Assessment on lattice thermal properties of two-dimensional

PHYSICAL REVIEW B 89, 035422 (2014)

Assessment on lattice thermal properties of two-dimensional honeycomb structures: Graphene, h-BN, h-MoS2 , and h-MoSe2 C. Sevik* Department of Mechanical Engineering, Faculty of Engineering, Anadolu University, Eskisehir, TR 26555, Turkey (Received 21 October 2013; revised manuscript received 30 December 2013; published 21 January 2014) The linear thermal expansion coefficients of two-dimensional honeycomb structures graphene, h-BN, h-MoS2 , and h-MoSe2 are systematically studied by using first-principles based quasiharmonic approximation. This approach is first tested on diamond crystal and excellent agreement with the available experimental data is achieved. Our simulations show that the linear thermal expansion coefficients of graphene and h-BN are more negative than that of their multilayered counterparts graphite and white graphite. In addition, there is a remarkable distinction between the coefficients of these two materials in particular at low temperatures. Contrary to graphene and h-BN, lattice thermal expansion coefficient of MoS2 and MoSe2 are always positive, and the values are comparable with those predicted for diamond. DOI: 10.1103/PhysRevB.89.035422

PACS number(s): 81.05.ue, 63.22.Kn, 63.22.Np, 65.80.−g

I. INTRODUCTION

Graphene and h-BN attracted significant interest as next generation electronic materials due to their potential applications [1–4]. In particular, their superior electronic [3,5–12], thermal [13–18], and mechanical [19–21] properties put these nanomaterials at the forefront of current nanoscience research. The encouraging performance of these two-dimensional materials has stimulated strong interest in alternative honeycomb structures such as h-MoS2 and h-MoSe2 . Their outstanding electronic and optical properties, which are of great relevance to technological applications, have been reported by several authors [8–12,21–23]. A comprehensive list of the physical properties of all the aforementioned honeycomb structures has been extensively studied through both theoretical and experimental analysis. However, as a fundamental physical property the linear thermal expansion coefficient (LTEC) of these materials has not been elaborately examined so far. Several authors focused on only graphene structure and predicted negative LTEC as previously observed in graphite in the temperature range of 0–700 K. Mounet et al. [24] reported the LTEC of graphene as around −3.6 × 10−6 K−1 at room temperature by means of first-principles based quasiharmonic approximation (QHA) [25] simulations. They also showed that graphene has a negative LTEC up to 2500 K. However, Zakharchenko et al. [26,27] estimated the average value to be −4.8 × 10− 6 K−1 between the temperature ranges 0–300 K and predicted a negative trend up to only 900 K by Monte Carlo simulations, based on the empirical bond order potential, LCBOPII [28]. In an experimental study, Bao et al. [29] determined the LTEC of graphene as negative only up to 350 K and measured the room temperature value as approximately two times larger than the theoretical estimations, −7 × 10−6 K−1 . In addition, Yoon et al. [30] determined the LTEC of graphene within the temperature range of 200– 400 K by careful exclusions of the substrate effects and reported significantly different values when compared with aforementioned studies. On the other hand, Pan et al. [31] predicted the LTEC of graphene by performing an experiment

*

[email protected]

1098-0121/2014/89(3)/035422(5)

on graphene/BN heterostructure and found compatible results with the theoretical calculations reported by Zakharchenko et al. [26,27] and Mounet et al. [24]. In contrast to graphene, the LTECs of other well-studied graphene related low-dimensional materials, h-BN, h-MoS2 , and h-MoSe2 are not investigated so far. Indeed, having extensive knowledge of such a fundamental physical property of these materials is of course essential due to their potentials in future high-speed integrated electronic device applications. Also, acquiring information about the difference between the LTECs of graphene and these alternative honeycomb structures is particularly important because of the proposed device applications regarding the hybrid systems [32–34] such as graphyne/BN and graphyne/MoS2 . Thermodynamic properties of the bulk semiconductors can be accurately predicted from first principles using QHA by only considering vibrational free energy [24]. Indeed, our test calculations for diamond (Egap = 5.5 eV) without electronic excitations are in very good agreement with the experimental measurements up to high temperatures. Thus, in this study, LTECs of graphene, h-BN, h-MoS2 , and h-MoSe2 are systematically studied with accurate phonon frequency calculations by density functional perturbation theory (DFPT). II. COMPUTATIONAL DETAILS

