The Two-Dimensional Brillouin Zone of Uniaxially Strained Graphene

Home , Brillouin zone

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Berlin, t ysit fteRmnactive Raman the of shifts cy eto fsri.Tu h muto tancnb directly be can strain measurement. of Raman amount single a the from Thus determined strain. of rection o ifrn tandrcin r on,tefrequencies active the Raman found, are directions structu strain band different phononic for and electronic the in changes specific I.1 Clroln)()Teui elvectors cell unit The (a) online) (Color 1: FIG. rpee h ro niae h ieto fteapids applied the of direction the indicates ϑ arrow The graphene. oeo nailysrie rpeei h direction the in graphene strained uniaxially of zone on ru eue to reduces group point os cBilunzn fuixal tandgahn in graphene strained uniaxially tion of zone (c)Brillouin tors. rloi oewti cm 5 whole within within the converged zone from frequencies Brillouin are phonon and differences better, or Energy meV/atom 5 graphene. spa for vacuum interlayer ing the and matrix dynamical of the number culating the zone, Brillouin the of sampling aefnto uof h hredniyctf,the cutoff, density charge the th cutoff, to function respect with differen wave frequencies between checked phonon carefully differences the We and energy configurations used. the was in Ry convergence 0.02 the of width smea a Gaussian with A ing fo functional[17]. parameterization exchange-correlation Ernzerhof and the Burke Perdew, the in tion approx gradient generalized the and pseudopotentials[16] h rloi oe h yaia arcswr calculated were matrices dynamical a The on zone. Brillouin the addi ln aebsswt neeg uofo 0Ry. 60 of cutoff energy an with basis A wave plane a in panded hoy oc osat eeotie i ore transfo Fourier a via obtained were constants Force theory. E SPRESSO oa adsrcueo nailysrie graphene strained uniaxially of structure band ional 42 acltoswr efre ihtecd Q code the with performed were Calculations 0 = a (b) (a) D ϑ 12 6h ri a edrcl eemndfo single a from determined directly be can train × 20 = ◦ × 42 tan.Tedfraino h ia cone Dirac the of deformation The strains. l y lascrepnst h izgdrcin (b)Brillouin direction. zigzag the to corresponds always 12 × ◦ 1] eue ln-aebssst RRKJ set, basis plane-wave a used We [15]. h on ru eue to reduces group point The . a n h em rsigmvsaa from away moves crossing Fermi the in, × 2 ϑ 1 E apiggi a sdfrteitgainover integration the for used was grid sampling 1 x 1 2 q n hita Thomsen Christian and g gi sn h mlmne linear-response implemented the using -grid a 1 and K D 5 2 M 2 M d D D 4 5 K . K E − 2d 6 4 b oe r needn ftedi- the of independent are modes M 2 b 2 1 ~ M graphene g 1 h aec lcrn eeex- were electrons valence The . 6 3 , K K and b ~ 3 ϑ=0° ,ε=0.3 1 2 M M 1 2 r h eirclltievec- lattice reciprocal the are K 2 b 2 1 D C oe are modes 2 (c) h 1 ~a b 2 1 C q ~a, pit o cal- for -points 2h 2 ϑ=20° ,ε=0.3 ϑ fhexagonal of UANTUM 0 = h direc- the b k 1 ◦ -point fthe of The . train. ima- c- re r- r- e - r t 2 mation and interpolated to obtain phonons at arbitrary points ment with experimental tension measurements on pyrolytic in the Brillouin zone. All frequencies were multiplied by a graphite that yield a value of ν=0.163.[18] Strain reduces the constant factor to match the experimental Raman frequency symmetry of the hexagonal system. For strain in arbitrary di- of graphene. rections the point group is reduced from D6h to C2h, only ◦ the C2 rotation (rotation by 180 ) and the inversion are re- tained (and the trivial mirror plane). For strain along the 0◦or 30◦directions additionally mirror planes remain, resulting in (a) D2d-symmetry. A sketch of the hexagonal lattice and the corresponding strain in real space can be seen in Fig. 1(a). Due 4 K2 to the hexagonal symmetry only strain in the range between M2 ϑ=0◦and 30◦are physically interesting. 2 M3 K3 ΓK M Γ 2 2 In Figs. 1 (b) and (c) the resulting Brillouin zones for strain ΓK M Γ K2M2 3 3 ◦ ◦ K M 0 in the directions 0 and 20 are shown. For clarity we have 0 3 3 unstrained used an exaggerated strain of ǫ = 0.3. Like in unstrained Energy [eV]

