Molecule–substrate interaction in functionalized graphene

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Molecule–substrate interaction in functionalized graphene Gunnar Bergha¨user *, Ermin Malic´ Institut fu¨r Theoretische Physik, Technische Universita¨t Berlin, Hardenbergstr. 36, 10623 Berlin, Germany

A R T I C L E I N F O

A B S T R A C T

Article history:

The study of carbon-based hybrid nanostructures is an emerging field of current research.

Received 12 September 2013

In particular, photo-active molecules have been shown to considerably influence optical

Accepted 22 December 2013

properties of carbon nanotubes suggesting realization of molecular switches. Here, we

Available online 31 December 2013

focus on the qualitative nature of molecule–substrate coupling within carbon-based hybrid nanostructures including nanoribbons and graphene. Our theoretical approach is based on density-matrix formalism and predicts a molecule-induced splitting of the pristine spectral resonances combined with a considerable spectral shift. Both effects strongly depend on the electronic bandstructure of the substrate. Furthermore, we investigate the impact of the substrate dimension on the coupling by increasing the width of nanoribbons from the very narrow up to graphene. Our calculations reveal a clear increase of the optical absorption of graphene in the vicinity of the Dirac point and a peak broadening at the saddle point due to the appearance of a high-energy shoulder. Our results give new insights into the molecule–substrate coupling and can guide future experiments towards the realization of tailored hybrid materials with desired optical properties. Ó 2013 Elsevier Ltd. All rights reserved.

1.

Introduction

Nowadays the study of carbon nanostructures is a major field in condensed matter research [1–6]. It has gained a tremendous boost since the discovery [7] of graphene in 2004 and it has been growing ever since. Graphene is considered as one of the most promising future materials with potential applications in spintronics, photovoltaics, batteries, touch screens, sensors and many more [4,8,9,6]. In this Article, we study carbon-based hybrid nanostructures consisting of a low-dimensional carbon substrate (nanoribbons and graphene) functionalized with molecules. Nanoribbons are thin slices of graphene and can be considered as graphene’s one dimensional equivalent [10]. Their advantage lies in the variable band gap, which can be controlled by the ribbon’s width [11]. Both, nanoribbons and graphene as mono-layers of carbon atoms, are very sensitive to changes in the surrounding material. This opens the

possibility to manipulate their properties by the control of the surrounding medium and motivates the functionalization approach [12–14]. The long-term aim is the design of new materials with desirable characteristics [15–18]. This idea has drawn researchers’ attention in recent years for many different reasons [19–23]. In graphene, the conduction and valence band intersect at the Dirac points. This makes graphene unsuited for semiconductor devices, where a defined band gap is needed. That is why a band gap opening without the loss of characteristic properties of graphene is one of the main goals of current research. Functionalization with molecules is believed to be a potential strategy to overcome this problem [24,25]. Furthermore, graphene functionalized with bio-molecules (e.g. proteins, DNA) opens a way for bio-applications of graphene-based materials [26], e.g. it could be shown that graphene can be used for DNA analysis [27]. Recently, electronic transport and optical properties of carbon nanostructures functionalized with photo-active molecules

* Corresponding author. E-mail addresses: [email protected] (G. Bergha¨user), [email protected] (E. Malic´). 0008-6223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.carbon.2013.12.063

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have been studied experimentally and theoretically [14,16,28– 33,20]. The focus of this work lies on the fundamental understanding of the molecule–substrate coupling (MSC) in hybrid nanostructures consisting of a carbon-based substrate and an adsorbed photo-active molecule inducing a strong dipole field, cf. Fig. 1. Within a microscopic approach based on the density-matrix formalism [34,35], we address the question of how an external dipole field influences the carriers in the substrate. The theoretical investigation is motivated by experimental measurements performed on carbon nanotubes [32,33]. Our calculations reveal the appearance of a considerable splitting combined with a spectral shift of resonances in the absorption spectrum of the substrate. Due to a dipole-induced momentum transfer, which allows indirect optical transitions, the carriers can reach states energetically below or above the excitation frequency. Furthermore, we shed light on the role of the substrate dimension. We studied carbon nanoribbons of increasing width up to the two-dimensional graphene. We find considerable changes in the absorption spectrum of the graphene-based hybrid nanostructure including a clearly enhanced absorption in the vicinity of the Dirac point and a peak broadening at the saddle point.

