretical predictions of unusual magnetic interactionsmediated by black and blue phosphorenes.
Acknowledgements
We thank Prof. Zhenyu Zhang for helpful discussions.This work was supported by the National Basic ResearchProgram of China (Grant No. 2012CB921300), Na-tional Natural Science Foundation of China (Grant Nos.11274280 and 11304288), and National Research Foun- dation of Korea (Grant No. 2015R1A2A2A01003248). ∗ e-mail address:[email protected] † e-mail address:[email protected] A. K. Geim and K. S. Novoselov, The rise of graphene,
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