Electronic energy [eV] graphene, the six corner points of the Brillouin zone corre- -2 K 1/16 KM spond to the K-points. In the unstrained lattice these K- points, e.g., 1 can be expressed via 1 b1 b2, -4 K K = 1/3 +2/3 where bi are the reciprocal lattice vectors. The same defini- Γ KM Γ tion does not hold anymore for the strained lattice, where the K-points can be obtained by constructing the perpendicular bisectors to all neighboring lattice points and determine their intersection point. The usual way to plot band structures is along the high sym- (b) metry directions. For unstrained graphene this corresponds to 1500 Γ − K − M − Γ. In strained graphene, as the 6-fold sym- ΓK M Γ ] 2 2 metry is broken these lines are not sufficient to cover all high- -1 ΓK M Γ ◦ 3 3 symmetry lines. For strain in the ϑ =0 direction we choose unstrained 1000 the lines Γ−K2 −M2 −Γ and Γ−K3 −M3 −Γ, the remaining ones can be found by symmetry. In Fig. 2 we plot the electronic band structure and the phonon dispersion curves for 2% strain and the relaxed graphene. The most obvious 500 Phonon frequency[cm changes in the electronic band structure include an increase in energy of the π-type valence bands along the K − M direction. The splitting of the σ-type valence bands at E = −3 eV 0 Γ KM Γ can clearly be seen. A closeup of the vicinity of the K-point reveals that the crossing of the Fermi level only takes place between K3 and M3. To further investigate the position of the Fermi crossing we evaluate the direct optical transition (DOT) FIG. 2: (Color online) (a) The electronic band structure and (b) the energy of the - ∗ -bands in the vicinity of the -point. In phonon dispersion curves of relaxed and ǫ = 0.02-strained graphene π π K1 = 0◦ Fig. 3 we show energy contour plots of the DOT energy of along the ϑ -direction. The corresponding paths in the Brillouin ∗ zone are indicated. A closeup for the electronic bands along KM is the π-π -bands. Each plot is centered at its respective K1- shown in the inset of (a). point. The dimension of the sides of the square is 0.02 π/a0, where a0 is the lattice constant . We show a relaxed and a ◦ ◦ The general 3-dimensional Hooke’s law connects the stress 2%-strained configuration for ϑ = 0 and 20 . Although our tensor σ and the strain tensor ǫ via the stiffness tensor c: plots show only the vicinity of the K1 point, the remaining σij = cijkl ·ǫkl. As we are only interested in planar strain, the plots can easily be constructed. Three of the six K-points are tensile strain along the x-axis can be expressed via the two- equivalent: This can be seen by apparent simple arguments: dimensional strain tensor ǫ = ((ǫ, 0), (0, −ǫν)), where ν is an arbitrary K-point, Ki corresponds to a set of different of the Poisson’s ratio. Since we want to apply arbitrary strain K-points in the adjacent Brillouin zones (Ki+2,Ki+4). The − directions the strain tensor has to be rotated ǫ′ = R 1ǫR, equivalence to the remaining three K-points follows by the where R is the rotational matrix. Here the x-axis corresponds C2 rotation (or inversion), where Ki maps to Ki+3. Thus the to the zigzag direction of graphene. After applying a finite contour plots of Ki i=1,3,5 are identical, the remaining con- amount of strain we relax the coordinates of the basis atoms tour plots Ki i=2,4,6 are found by inversion. until forces are below 0.001Ry/a.u. and minimize the total As can be seen, for the relaxed configuration the Dirac energy with respect to the Poisson’s ratio ν. For small strain cones coincide with the K-point. For the strained configu- ◦ values we obtain a Poisson’s ratio ν=0.164, in excellent agree- ration at ϑ =0 the crossing moves along the K3 − M3-line. 3

40 40 40 0.216 0.192 0.216 35 (a) 35 (b) 35 (c) 0.192 0.192 0.168 30 30 0.16830 0.168 0.144 0.144 25 25 25 0.144 0.120 2γ 0.120 20 20 20 0.120 0.096 0.096 0.096 15 15 15 0.072 0.072 0.072 10 10 10 0.048 0.048 0.048

5 5 5 0.024 0.024 0.024

0 0.0000 0.0000 0.000 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40

FIG. 3: (Color online) Contour plots of the direct optical transition energy of the π-π∗ -bands. The energy is given in eV. Each is plot centered at its respective K-point. The length of each side is 0.02π/a. Shown here is the plot around K1. Plots around Ki (i=3,5) are identical. Ki ◦ (i=2,4,6) can be found by inversion. (a) Relaxed configuration, (b) 2%-strained configuration for ϑ = 0 and (c) a 2%-strained configuration ◦ for ϑ = 20 .