The equilibrium in-plane lattice parameters at any temperature T is determined by fitting the Birch-Murnaghan equation of states [35] to Helmholtz free energy calculated by the following equation, i ω F (ai ,T ) = E[ai ] + 2 ωi ρ(ωi )dωi , ln 1 − exp − + kB T kB T (1) where i is the structure index, E[ai ] is the first-principles ground state energy of the structure i, ω is the phonon frequency, and ρ(ωi ) is the phonon density of states of the structure i. During the fitting procedure, twelve different lattice

035422-1

©2014 American Physical Society

C. SEVIK


TABLE I. Calculated lattice constants of the structures with DFT and QHA together with the percent change. DFT

Diamond Graphene BN MoS2 MoSe2

QHA

%

GGA

LDA

GGA

LDA

GGA

LDA

3.5714 2.4665 2.5119 3.1832 3.3178

3.5332 2.4452 2.4876 3.1301 3.2524

3.5863 2.4747 2.5197 3.1878 3.3223

3.5476 2.4531 2.4953 3.1339 3.2561

−0.42 −0.33 −0.31 −0.14 −0.14

−0.41 −0.32 −0.31 −0.12 −0.11

parameters, ai , around first-principles equilibrium lattice constant are considered (between 0.995 and 1.008 a0 for diamond, 0.995 and 1.006 a0 for graphene and h-BN, and 0.9875 and 1.0150 a0 for h-MoS2 and h-MoSe2 structures). For each structure with the lattice constant ai , the exact vibrational frequencies and corresponding density of states (ismear parameter of the code = 5) are obtained by using PHONOPY [36] code, which can directly use the force constants calculated by DFPT as implemented in the Vienna ab initio simulation package (VASP) [37–39]. Finally, the LTECs of the materials are predicted as follows, α(T ) =

1 da(T ) , a(T ) dT

(2)

where, a(T ) is the equilibrium lattice parameter corresponding to the minimum Helmholtz free energy at any temperature. The ab initio calculations are performed using the VASP code which is based on density functional theory [40]. The projector augmented wave pseudopotentials (PAW) [41,42] from the standard distribution are incorporated in the calculations. For electronic exchange-correlation functional, both the generalized gradient approximation (GGA), in its PBE parametrization [43], and local density approximation (LDA) [44] are used. Since the strong influence of the chosen plane wave cutoff energy, Brillouin zone sampling, and supercell size on thermal properties, the following parameters are adopted after a series of test simulations: 750 eV for C, BN, and diamond and 500 eV for MoS2 and MoSe2 structures as plane wave cutoff energy, 3 × 3 × 3 for diamond and 5 × 5 × 1 for the rest as supercell structure, and 6 × 6 × 6 for diamond and 8 × 8 × 1 for the rest as k-point mesh for the Brillouin zone sampling of the super cell structures. III. RESULTS

In the context of the present study, we first focus on the calculation of LTEC of diamond crystal to test the accuracy of our QHA implementation. The equilibrium lattice constant values obtained by first-principle simulations and acquired by the Helmholtz free energy minimization at zero temperature are listed in Table I. The expansion in the lattice parameters due to zero-point energy is around 0.4% for both LDA and GGA pseudopotentials. Figure 1 shows the calculated coefficients of linear thermal expansion up to 2500 K. The predicted LTEC values of diamond closely match with the experimental values [45], and the agreement remains excellent up to 1200 K. In a recent study, Stoupin et al. [46] reported highly accurate

FIG. 1. (Color online) LTEC for diamond as a function of temperature. Our QHA-LDA (black solid line) and QHA-GGA (blue dashed line) results are compared with the ab initio calculations (Ref. [24], green dash dot dot line) and experimental measurements (Ref. [45], black solid circles). The inset shows the results up to 300 K. Here, we also compare our results with very recent experiments (red solid squares and green solid diamonds) reported in Ref. [46].

