Dirac-Electron Behavior for Spin-Up Electrons in Strongly Interacting Graphene on Ferromagnetic Mn$_5$Ge$_3$

TL;DR

This study demonstrates that in graphene on ferromagnetic Mn$_5$Ge$_3$, only spin-up electrons retain Dirac-like behavior near the Fermi level, revealing potential for spintronic applications due to spin-dependent electronic properties.

Contribution

The paper introduces a method to synthesize graphene on Mn$_5$Ge$_3$ and shows that only spin-up electrons exhibit Dirac-electron-like behavior, a novel spin-dependent electronic feature.

Findings

Spin-up electrons in graphene preserve Dirac-like behavior near Fermi level.

Strong exchange splitting observed in the graphene-derived π band.

Potential application in spintronics due to spin-selective electronic properties.

Abstract

An elegant approach on the synthesis of graphene on the strong ferromagnetic (FM) material Mn$_5$Ge$_3$ is proposed via intercalation of Mn in the graphene-Ge(111) interface. According to the DFT calculations, graphene in this strongly interacting system demonstrates the large exchange splitting of the graphene-derived $\pi$ band. In this case only spin-up electrons in graphene preserve the Dirac-electron-like character in the vicinity of the Fermi level and the K point, whereas such behavior is not detected for the spin-down electrons. This unique feature of the studied gr/FM-Mn$_5$Ge$_3$ interface which can be prepared on the semiconducting Ge can lead to its application in spintronics.

Full text

This document is the unedited author’s version of a Submitted Work that was subsequently accepted for publication in J. Phys. Chem. Lett., copyright $\copyright$ American Chemical Society after peer review. To access the final edited and published work see doi: 10.1021/acs.jpclett.9b00893.

The recent progress in the intensive studies of the graphene-metal (gr-M) interfaces placed these systems in the forefront of the 2D-materials and particularly of the graphene-related research Geim (2009); Batzill (2012); Dedkov and Voloshina (2015). This situation is due to not only the perspectives of the mass-production of the high-quality graphene on metals Bae et al. (2010); Ryu et al. (2014), but mainly because of the intriguing properties of the graphene-metal interface itself and that different gr-M systems are used as a playground in fundamental studies of the properties of 2D systems and as a basis for the preparation of more complex 2D-based heterosystems Meng et al. (2012); Xie et al. (2018); Ehlen et al. (2019). Among the gr-M systems the distinct attention is paid to the graphene-ferromagnet (gr-FM) interfaces and significant understanding of the properties of these systems has been achieved Dedkov and Fonin (2010); Voloshina and Dedkov (2011); Usachov et al. (2015). However, there are some factors which limit the further progress in the fundamental studies and in applications of these interesting objects. For example, the strong interaction at the gr-Ni(111) or gr-Co(0001) interfaces leads to the high degree of the $\pi-3d$ hybridization of the valence band states at the interface between graphene and FM metal and, as a consequence, graphene loses its free-standing character and the carriers in the vicinity of the Fermi ($E_{F}$) level cannot be described as mass-less Dirac particles Karpan et al. (2007); Dedkov and Fonin (2010); Voloshina and Dedkov (2011); Usachov et al. (2015). Yet, the same reason causes appearance of induced magnetic moments in graphene Weser et al. (2010); Matsumoto et al. (2013). In this case the states at $E_{F}$, which are significant for the description of the charge and spin transport properties of the junction, mainly have FM $3d$ character, substantially changing the band dispersion, states character and Fermi velocity of the carriers. Following works demonstrate the possibility to decouple graphene from the underlying FM material via intercalation of different species Batzill (2012); Dedkov and Voloshina (2015), giving a route to restore the free-standing character of graphene, however losing the possibility of the induced spin polarization in a graphene layer Voloshina et al. (2011); Vita et al. (2014); Omiciuolo et al. (2014).