Now the origin of the bands in Fig. 2(a) becomes clear: the stems from the isotropy of the hexagonal lattice. line along K2 − M2 cuts the Dirac cones away from the cen- In graphene the shape of the double-resonant 2D mode ter, resulting in a small opening of the bands. gives information on the number of layers[20]. This mode For the strained configuration at ϑ = 20 ◦ shown in is energy-dependent and double resonant[13, 14]. The mech- Fig. 3(c) the Fermi crossing moves away from the zone-edge anism of a DR-process is shown in Fig. 4(a). Here the phonon into the Brillouin zone. The question of whether or not a gap energy is assumed to be zero. The contributing phonon opens in graphene under small strains must be answered by branch, the TO-mode, is the mode with the highest energy looking into the appropriate direction in reciprocal space. The between K and M [see Fig. 2(b)]. Although the 2D mode tip of the Dirac cones, according to our DFT calculations, contains contributions from all over the Brillouin zone, the lie on lines through K that enclose an angle of 2ϑ and the main contributions come from the K − K′ valleys as shown K − M line [see Fig. 3(c)]. Looking at the band structure by Narula and Reich[21]. Thus the one-dimensional treatment along the strained high-symmetry directions, in contrast, cor- of the DR gives a good approximation. responds to cuts of the Dirac cones not centered at the Fermi The changes of the electronic bands and the vibrational level crossing, seemingly suggesting an energy gap. This be- bands both influence the 2D-mode. Contour plots of the TO- comes important for strain in directions other than ϑ = 0◦ mode energy are shown in Fig. 5 for the relaxed and the 2%- and ϑ = 30◦, when the Fermi level does not coincide with the strained configuration for ϑ = 0◦ and 20◦. Each plot is cen- Brillouin zone edges. tered at its respective K1-point. Although the general shape of We now turn our attention to the changes in the vibra- the vibrational mode differs more strongly than the electronic tional spectrum when strain is applied. The vibrational bands bands under strain, the stronger contribution for the shift of show the general trend of softening under strain. In con- the 2D mode comes from changes in the electronic structure. trast, the lowest-energy acoustic mode hardens. This mode, Figure 4(b) shows a contour plot of the lowest-energy which shows a quadratic dependence for q → 0 for unstrained conduction band and two different K − K′ paths for the graphene, is an out-of-plane mode and therefore referred to ϑ =0◦direction. In Fig. 4(c) we plot the electronic bands as ZA-mode[19]. The hardening only occurs for the line in along these two paths. Depending on the wave vector of the the Γ − K2 direction. Also the quadratic dependence for electronic transition a different phonon wave vector is doubly q → 0 changes into a linear dependence. This line describes resonantenhanced. This leads to differencesin phonon energy propagating waves along the direction of the applied strain. of up to 10cm−1/% strain and explains the broadening of the Thus the hardening can be compared to the frequencyincrease 2D mode under strain[6, 7]. Experimentally the broadening when tension is applied to a string. is found to be around 13cm−1/% strain[6]. −1 At the Γ-point the high-energy E2g-phonon at 1581 cm In Table I we show a summary of the determined shift rates is two-fold degenerate for graphene. This degeneracy is lifted for the G+, G− and 2D modes. These rates are for small under strain. As shown in Ref. 6 and Ref. 7 the E2g-mode strain values up to 2%. Our calculated values are in excel- of graphene splits into two distinct modes. These two modes lent agreement with the experimental results from Ref .[6], al- possess eigenvectors parallel and perpendicular to the strain though the calculated values are a bit higher. In contrast to the direction. The mode parallel (perpendicular) to the strain di- G+ and G−-modes, the 2D mode behaves slightly non-linear rection undergoes a larger (smaller) redshift and is therefore for strain greater than 1%. Therefore we suggest to measure − entitled G− (G+). As discussed in Ref. 6 the strain rate of the the G+ and G -modes for determining the strain rather than G− (G+) -mode is independenton the direction of strain, a re- the 2D mode. sult which we find confirmed in our calculations. This result In summary, we have presented an in-depth analysis 4 of the electronic and vibrational properties of uniaxially + − 2 strained graphene. We demonstrated that graphene remains TABLE I: Shift rates for the G , the G and the D mode in strained graphene (in cm−1/% strain). For the excitation energy dependent semimetallic under small strain. The change of the Fermi sur- 2D mode the excitation energy is written parenthesis. face suggests favored directions for electronic transport de- Raman mode Ref.[5] Ref.[6] Ref.[7] this work pending on the direction of strain. Our calculated shift rates of G+ -14.2 -10.8 -5.6 -14.5 − the Raman-active G and 2D-band will help experimentalists G – a -31.7 -12.5 -34.0 to determine the strain. Due to a deformation of the three-fold 2D (2.41 eV) – -64 – -46..54 Dirac cone around the K-point the double resonance condi- 2D (2.33 eV) -27.8 – -21 -46..54 tion changes and gives rise to a broadening of the 2D mode. 2D (1.96 eV) – – – -46..55 ano distinction between G+ and G− q 40 K2 (a) (b) 5.6 35