measurements of the LTEC of diamond crystals at low temperatures. As shown in the inset, our calculations are in excellent agreement with the reported results. In addition, the same authors detected an abnormal change in LTEC in the temperature range 0–50 K and attributed this anomaly to the influence of defects in the crystal. Our predictions without such an anomaly parallel with the explanation and support the irrelevance of the mean Grüneisen parameter with the observed abnormal change in LTEC at low temperatures. Following the diamond crystal, we apply the same method on the graphene by constraining the out-off-plane lattice ˚ We predict an constant of three-dimensional structure as 12 A. enhanced thermal contraction in this material when compared with the bulk graphite. The same phenomenon was previously determined by Mounet et al. [24] and explained this by the existence of more negative Grüneisen parameters (γv (q) = v (q) ]) corresponding to the lowest transverse acous[ 2ωav (q) ][ dωda tic (ZA) modes in graphene, see the inset in Fig. 2. Our LTEC results for graphene, depicted in Fig. 2, are parallel with their predictions up to 450 K. However, for the larger temperatures, the values obtained by our implementation is slightly larger than those reported by the authors. The results obtained with two different pseudopotentials, LDA and GGA, for graphene that are in broad agreement with each other are also compared with the experimental results [29–31] as seen in Fig. 2 up to 1500 K. The measured values by Bao et al. [29] and Yoon et al. [30] in the temperature range 200–400 K are strikingly different when compared with the theoretical values obtained by Mounet et al. [24] and in this study. QHA generally works quite well in bulk semiconductors as shown in the diamond case, and it is expected to yield results comparable with the experiments for graphene as well. However, due to the strong anharmonic character of graphene structure [51], which is not taken into account in QHA calculations, a prominent difference may arise. With this intention, Zakharchenko et al. [26,27] study the LTEC of

035422-2

ASSESSMENT ON LATTICE THERMAL PROPERTIES OF . . .


FIG. 2. (Color online) LTHE for graphene as a function of temperature. Our QHA-LDA-VASP (black solid line), QHA-GGA-(red dash dot dot line), are compared with the ab initio calculations (Ref. [24], green dash dot line) and experimental measurements for graphene (Ref. [30], orange solid circles, Ref. [29], red solid up triangles, Ref. [31] green solid down triangles), and graphite (Ref. [47] blue solid diamonds). Inset shows the calculated mode Grüneisen parameters for graphene.

FIG. 3. (Color online) LTEC for single layer h-BN (QHA-GGA black solid line, QHA-LDA red dashed line) and graphene (QHAGGA green dotted line, QHA-LDA blue dash dot dot line) as a function of temperature. Our results are compared with the experimental measurements for hexagonal BN (Ref. [48], red solid squares, Ref. [49], orange solid circles, and Ref. [50], blue up-triangles). Inset shows the calculated mode Grüneisen parameters of single layer h-BN (black solid line) in comparison with graphene (green dashed line).

graphene by means of Monte Carlo simulations directly taking into account all anharmonic effects. As a result, they estimate the average LTEC value of graphene between 0–300 K as about half of the experimental value, α ≈ −8.0 ± 0.7 × 10−6 K−1 at 300 K [30], which is predicted by carefully exclusion of substrate effects. Therefore, the sharp gap between the simulations and these two experiments can not be directly related with the strong anharmonic character of graphene structure. On the other hand, the measured LTEC values by Pan et al. [31] is qualitatively consistent with the aforementioned theoretical results [24,26,27] and our predictions. Using the same constant out-of-plane lattice spacing in the ˚ we calculate the LTEC of a single layer graphene case (12 A), h-BN crystal. The expansion in in-plane lattice constant of this material, 0.3%, due to zero-point energy is similar to that of graphene owing to their comparable phonon spectrums. However, negative LTEC of this material is considerably larger than that of graphene in a broad range of temperature and more sensitive to temperature in the range 0–300 K, as seen in Fig. 3. Therefore, in graphene/h-BN hybrid structures and devices, structural changes, leading to variation of electronic properties, may arise with temperature changes [31,52]. The basis of the distinction between the graphene and h-BN can be clearly figured out from the transverse acoustic mode Grüneisen parameter comparison of these two materials depicted as inset in the figure. We also compare our results with the experimental measurements for bulk h-BN crystal [48]. Surprisingly, the order of change in LTEC of single layer h-BN with the lack of out-of-plane interaction is quite similar to the change in graphene, see Figs. 2 and 3. As a final analysis, we investigate vibrational and lattice thermal properties of single layer MoS2 and MoSe2 structures which are considered as a new class of two-dimensional materials whose electronic properties [9] excite as much as those of graphene [8,12]. These honeycomb two-dimensional