Further progress in the graphene synthesis on different substrates and desire to implement graphene in the modern semiconductor technology led to the recent discovery of the graphene synthesis on the catalytically active semiconducting Ge substrates of different orientations Lee et al. (2014). This and subsequent studies demonstrate that Ge$(110)$ and Ge$(111)$ planes are more suitable for the graphene growth Kiraly et al. (2015); Wang and Yu (2016); Tesch et al. (2018), whereas in case of Ge(001) a significant surface faceting under graphene layer Pasternak et al. (2016); Dabrowski et al. (2017) was found limiting further technological processing of the later interface. The recent progress in this area is mainly focused on the studies of the growth mechanism of graphene on Ge surfaces and the studies of the electronic structure of gr-Ge interface or its modification are very rare Dabrowski et al. (2017); Tesch et al. (2018); Zhou et al. (2018); Ahn et al. (2018).

In the present work, we propose the elegant approach to prepare new gr-FM interface, namely gr-Mn5Ge3, which combines several well approved steps used in the graphene-related studies. We performed density functional theory (DFT) calculations of the lattice-mismatched gr-Mn5Ge3(0001) interface in the super-cell geometry and it is computationally shown that in this strongly interacting system the graphene $\pi$ states undergo strong exchange split leaving only spin-up electrons in the vicinity of $E_{F}$ which preserve the Dirac-electron-like character for the carriers with the linear energy band dispersion. At the same time, for the spin-down electrons a series of localized interface states is formed that strongly reduces the carriers’ mobility for this spin channel. Such difference for the electron mobilities can lead to the effective spin-filtering effect in such single-layer graphene-based junction. We believe that the described simple approach, which can be used for the preparation of this gr-FM interface, will be extremely useful in the future studies and applications, where 2D graphene as well as FM and semiconductor materials are included in one heterosystem. Details of DFT calculations and additional data are presented as the Supporting Information.

The preparation of the gr-Mn5Ge3(0001) interface can be accomplished in two steps (Figure 1). On the first step graphene is grown via chemical vapour deposition (CVD) or molecular beam epitaxy (MBE) on semiconducting Ge(111). As was demonstrated earlier Lee et al. (2014), in this case graphene is aligned on this surface with the graphene zig-zag edges along the Ge$\langle 1\bar{1}0\rangle$ and the arm-chair edges along Ge$\langle 11\bar{2}\rangle$. On the second preparation step a thin layer of FM metal Mn5Ge3(0001) can be grown under graphene via well established intercalation procedure, where thin Mn layer is deposited on graphene and then subsequently annealed at the elevated temperature. It is well known that FM Mn5Ge3(0001) can be epitaxially grown on Ge(111) in the temperature range $300-650^{\circ}$ C Zeng et al. (2003); Dedkov et al. (2009); Grytzelius et al. (2011) and it has relatively high Curie temperature of $296$ K, which can be increased well above room temperature via chemical doping Gajdzik et al. (2000); Slipukhina et al. (2009); Stojilovic et al. (2013); Petit et al. (2015). Also previous experiments demonstrate the gr/Ge interface is stable at temperatures, at least, up to $800^{\circ}$ C Kiraly et al. (2015); Tesch et al. (2017, 2018). Thus, the proposed preparation routine can lead to the epitaxial heterostructure consisting of 2D graphene, FM Mn5Ge3, and semiconducting Ge.