4.8 30

q 4.0 EL 25 Efermi 3.2 20 K3

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0 0.0 0 5 10 15 20 25 30 35 40 (c) 3 K2 M K' K3 M K' 2 [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,

1 Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004). 0 [2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. -1 Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005). Electronic energy [eV] -2 [3] Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature 438, -3 201 (2005). KM [4] K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and FIG. 4: (Color online) (a) Double resonance mechanism for the 2D A. K. Geim, Science 315, 1379 (2007). mode. The energy of the scattering phonons is neglected. (b) Con- [5] Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. tour plot of the lowest-energy conduction band. (c) Band structure Shen, ACS Nano 2, 2301 (2008). ′ between K and K between different K-points. The correspond- [6] T. M. G. Mohiuddin, A. Lombardo, R. R. Nair, A. Bonetti, ing paths are shown in (b). Depending on the electronic transition G. Savini, R. Jalil, N. Bonini, D. M. Basko, C. Galiotis, a different wave vector of the scattered phonon becomes resonantly N. Marzari, et al., Phys. Rev. B 79, 205433 (2009). enhanced. This leads to a broadening of 2D mode, as contributions [7] M. Huang, H. Yan, C. Chen, D. Song, T. F. Heinz, and J. Hone, from the vicinity of all K-points are added up. PNAS 106, 7304 (2009). [8] G. Gui, J. Li, and J. Zhong, Phys. Rev. B 78, 075435 (2008). [9] Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, ACS Nano 22, 483 (2009). 40 40 131840 1310

1318

1310 1329 35 (a) 35 (b) 131235 (c) 1304

1329 [10] V. M. Pereira, A. H. C. Neto, and N. M. R. Peres 1323 1306 1298 30 30 30

1300 1292 (arXiv:0811.4396v3). 1317 25 25 25 1294 1286 20 131120 20 [11] C. Thomsen, S. Reich, and P. Ordej´on, Phys. Rev. B 65, 073403 1288 1280 15 130515 15 1282 1274 (2002). 10 129910 10 1276 1268

5 12935 5 [12] F. Liu, P. Ming, and J. Li, Phys. Rev. B 76, 064120 (2007). 1270 1262

0 12870 12640 1256 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40

1256

1264 1287 [13] C. Thomsen and S. Reich, Phys. Rev. Lett. 85, 5214 (2000). [14] J. Maultzsch, S. Reich, and C. Thomsen, Phys. Rev. B 70, FIG. 5: (Color online) Contour plot of the fully symmetric TO 155403 (2004). phonon branch centered at the K1-point. The TO mode contributes [15] P. Giannozzi et al., http://www.quantum-espresso.org. exclusively to the D-mode. The dimension of the sides of the square [16] A. M. Rappe, K. M. Rabe, E. Kaxiras, and J. D. Joannopoulos, is 0.2 π/a0 (a) Relaxed configuration, (b) 2%-strained configuration Phys. Rev. B 41, 1227 (1990). ◦ ◦ for ϑ = 0 and (c) 2%-strained configuration for ϑ = 20 . [17] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). [18] O. L. Blakslee, D. G. Proctor, E. J. Seldin, G. B. Spence, and JM and CT acknowledge support by the Cluster of Excel- T. Weng, J. Appl. Phys. 41, 3373 (1970). lence ’Unifying Concepts in Catalysis’ coordinated by the TU [19] M. Mohr, J. Maultzsch, E. Dobardˇzić, S. Reich, I. Miloˇsević, Berlin and funded by DFG. M. Damnjanović, A. Bosak, M. Krisch, and C. Thomsen, Phys. Rev. B 76, 035439 (2007). [20] A. C. Ferrari, J. C. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri, F. Mauri, S. Piscanec, D. Jiang, K. S. Novoselov, S. Roth, et al., Phys. Rev. Lett. 97, 187401 (2006). ∗ Electronic address: [email protected] [21] R. Narula and S. Reich, Phys. Rev. B 78, 165422 (2008).