crystals, known as 1H structure, are built up from a layer consisting of a hexagonal atomic plane of Mo sandwiched between two hexagonal atomic planes of S/Se in a trigonal prismatic arrangement. In this respect, the lattice vibrations of these single layer crystals are characterized by nine phonon branches including three acoustic and six optical branches. Therefore, the vibrational properties of these two materials are quite different from graphene and h-BN structures (Fig. 4) with the most discernible distinction being the well-separated acoustic and optical modes. The only slight resemblance between the phonon dispersions of these two classes of materials is the linear nature of LA and TA acoustic branches

FIG. 4. (Color online) Calculated phonon dispersions of (a) graphene, (b) h-BN, (c) h-MoS2 , and (d) MoSe2 structures along high symmetry lines of the Brillouin zone.

035422-3

C. SEVIK


FIG. 5. (Color online) LTEC for single layer h-MoS2 (QHAGGA black solid line, QHA-LDA red dashed line) and h-MoSe2 (QHA-GGA blue dash dot line, QHA-LDA green dash dot dot line) crystals as a function of temperature. Inset shows the calculated mode Grüneisen parameters of h-MoS2 (black solid line) and h-MoSe2 (green dashed line).

and quadratic nature of ZA acoustic branch in the vicinity of q = 0. After predicting phonon dispersions of MoS2 and MoSe2 structures that are in broad agreement with the previous calculations [53–55], we investigated LTECs of these materials. Accordingly, the expansion in the calculated lattice parameters due to zero-point energy is around only 0.1% for both materials as seen in Table I. These results are quite reasonable due to the low maximum phonon frequencies (around 600 cm−1 ) of these materials when compared with the others (around 1600 cm−1 ) considered in this study. As seen in Fig. 5, positive LTEC is observed for both materials over a wide range of temperatures. The found lack of length contraction, compatible

[1] A. K. Geim, Science 324, 1530 (2009). [2] P. Avouris, Nano Lett. 10, 4285 (2010). [3] Y. Shi, C. Hamsen, X. Jia, K. K. Kim, A. Reina, M. Hofmann, A. L. Hsu, K. Zhang, H. Li, Z.-Y. Juang, M. S. Dresselhaus, L.-J. Li, and J. Kong, Nano Lett. 10, 4134 (2010). [4] D. Golberg, Y. Bando, Y. Huang, T. Terao, M. Mitome, C. Tang, and C. Zhi, ACS Nano 4, 2979 (2010). [5] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature (London) 438, 197 (2005). [6] Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature (London) 438, 201 (2005). [7] M. Neek-Amal, J. Beheshtian, A. Sadeghi, K. H. Michel, and F. M. Peeters, J. Phys. Chem. C 117, 13261 (2013). [8] S. Larentis, B. Fallahazad, and E. Tutuc, Appl. Phys. Lett. 101, 223104 (2012). [9] J. S. Ross, S. Wu, H. Yu, N. J. Ghimire, A. M. Jones, G. Aivazian, J. Yan, D. G. Mandrus, D. Xiao, W. Yao, and X. Xu, Nat. Commun. 4, 1474 (2013).

