Graphene Nanoribbons

Armchair graphene nanoribbons (AGNRs) and zigzag graphene nanoribbons (ZGNRs) form by cutting infinite graphene along the high-symmetry directions, with ribbon edges oriented parallel to armchair or zigzag directions of graphene, respectively. Cutting along arbitrary directions leads to chiral GNRs, where edges are composed of a mixture of arm-chair and zigzag elements. A number of possible edge configurations are depicted in Figure 2.4.

Armchair graphene nanoribbons

An AGNR with widthN is composed ofN lines of carbon dimers, with each dimer built from two atoms belonging to the A and B sublattice, respectively [Figure 2.4]. Trans-lational symmetry is given inx-direction along~aac, whereas the system is confined in y-direction. A theoretical treatment of AGNRs can be performed using tight binding cal-culations taking into account nearest neighbor hopping and single hydrogen terminated edges [20, 21, 46]. The hydrogen termination aty=0 andy=(N+1)aG/2=W saturates the dangling bonds and ensures the absence of states near the Fermi level thus allowing to write boundary conditions in the form of vanishing wave functions at the positions of the hydrogen atoms [46]:

ψA,0=ψB,0=ψA,N+1=ψB,N+1=0 (2.18)

0

Figure 2.6 | Band structure of armchair graphene nanoribbons with various widths.The band structure is calculated according to equation2.19for semiconducting AGNRs with(a)N=9,(b)N=10 and(c)for a metallic AGNR withN=11.

A and B refer to the two sublattices of graphene. An analytic derivation of the dispersion of AGNRs leads to the following equation [46]:

E= ±t Here,±refers to the conduction and valence band,t to the hopping energy andkto the longitudinal wavevector of the ribbon (withk= −_|~a^π_ac|.._|~a^π

ac|) parallel to thekx-direction of the infinite graphene layer. The transverse wavenumberparises from the edge bound-ary condition and leads to discrete values of the momentum withp=_Nⁿ₊₁^π ;n=1, 2, 3, ..,N. An example for the dispersion relation of an AGNR withN=29 is plotted in Figure 2.5.

In the limit of infinite width the transverse wavenumber becomes continuous with p→ ^ky

p3a

2 and the solution in equation 2.19 coincides with the bulk graphene disper-sion relation in equation 2.5. One can see that in the case of AGNRs, the electronic band structure corresponds to the approximation gained from the zone-folding tech-nique [48], i.e. the slicing of the bands along the directionkx [Figure 2.5 (a-b)] followed by a subsequent projection onto the latter axis [Figure 2.5 (c)]. The initial touching points ofπandπ^∗bands of infinite graphene become backfolded tok=0 in the AGNR band structure.

In Figure 2.6 the band structure of several AGNRs withN =9, 10 and 11 is presented.

The band structure is composed of N bands and reveals either semiconducting [Fig-ure 2.6 (a-b)] or metallic [Fig[Fig-ure 2.6 (c)] behavior depending on the width of the ribbon.

Metallic AGNRs form in the case of ribbon widths ofN=3m−1, withmbeing an integer [20, 21]. In this case the slices directly cut the K and K’ points and lead to the absence of a band gap due to the presence of a band with linear dispersion. All other ribbon widths

2.2 Confinement in Graphene Structures 15

(c)

(b) (d) (e)

−2π/3 _ka 2π/3

G

−π 0 π

E

−1 0 1 2 (a)

Figure 2.7 | Band structure and edge states of zigzag graphene nanoribbons. (a)Calculated band dispersion of a ZGNR.(b-e)Localization of the edge wave functions in a semi-infinite zigzag terminated graphene layer: (b)ka_G=π, (c)ka_G=8π/9, (d)ka_G=7π/9and (e)ka_G=2π/3. The radius of the circle represents the charge density on the particular site. (a-e) Reprinted (adapted) from [49].

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do not feature a metallic band since the slices do not exactly cut the K-point and thus feature a band gap. The size of the band gap in semiconducting AGNRs increases with decreasing width and is in the order of 1 eV for ribbon widths below 1 nm [22].

Zigzag graphene nanoribbons

In the case of ZGNRs with widthN, the carbon atoms are arranged inN zigzag lines with translational symmetry along~azz [Figure 2.4]. Applying a simple projection of the bulk graphene bands onto theky direction places the touching points ofπandπ^∗ bands at k= ±_3a^2π_G [21]. However, in contrast to AGNRs the tight-binding solution of the band structure of ZGNRs shows that the bands in ZGNRs are not well approximated by a sim-ple zone-folding technique, i.e. the projection of the sliced bulk band structure [21]. This is because the terminating edge atoms of opposite sides always belong to different sub-lattices leading to boundary conditions of the form [46]

ψA,0=ψB,N+1=0 (2.20)

which imply a transverse wavenumber (or band index)pdependent onk[46]. Calcula-tions of the band structure [20, 21, 46] lead to a set of extended bands that show a large band gap at the projected K-point atk= ±_3a^2π_G [Figure 2.7 (a)]. In addition, a peculiar single band within the band gap is observed and shows a nearly flat dispersion between

Figure 2.8 | Triangular graphene nanoflake. Spatial distribution of the spin density of states. Reprinted with permission from [23]. Copyright 2008 American Chemical Society._→Online.

|k| =_3a^2π_G.._a^π

G giving rise to metallic behavior.

This peculiar flat band corresponds to a localized state. Calculations of a semi-infinite zigzag edge [20] show that the local density of states (LDOS) of the wave function is con-centrated at the edge carbon atoms [Figure 2.7 (b-e)]. The localization is largest fork=_a^π_G with the LDOS concentrated at the edge carbon atoms only [Figure 2.7 (b)]. Fork→_3a^2π_G the localization extends increasingly into the interior of the graphene sheet [Figure 2.7 (c-e)]. Due to the strong edge localization, this state is referred to asedge state. The LDOS of the edge state is limited to the sublattice of the edge carbon atoms. The flat dispersion of the edge state close to the Brillouin zone boundary leads to a sharp peak in the ribbon DOS atE=EF[21].

Important for the experimental realization of edge states is the theoretical result that the edge state survives even for arbitrary edges except perfect armchair configurations.

The termination of graphene with a sequence of zigzag and armchair segments is suffi-cient to give rise to the edge state [50]. However, as one would expect, the LDOS of the edge state scales with the number of consecutive zigzag elements. A maximum value of the ribbon edge state DOS is reached for perfect zigzag edge termination. A continuous decrease is observed for increasing misalignment from a pure zigzag orientation [21, 50].

The flat dispersion of the edge state and the resulting large ribbon DOS atE=E_Fare responsible for a magnetic ordering in ZGNRs [20]. The magnetic moment scales with the LDOS of the edge state and hence a maximum magnetic moment is found at the rib-bon edges. The magnetic moments of the two sublattices couple antiferromagnetically and lead to opposite magnetic moments on opposite ribbon edges [compare Figure 2.4].

An identical number of atoms in each sublattice leads to a zero net magnetic moment of ZGNRs [20, 44, 43, 45].