2. Theoretical approach substrate coupling (MSC)

for

molecule–

The substrate is chosen to lie in the xy-plane and in the case of 1-dim nanoribbons, we introduce a further spatial confinement along the y-direction, cf. Fig. 1. The edge state of nanoribbons is not in the focus of our study. Here, we investigate exemplary armchair graphene nanoribbons. Their width is given by ðNc þ 1Þ a20, where Nc denotes the number of carbon atoms along the width and a0 the lattice constant of graphene. This results in a constraint for the transversal wave m number k? ¼ ky ! ky ¼ 2mp=a0 ðNc þ 1Þ with m 2 f1; 2; ::; Nc g denoting the nanoribbon subbands. The electronic bandstructure corresponds to that of graphene sliced into Nc subbands with different curvatures and band gaps. For growing

537

Nc , the quantization goes over into a quasi-continuum and the ribbon becomes graphene. This allows us to study the transition from one-dimensional to two-dimensional substrates. As an exemplary molecule, we study spiropyran, a photoactive molecule, which can be reversibly switched into the merocyanine configuration via infrared light inducing a large dipole moment [36,33]. The molecules are non-covalently bound to the carbon surface via Van der Waals interaction leaving the electronic wave functions of the substrate unchanged to a large extent [36]. Therefore, we can consider the substrate to be located within a static field of the attached molecules. The obtained insights are applicable to a variety of other molecules inducing strong dipole fields. Furthermore, we assume a periodic self-assembled ordering of the molecules on the substrate, cf. Fig. 1. The molecules are expected to align depending on their size and polarity [37– 40]. In this work, the focus lies on the qualitative understanding of the molecule–substrate coupling. Therefore, we restrict our analysis to the case of periodically distributed molecules. Note however that the presented approach can be extended to random phases, as discussed in our previous study on carbon nanotubes [36]. To obtain thorough insights into the influence of the attached molecule on the optical properties of the substrate, we calculate the absorption coefficient aðxÞ / xIm½vðxÞ, which is proportional to the optical susceptibility vðxÞ representing the linear response to an optical perturbation that is described by the vector potential AðxÞ [34]. The susceptibility vðxÞ / PðxÞ=ðx2 AðxÞÞ is determined by the macroscopic polarP vc ization PðxÞ ¼ k pvc k ðxÞMk þ cc, which is a sum over the y microscopic polarizations pvc k ðtÞ ¼ havk ack iðtÞ weighted by the

interband carrier–light coupling element Mvc k [41–43]. To obtain the absorption coefficient aðxÞ, we solve the Heisenberg equation of motion for pvc k ðtÞ. This requires the knowledge of the Hamilton operator H ¼ H0 þ Hcl þ Hcd containing the P non-interacting carrier contribution H0 ¼ kk ekk aykk akk , the cary he0 P kk0 0 rier-light interaction Hcl ¼ im k;k–k0 Mk  AðtÞ akk akk , and the 0

Fig. 1 – (a) and (b) One- and two-dimensional carbon-based hybrid nanostructures, respectively. Photoactive molecules (here, merocyanine) are adsorbed on the surface of (a) a nanoribbon and (b) a graphene sheet. The molecules are ordered in a periodic self-assembled lattice. In the case of the one-dimensional nanoribbon, there is one row of molecules with the molecule–molecule distance DRx , while in the case of graphene the molecules form a periodic lattice with the lattice constants DRx and DRy . (c) Close-up of a randomly orientated merocyanine molecule on a hexagonal carbon surface. The molecule distance to the substrate is given by jRz j  0:34 nm corresponding to the Van der Waals diameter of carbon. The dipole vector d is determined by the molecular dipole moment d and the angles a; /d expressing the dipole orientation. (A colour version of this figure can be viewed online.)