with the experimentally predicted thermal expansion in layered bulk form of these materials [56], can be explained by the transverse acoustic mode Grüneisen parameters around the point predicted to be ∼−10 for MoS2 and ∼−8 for MoSe2 , which is much smaller than those of graphene and h-BN structures (as low as −80), see insets in Figs. 3 and 5. These relatively small parameters only lead to considerably small negative LTEC values between the temperatures 0 and 30 K. Despite the notable difference between the GGA and LDA results, the overall trend is quite consistent as seen in Fig. 5. In particular, a sharp initial change at low temperatures and a subsequent saturation around the room temperature emerged in the calculations carried out with both pseudopotentials are remarkable due to the potential of these materials in future nanoelectronics applications. However, the opposite nature of LTECs of these two structures and graphene might be considered in the design of hybrid structures like graphene/MoX2 . In summary, we present a detailed investigation of LTECs of two-dimensional honeycomb structures, graphene, h-BN, h-MoS2 , and h-MoSe2 by means of QHA. The calculated negative LTEC of h-BN structure is considerably larger than that of graphene in a broad range of temperature. However, the LTECs of h-MoS2 and h-MoSe2 crystals are predicted to be positive. For all materials, mode Grüneisen parameters are also investigated and the results are quite consistent with the nature of LTECs of those materials. ACKNOWLEDGMENTS

We particularly acknowledge the support from The Scientific and Technological Research Council of Turkey (TUBITAK 113F096) and Anadolu University (BAP1306F261 and -1307F281) to this project. We would also like to thank the ULAKBIM High Performance and Grid Computing Center for a generous time allocation for our projects.

[10] B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti, and A. Kis, Nat. Nano 6, 147 (2011). [11] S. Tongay, J. Zhou, C. Ataca, J. Liu, J. S. Kang, T. S. Matthews, L. You, J. Li, J. C. Grossman, and J. Wu, Nano Lett. 13, 2831 (2013). [12] Z. Yin, H. Li, H. Li, L. Jiang, Y. Shi, Y. Sun, G. Lu, Q. Zhang, X. Chen, and H. Zhang, ACS Nano 6, 74 (2012). [13] J. Serrano, A. Bosak, R. Arenal, M. Krisch, K. Watanabe, T. Taniguchi, H. Kanda, A. Rubio, and L. Wirtz, Phys. Rev. Lett. 98, 095503 (2007). [14] J. H. Seol, I. Jo, A. L. Moore, L. Lindsay, Z. H. Aitken, M. T. Pettes, X. Li, Z. Yao, R. Huang, D. Broido, N. Mingo, R. S. Ruoff, and L. Shi, Science 328, 213 (2010). [15] A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, Nano Lett. 8, 902 (2008). [16] A. A. Balandin, Nat. Mater. 10, 569 (2011). [17] S. K. Singh, M. Neek-Amal, S. Costamagna, and F. M. Peeters, Phys. Rev. B 87, 184106 (2013). [18] D. L. Nika and A. A. Balandin, J. Phys.: Condens. Matter 24, 233203 (2012).