The crystallographic structure, top and side views, of the respective interfaces discussed above are shown in Figure 2. FM Mn5Ge3 has a hexagonal structure and it epitaxially grows on the Ge(111) surface forming the $(\sqrt{3}\times\sqrt{3})R30^{\circ}$ structure with respect to $(1\times 1)$-Ge(111) (Figure 2a). As described earlier, the gr-Mn5Ge3(0001) interface is formed via intercalation of a Mn layer in the gr-Ge(111) system. Two possible arrangements of C-atoms on Mn5Ge3(0001) are considered, structure B and structure T (Figure S1). Both structures are very similar with respect to their energetical stability (the difference in the interaction energy of graphene with substrate is of only $7$ meV/C-atom). It is found that the discussed effects are valid for both structures and they are robust to the relative orientation of the layers at the gr-Mn5Ge3 interface. The resulting most energetically favourable interface (structure B) has the structure presented in Figure 2b, where the corresponding orientation of graphene with respect to the Ge(111) surface is preserved in order to avoid any layers rotations. For the formed gr-Mn5Ge3(0001) interface the unit cell of the obtained structure has a periodicity of 5 unit cells of graphene and this unit cell (as well as the primitive unit cell of graphene) is rotated by $30^{\circ}$ with respect to the unit cell of the underlying Mn5Ge3. The side view of the fully relaxed gr-Mn5Ge3(0001) interface is shown in Figure 2c with the mean distance between graphene and the top Mn layer of $2.144$ Å. Considering the fact that the surface of Mn5Ge3 is Mn-terminated with the relatively small density of these atoms in the surface layer one can expect the appearance of the substantial corrugation of a graphene layer in this system. However, our calculations show that here graphene is quite flat with the corrugation of only $0.122$ Å and the interaction energy of graphene with the Mn5Ge3(0001) surface is $149.5$ meV/C-atom, which places this system to the sub-class of a strongly interacting graphene with metallic substrate. The relatively small corrugation of graphene in the considered lattice mismatched system can be connected with the variation of the local environment for carbon atoms on the distance of not more than 2 carbon rings. The strong $sp^{2}$ character of carbon atoms in graphene and the rigidity of the $\sigma$ bonds prevents the strong long-distance corrugation effect for the graphene layer. The flatness of graphene in this system might be crucial for the use of this interface as a spin filter thus avoiding the strong charge and spin scattering due to the interface corrugation on the atomic level. The simulated scanning tunnelling microscopy (STM) images of the Mn5Ge3(0001) surface before and after graphene adsorption are shown in Figure S2 and can be used as a reference for the future experimental studies. The theoretical STM results for the clean surface are in very good agreement with the available experimental and theoretical data Zeng et al. (2003); Grytzelius et al. (2011); Zhang et al. (2013) confirming the correctness of the approach used in the present study.

Before adsorption of graphene on Mn5Ge3(0001) the density of states of this material is strongly exchange split (Figure 3a and Figure S3). The magnetic moment of the surface Mn-(S) atoms is $3.48\mu_{B}$, which is larger compared to the value of $3.11\mu_{B}$ for the subsurface Mn-(S-1) atoms, that can be explained by the reduced number of neighbours at the surface and the respective narrowing of the Mn $3d$ band. Adsorption of graphene leads to the substantial reduction of the magnetic moment of Mn-(S) and Mn-(S-1) to $3.11\mu_{B}$ and $2.98\mu_{B}$, respectively, which is reflected as the decrease of the exchange splitting of the respective states in Mn-atom projected partial DOS (PDOS) (Figure 3b). At the same time the $\pi$ states of graphene become spin polarized due to the presence of the FM Mn5Ge3 substrate (Figure 3c). The calculated DOS for structure T and magnetic moments for all atoms in the studied systems can be found in Figure S4, in Table S5, and in the Supplementary data files.

The space-, energy, and $k$-vector overlap of the valence band states of graphene and Mn5Ge3 leads to the effective hybridization of these states. At the same time the spin character of these states is also conserved in this process. This can be clearly seen in Figure S6 where large-energy scale band structures of the gr/Mn5Ge3(0001) interface obtained after unfolding procedure are shown (for comparison reasons, in case of structure B, DFT calculations were performed with and without inclusion of the spin-orbit interaction). Considering the DOS of Mn5Ge3 (Figure 3a,b and shadowed areas in Figure S6) we can clearly see that spin-polarized graphene $\pi$ bands hybridize with the Mn-atoms projected valence band states of the only respective spin character. This effect leads to the “huge” band gaps for both graphene $\pi$ bands of different spin characters around the $K$ point of the graphene-derived $(1\times 1)$ Brillouin zone: for spin-up states this gap is between $E-E_{F}\approx-3.8\mathrm{eV}\ldots-0.5\mathrm{eV}$ and for spin-down states this gap is between $E-E_{F}\approx-1.4\mathrm{eV}\ldots+2.1\mathrm{eV}$. At the same time the effect of hybridization leads to the appearance of the so-called interface states of the mixed graphene-$\pi$ and Mn $3d$ character, which manifest themselves as a series of flat bands of the respective spin character in the energy range $E-E_{F}\approx-3.8\mathrm{eV}\ldots+2.1\mathrm{eV}$. The similar effect of the interface states formation was observed earlier for other gr-FM interfaces, gr/Ni(111) and gr/Co(0001) Bertoni et al. (2004); Karpan et al. (2008); Dedkov and Fonin (2010); Voloshina et al. (2011); Voloshina and Dedkov (2011).