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Fig. 2 – Contour plot of the molecule–substrate coupling (MSC) elements of (a) and (b) 1-dim nanoribbons with increasing ~ uy ¼ 2up=DRy , where ~ nx ¼ 2np=DRx and q width up to (c) 2-dim graphene. The dots indicate the allowed momentum transfer q the color denotes the strength of the coupling (red to blue being equivalent to strong to weak). The contour lines reflect the momentum dependence of the MSC in Eqs. 2, 3 before considering the selection rules expressed by djk1x k2x j;q~ nx and djk1y k2y j;q~ uy . The transitions characterized by small momentum transfer q1xðyÞ clearly show the largest oscillator strength (red dots). In the 1-dim case, the width of the substrate determines the number of possible transitions along the ky -direction. (A colour version of this figure can be viewed online.)

molecule–substrate coupling Hcd ¼

P P kk0

kk0 y 0 kk0 gkk0 akk ak k0

with

0

. In this first study, we do not consider the MSC element gkk kk0 the effects of the Coulomb interaction [44], since we are interested in understanding the qualitative nature of the molecule–substrate coupling. Though Coulomb interaction is known to considerably influence the spectrum of nanoribbons and graphene [45,46,43], it can be expected to only have a quantitative impact on the molecule–substrate coupling in the investigated structures [36,31]. The electronic bandstructure ekk and the coupling element 0 Mkk k are obtained within the nearest-neighbor tight-binding approach [43]. The MSC element is given by the expectation value of the dipole potentials Udl ðrÞ of the adsorbed molecules [36]   * + X    k k d gk;k ¼ w ð r Þ U ð r Þ w ð r Þ : ð1Þ   l k kk  l  k Inserting the tight-binding wave functions, we obtain for 1dim hybrid nanostructures 0

gkk kk0 ;1D ¼

0 ie0 X 0 0 nx d nd;k;k k  k : 0 n¼0 jkx kx j;~qnx 1D

ð2Þ

and for two-dimensional structures 0

gkk kk0 ;2D ¼

X 0 ie0 X 0 ny djk k j;~q nx djk k j;~q nd;k;k k  k : 0 u¼0 1y 2y uy n¼0 1x 2x nx 2D

ð3Þ

Here, nxðyÞ ¼ Nl;xðyÞ =L is the molecule density on the substrate and DRxðyÞ is the molecule–molecule distance in x (y)-direction, cf. Fig. 1. The appearing confinement functions 0  0 nd;k;k 1D=2D k  k explicitly depend on the molecular dipole vector d and are inversely proportional to the momentum transfer 0 jk  k j, cf. the supplementary material for more details. Furthermore, 0 is the electric field constant and e0 the elementary charge. In Fig. 2, the momentum-dependent MSC element is shown for one dimensional and two dimensional hybrid nanostructures. The presence of the molecular dipole fields allow indirect optical transitions with a momentum ~nxðuyÞ ¼ nðuÞ2p=DRxðyÞ inversely proportional to the transfer q molecular distance DRxðyÞ , cf. the Kronecker deltas djk1x k2x j;~qnx

Fig. 3 – Schematic illustration of the molecule-induced indirect optical transitions within the two-dimensional graphene bandstructure. The black lines show the subbands of an exemplary nanoribbon. The MSC allows transitions between subbands of different band number. (A colour version of this figure can be viewed online.)

and djk1y k2y j;~quy in Eqs. 2 and 3. The dots in Fig. 2 show the al~nx and q ~uy . The color of the dots lowed momentum transfer q indicates the oscillator strength of the corresponding transition (red denoting strong and blue weak interactions). For 1dim substrates, the optical selection rule in perpendicular direction (here, y-axis) is not sharp and depends on the width of the substrate. The broader the substrate, the closer are the dots along the y-direction implying more allowed interband transitions. The contour lines in Fig. 2 indicate the interference of the adsorbed molecules having an influence on the strength of the coupling. Now, having analytic expressions for all relevant coupling elements, we can apply the Heisenberg equation of motion and derive the Bloch equation for hybrid nanostructures, which reads in the limit of linear optics X  vc vc cc vc ihp_ vc gvv ð4Þ k ¼ ekk pk þ k0 k rk0 k  gkk0 rkk0  Xkk ðtÞ k0

We find that the presence of the molecule-induced dipole field gives rise to indirect optical transitions described by 0

¼ hayk;k ak0 ;k0 i and Xkk0 ðtÞ ¼ iem00h Mvc the new quantity rk;k kk0  AðtÞ, kk0 being the Rabi frequency. This process is schematically illustrated in Fig. 3. The corresponding equation of motion reads

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vc 0 i hr_ vc kk0 ¼ ekk rkk0 þ

 X vc cc vc 0 0 gvv pk rpk0  gk0 p rkp  Xkk ðtÞdkk :

6 9 (2 0 1 4) 5 3 6–54 2

539

ð5Þ

p 0

The dynamics of the diagonal terms (k ¼ k ) corresponds to the microscopic polarization from Eq. 4. Here, ekk0 ¼ evk  eck0 þ ic is the energy difference between the initial state eck and the final state evk0 . Furthermore, c is a phenomenological dephasing rate1 taking into account scattering terms beyond the microscopically considered Hartree Fock contributions. It determines the width of the absorption resonances and has no influence on their position.

3.

One dimensional hybrid systems

To obtain a microscopic access to the optical properties of investigated hybrid nanostructures, we evaluate the derived Bloch equation. The dipole moment of the adsorbed merocyanine molecules has been determined in previous work [31] to be approximately 16 Debye. Since in this study, we are interested in the qualitative nature of the molecule– substrate coupling, we do not consider different dipole orientations or different molecular coverages.2 The optical absorption of a functionalized nanoribbon is shown for the two energetically lowest transitions in Fig. 4. The molecule–substrate coupling introduces an asymmetric peak splitting leading to a considerable red-shift of the pristine absorption peaks including the appearance of a side peak energetically above the pristine resonance. For the energetically higher E22-transitions, this feature is even more pronounced, cf. Fig. 4(c). Since the MSC element has the same strength for both transitions, this observation implies the importance of the electronic bandstructure on the molecule–substrate coupling. In Fig. 4(b) and (d), the underlying optical transitions are schematically shown. The presence of a molecule-induced external dipole field gives rise to indirect optical transitions, which are responsible for the pronounced multi-peaked structure of the absorption spectra of functionalized carbon nanoribbons. To obtain profound insights into the MSC, we reduce the complexity of Eqs. 4 and 5 to the predominant terms allowing us to find analytical solutions for the position of the most pronounced peaks sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi De2q~1x Deq~1x e ¼ e0 þ  2jg0;~q1x j2 þ : ð6Þ 2 4 This analytic expression captures well the main features of the absorption spectrum, cf. Fig. 4(b) and (d). It explains in a straight-forward way the peak splitting in the presence of an external dipole field. The strength of the splitting is determined by the MSC element g0;~q1x and by the band structure via Deq~1x expressing the energy difference between the final states of direct and indirect optical transitions, cf. Fig. 4(b) and (d). The latter is characterized by a momentum transfer ~1x and also determines the overall shift of the spectrum. of q

Fig. 4 – Absorption spectra of the energetically lowest (a) E11 and (b) E22 transitions in an exemplary nanoribbon with a width of 2.5 nm. The absorption of the hybrid nanostructure consisting of a merocyanine-functionalized nanoribbon clearly differs from that of the pristine nanoribbon (shaded peak). (b), (d) Schematic illustration of the most relevant molecule-induced indirect optical transitions. Here, e and eþ denote the analytically predicted resonances according to Eq. 6. (A colour version of this figure can be viewed online.)