The effect of the spin-conservation during hybridization of the valence band states at the gr-Mn5Ge3(0001) interface leads to the interesting results, which have dramatic implications on the spin-filtering properties of this interface. Analysis of the zoomed region around the $K$ point and in the vicinity of $E_{F}$ of the band structure of this system showed in Figure 3d,e demonstrates that only spin-up electrons in graphene retain the Drirac-electron-like character with the linear energy dispersion for these electronic states. There is a clear anisotropy for the band dispersion along the $\Gamma-\mathrm{K}$ and $\mathrm{M}-\mathrm{K}$ directions and the linear fit of the graphene $\pi$ band around $E_{F}$ gives the Fermi velocity of $v_{F}=0.(92\pm 0.01)\times 10^{6}\mathrm{m/s}$ and $v_{F}=0.(71\pm 0.01)\times 10^{6}\mathrm{m/s}$ for $\Gamma-\mathrm{K}$ and $\mathrm{M}-\mathrm{K}$, respectively, which is very close to the value characteristic for free-standing graphene and which is much higher compared to the value which was extracted for the strongly hybridized states formed at the gr/Co(0001) interface Usachov et al. (2015). Moreover, this spin-up state has a linear dispersion in the energy range of $\approx 1\mathrm{eV}$ around $E_{F}$ that significantly simplifies the energy position control for this state. For spin-down electrons, there is a relatively large gap between $E-E_{F}\approx-1.2\mathrm{eV}\ldots+0.1\mathrm{eV}$ at the $K$ point. Above and below this energy window a series of the interface states is formed which was described above. The carriers on these states have very small group velocity due to the relatively large Mn $3d$ character. All these factors allow us to conclude that the contact of graphene with FM-M5Ge3 will have a strong influence on its spin-transport properties.

In conclusion, we have discovered a new interface of graphene with strong ferromagnetic material Mn5Ge3, which valence band states are strongly exchange split. This system can be easily fabricated using two steps preparation routine: CVD or MBE of graphene on the catalytically active Ge(111) surface and then intercalation of Mn in the gr-Ge(111) interface with the formation of the ordered gr-Mn5Ge3(0001) interface. Our systematic large-scale DFT calculations for this interface demonstrate that graphene is relatively strongly bonded to the FM Mn5Ge3 substrate that leads to the substantial exchange splitting of the graphene $\pi$ states. It is found that the effect of the spin-conservation during hybridization of the graphene $\pi$ and Mn $3d$ valence band states leads to the appearance of the large band gaps in the energy dispersion of the graphene $\pi$ states for both spin channels. The unique combination of all factors leads to the observation of the only spin-up electrons of graphene at the $\mathrm{K}$ point and at $E_{F}$ which conserve the Dirac-electron-like character. At the same time for the spin-down electrons the band gap is found at $E_{F}$. We expect that our study will stimulate future experimental effort to create graphene-semiconductor and graphene-FM structures and we can expect that the possible 2D graphene – FM Mn5Ge3 – semiconducting Ge systems will be used in the charge and spin transport studies and applications.

The North-German Supercomputing Alliance (HLRN) is acknowledged for providing computer time.