The solution of Eq. 6 gives qualitative insights into the molecule–substrate coupling for a variety of different hybrid nanostructures. Here, jg0;~q1x j is directly proportional to the molecular dipole strength d. Modern chemistry offers a large range of switchable polar molecules, e.g. dihydroazulene molecules exhibiting large dipole moments are expected to induce pronounced spectral shifts [47]. Furthermore, organic PTCDA molecules forming self-assembled layers with a considerable dipole moment [39,48] can have a measurable influence on the optical properties of the substrate. Another functionalization parameter is the molecular dipole distribu~1x intion. For a high dipole density, the momentum transfer q versely scaling with the dipole distance is large. As a result, the difference between the final states of indirect and direct transitions Deq~1x increases giving rise to an enhanced overall spectral shift of the absorption spectrum, cf. Eq. (6). Therefore, the variation of molecular dipole densities can be exploited to manipulate the absorption spectrum of the substrate, e.g. large densities are preferable to enhance blueshifted resonances, while small densities are suited to achieve predominantly red-shifted resonances. For more details on the analytical solution of Eq. 6, cf. the Supplementary Material.

1 For nanoribbons, we take a typical value of c ¼ 50 meV, while for graphene c ¼ 125 meV taking into account the broader experimental spectrum of graphene. 2 The molecular dipole orientation is chosen as ad ¼ 0 ; /d ¼ 0 and the coverage is determined by the molecular distance DRx ¼ 4 nm and DRy ¼ 2 nm.

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1

3

0.4

2

0.2 0 −0.2 E1212 E1010 E44

E66

4.

0 −1

E22

E11

−3 −4

−0.8 −1

1

−2

−0.4 −0.6

coupling, we have also derived analytical solutions for the limiting case of a simplified parabolic system addressing the importance of the band curvature, cf. the Supplementary Material.

(b)

4

0.6 Energy [eV]

Shift [eV]

0.8

5

(a)

6 9 ( 2 0 1 4 ) 5 3 6 –5 4 2

← 0 increasing c →

−5 −1

0 kx [a0]

1

Fig. 5 – a) Spectral peak shift of an exemplary nanoribbon with a width of 2.5 nm after functionalization showing several characteristic transitions Eii as a function of the curvature c of the involved subbands. Triangles with tip pointing down (up) correspond to peaks located energetically below (above) the pristine resonance (blueshaded area). The triangle size reflects the peak intensity with respect to the pristine peak. (b) The corresponding conduction and valence bands with colors matching the transitions plotted in the part (a). (A colour version of this figure can be viewed online.)

Two dimensional hybrid systems

Besides the bandstructure, we find that the dimensionality of the hybrid system plays a significant role. In contrast to nanoribbons, the two-dimensional graphene substrate is only affected by the MSC around the high symmetry points at the edges of the Brillouin zone, namely the Dirac (K point) and the saddle point (M point). While the absorption is clearly enhanced at the Dirac point, it is slightly reduced at the saddle point, where we also observe a shoulder at the high-energy side resulting in a broadening of the absorption resonance. Fig. 6 shows the entire spectrum of nanoribbons with widths of 2.5 nm, 6.5 nm and of graphene. With the increasing dimension of the substrate, the number of appearing peaks increases and the spectrum broadens. As shown in the previous section, the molecule–carrier coupling induces

(a)

To better understand the changes in the optical properties of the substrate, we show the position and the intensity of occurring peaks for several intersubband transitions Eii as a function of the corresponding band curvature c, cf. Fig. 5(a). The E1212 transition takes place between negatively curved subbands with m ¼ 12, cf. Fig. 5(b). The corresponding absorption spectrum shows a number of well pronounced red-shifted peaks indicated by red arrows in Fig. 5(a). The arrow size denotes the peak intensity that is found to clearly increase close to the pristine resonance due to the enhanced density of states. The E1010 transition involving dispersionfree bands is not affected by the MSC, i.e. there is no peak splitting and no spectral shift. The peak position exactly corresponds to the pristine resonance (blue-shaded area). The energetically highest valence and lowest conduction subband are located close to the Dirac point and have the strongest positive curvature. For the E11 resonance denoting the transition between these subbands, the red-shifted peak clearly dominates the spectrum. For transitions involving subbands with a smaller positive curvature, the blue-shifted peaks become more pronounced, cf. blue arrows in Fig. 5(a). In general, stronger curvatures lead to a larger energetic difference bevc tween the final states of pvc k and rkk0 , cf. Fig. 4(b) and (d). As a result, the spectrum shifts towards higher energies or in the case of a negatively curved bandstructure towards lower energies, as observed for the E1212 transition. The electronic bandstructure plays a crucial role for the impact of the molecule–substrate coupling on the optical absorption of the substrate. The curvature of the involved subbands is the key feature that determines the peak splitting and the oscillator strength of the generated peaks. To gain thorough qualitative insights into the molecule–substrate

(b)

(c)

Fig. 6 – The influence of the adsorbed molecule on the absorption spectra of 1-dim nanoribbons with increasing width up to the 2-dim graphene. While the absorption of functionalized nanoribbons significantly differs from the pristine system along the entire spectral region, the spectrum of the functionalized graphene shows clear changes only at the high symmetry points of the Brillouin zone (Dirac and saddle point). (A colour version of this figure can be viewed online.)

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a significant peak splitting, which even further accelerates the width-dependent broadening towards the graphene spectrum. We also observe that the pronounced peak around 5 eV characterizing the spectrum of narrow nanoribbons is suppressed by the molecule–substrate coupling. This is surprising since the peak stems from a transition between dispersion-free subbands, where the MSC has no influence, cf. Fig. 5. However, the coupling to the adsorbed molecules considerably increases in 2-dim substrates due to the dipole-induced transitions to neighboring subbands. In general, we observe that the absorption of 1-dim nanoribbons is affected by the MSC throughout the entire spectral region (cf. Fig. 6(a) and (b)), while the spectrum of functionalized 2-dim graphene is only changed at the edges of the Brillouin zone, i.e. in the vicinity of the Dirac and the saddle point, cf. Fig. 6(c). Our calculations predict a clearly enhanced absorption at the Dirac and a broadening of the peak at the saddle point. Close to the Dirac point, the bands are linear and they are characterized by a maximum slope. In analogy to the curvature investigation from Fig. 5(a), we find that the larger the band slope, the closer are the red-shifted peaks to the pristine resonance and the higher is their intensity. This results in a constructive superposition of these MSC-induced peaks and gives rise to the observed enhanced absorption at the Dirac point. The M point is a two-dimensional saddle point with a positive curvature in kx -direction and a negative curvature in ky direction, cf. Fig. 3. As discussed in the previous section, positively curved bands shift the spectrum towards higher energies, whereas negative curvatures account for a shift towards lower energies, cf. Fig. 5. As a result, there is an overlap of these two MSC-induced effects resulting in a slight reduction and a broadening of the peak at the saddle point. The broadening can be ascribed to the appearance of a high-energy shoulder.

5.

Conclusion

In conclusion, we have investigated the molecule–substrate coupling in nanoribbons of increasing widths up to graphene. In particular, we have addressed the role of the electronic bandstructure and substrate dimension for the molecule-induced changes of optical properties. We find that the presence of an external dipole field gives rise to indirect optical transitions within the substrate resulting in a considerable splitting of the pristine spectral resonances combined with a considerable spectral shift. Both effects do not only depend on the molecular properties, such as dipole moment, dipole orientation, and molecular coverage, but they are also strongly influenced by the electronic bandstructure of the substrate. To gain fundamental insights into the molecule–substrate coupling, we have also derived simplified analytical solutions. Furthermore, our calculations reveal the significant role of the dimensionality of the substrate. The increased number of involved states of two-dimensional graphene increases the impact of the adsorbed molecules. At the same time, the molecular interference within the graphene surface becomes more important leading to a spectrally narrow molecule–

541

substrate coupling. As a result, the predicted peak splitting and the spectral shift smear out within the graphene continuum. However, at the Dirac and the saddle point, we observe clear changes in the absorption spectrum of graphene. Our results shed light on the molecule–substrate coupling in functionalized nanoribbons and graphene. The gained insights can guide future experiments in this growing research field and can be applied to other one- and two-dimensional substrates under the influence of a molecular dipole field.

Acknowledgments We acknowledge the support by the Einstein Foundation Berlin and the Deutsche Forschungsgemeinschaft (DFG) through SFB 658. Furthermore, we thank A. Knorr, F. Wendler, and P. Munnelly for helpful discussions.

Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.carbon. 2013.12.063.

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