Computational Details

Spin-polarised DFT calculations based on plane-wave basis sets of $500$ eV cutoff energy were performed with the Vienna ab initio simulation package (VASP) Kresse:1994 ; Kresse:1996 . The Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional Perdew:1996 was employed. The electron-ion interaction was described within the projector augmented wave (PAW) method Blochl:1994 with C ($2s$,$2p$), Ge ($4s$,$4p$), and Mn ($3d$,$4s$) states treated as valence states. The Brillouin-zone integration was performed on $\Gamma$-centred symmetry reduced Monkhorst-Pack meshes using a Methfessel-Paxton smearing method of first order with $\sigma=0.15$ eV, except for the calculations of total energies. For these calculations, the tetrahedron method with Blöchl corrections Blochl:1994vg was employed. The $k$ mesh for sampling the supercell Brillouin zone are chosen to be as dense as $18\times 18\times 24$, $18\times 18\times 1$ and $12\times 12\times 1$ in the case of Mn5Ge3 bulk, Mn5Ge3 (0001) surface and gr/Mn5Ge3(0001) interface, respectively. Dispersion interactions were considered by adding a $1/r^{6}$ atom-atom term as parameterised by Grimme (“D2” parameterisation) Grimme:2006 . For comparisons reasons, some calculations (results are marked in the respective figures) were performed with spin-orbit interaction taken into account. The calculated band structures were unfolded (where necessary) to the graphene ($1\times 1$) primitive unit cells according to the procedure described in Refs. 7, 8 with the code BandUP.

Bulk Mn5Ge3 is a ferromagnetic intermetallic compound. It crystallises in the hexagonal $D8_{8}$ structure (prototype Mn5Si3 and space group $P6_{3}/mcm$) and has lattice parameters of $a=7.184$ Å and $c=5.053$ Å Forsyth:1990ia . The hexagonal cell contains $10$ Mn (two sub-lattices, Mn1 and Mn2) and $6$ Ge atoms. Four atomic planes are situated perpendicular to the axis $z$. Planes $z=0$ and $z=1/2$ contain only Mn1-atoms, planes $z=1/4$ and $z=3/4$ contain Ge- and Mn2-atoms. Mn1-atoms are surrounded by $6$ Ge-atoms and Mn2-atoms are surrounded by $5$ Ge-atoms. The atomic positions are: Mn1 in 4(d) site: $\pm(1/3,2/3,0;\,2/3,1/3,1/2)$; Mn2 in 6(g) site: $\pm(x_{1},0,1/4;\,0,x_{1},1/4;\,-x_{1},-x_{1},1/4)$ with $x_{1}=0.2397$, Ge in 6(g) site: $\pm(x_{2},0,1/4;\,0,x_{2},1/4;\,-x_{2},-x_{2},1/4)$ with $x_{2}=0.6030$. The optimised lattice parameters ($a=7.145$ Å, $c=4.976$ Å, $x_{1}=0.2435$, $x_{2}=0.6059$) were used in order to construct models for the (0001) surface as well as interface with graphene.

The Mn5Ge3(0001) surface was modelled by a slab containing $12$ atomic layers and a vacuum gap of approximately $23$ Å. The atoms of the $8$ bottom layers were fixed at their bulk positions during the structural optimisation procedure, whereas the positions ($x$, $y$, $z$-coordinates) of all other atoms were fully relaxed until forces became smaller than $0.02$ eV Å*-1*.

To model the gr/Mn5Ge3(0001) interface, a $\left(\sqrt{3}\times\sqrt{3}\right)R30^{\circ}$ structure was formed with respect to Mn5Ge3(0001) slab and its top side was covered by a graphene layer consisting of $(5\times 5)$ unit cells. Thus, the resulting supercell has the size $(12.375\,\textrm{\AA}\times 12.375\,\textrm{\AA})$ and consists of $50$ C-atoms, $90$ Mn-atoms and $54$ Ge-atoms. In the resulting structure the graphene lattice was stretched by $0.6$% compared to the equilibrium lattice constant of $2.461$ Å. Two possible arrangements of C-atoms above the Mn5Ge3(0001) surface were considered - structure B and structure T. In structure B (which is energetically more favourable one) all Mn atoms of the interface layer are bridged by the C-atoms of the graphene layer. Structure T is obtained by shift of a graphene layer in the $(x,y)$ plane in such a way, that C atoms of graphene occupy all possible high-symmetry adsorption positions with respect to the interface Mn-layer. As for the clean surface, the atoms of the $8$ bottom layers were fixed at their bulk positions during the structural optimization procedure, whereas the positions ($x$, $y$, $z$-coordinates) of all other atoms were fully relaxed until forces became smaller than $0.02$ eV Å*-1*.