035422-4

ASSESSMENT ON LATTICE THERMAL PROPERTIES OF . . . [19] F. Scarpa, S. Adhikari, and A. S. Phani, Nanotechnology 20, 065709 (2009). [20] C. Li, Y. Bando, C. Zhi, Y. Huang, and D. Golberg, Nanotechnology 20, 385707 (2009). [21] C. Ataca, H. S¸ahin, E. Aktürk, and S. Ciraci, J. Phys. Chem. C 115, 3934 (2011). [22] J. N. Coleman, M. Lotya, A. O’Neill, S. D. Bergin, P. J. King, U. Khan, K. Young, A. Gaucher, S. De, R. J. Smith, I. V. Shvets, S. K. Arora, G. Stanton, H.-Y. Kim, K. Lee, G. T. Kim, G. S. Duesberg, T. Hallam, J. J. Boland, J. J. Wang, J. F. Donegan, J. C. Grunlan, G. Moriarty, A. Shmeliov, R. J. Nicholls, J. M. Perkins, E. M. Grieveson, K. Theuwissen, D. W. McComb, P. D. Nellist, and V. Nicolosi, Science 331, 568 (2011). [23] O. Lopez-Sanchez, D. Lembke, M. Kayci, A. Radenovic, and A. Kis, Nat. Nano 8, 497 (2013). [24] N. Mounet and N. Marzari, Phys. Rev. B 71, 205214 (2005). [25] P. Pavone, K. Karch, O. Schütt, D. Strauch, W. Windl, P. Giannozzi, and S. Baroni, Phys. Rev. B 48, 3156 (1993). [26] K. V. Zakharchenko, M. I. Katsnelson, and A. Fasolino, Phys. Rev. Lett. 102, 046808 (2009). [27] K. V. Zakharchenko, J. H. Los, M. I. Katsnelson, and A. Fasolino, Phys. Rev. B 81, 235439 (2010). [28] J. H. Los, L. M. Ghiringhelli, E. J. Meijer, and A. Fasolino, Phys. Rev. B 72, 214102 (2005). [29] W. Bao, F. Miao, Z. Chen, H. Zhang, W. Jang, C. Dames, and C. N. Lau, Nat. Nano 4, 562 (2009). [30] D. Yoon, Y.-W. Son, and H. Cheong, Nano Lett. 11, 3227 (2011). [31] P. Wei, X. Jianliang, Z. Junwei, Y. Chenxi, Z. Gang, N. Zhenhua, W. K., T. T., S. Yi, and W. Xinran, Sci. Rep. 2, 893 (2012). [32] K. Roy, M. Padmanabhan, S. Goswami, T. P. Sai, G. Ramalingam, S. Raghavan, and A. Ghosh, Nat. Nano 8, 826 (2013). [33] Z. Liu, L. Ma, G. Shi, W. Zhou, Y. Gong, S. Lei, X. Yang, J. Zhang, J. Yu, K. P. Hackenberg, A. Babakhani, J.-C. Idrobo, R. Vajtai, J. Lou, and P. M. Ajayan, Nat. Nano 8, 119 (2013). [34] M. P. Levendorf, C.-J. Kim, L. Brown, P. Y. Huang, R. W. Havener, D. A. Muller, and J. Park, Nature (London) 488, 627 (2012).

PHYSICAL REVIEW B 89, 035422 (2014) [35] F. Birch, Phys. Rev. 71, 809 (1947). [36] A. Togo, F. Oba, and I. Tanaka, Phys. Rev. B 78, 134106 (2008). [37] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993). [38] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). [39] G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996). [40] R. M. Martin, Electronic Structure (Cambridge University Press, Cambridge, England, 2004). [41] P. E. Blöchl, Phys. Rev. B 50, 17953 (1994). [42] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). [43] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). [44] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). [45] G. A. Slack and S. F. Bartram, J. Appl. Phys. 46, 89 (1975). [46] S. Stoupin and Y. V. Shvyd’ko, Phys. Rev. Lett. 104, 085901 (2010). [47] A. C. Bailey and B. Yates, J. Appl. Phys. 41, 5088 (1970). [48] G. Belenkii, E. Salaev, R. Suleimanov, N. Abdullaev, and V. Shteinshraiber, Solid State Commun. 53, 967 (1985). [49] W. Paszkowicz, J. Pelka, M. Knapp, T. Szyszko, and S. Podsiadlo, Appl. Phys. A 75, 431 (2002). [50] B. Yates, M. Overy, and O. Pirgon, Philos. Mag. 32, 847 (1975). [51] N. Bonini, M. Lazzeri, N. Marzari, and F. Mauri, Phys. Rev. Lett. 99, 176802 (2007). [52] A. Locatelli, C. Wang, C. Africh, N. Stojić, T. O. Mentes¸, G. Comelli, and N. Binggeli, ACS Nano 7, 6955 (2013). [53] C. Ataca, H. S¸ahin, and S. Ciraci, J. Phys. Chem. C 116, 8983 (2012). [54] S. Horzum, H. S¸ahin, S. Cahangirov, P. Cudazzo, A. Rubio, T. Serin, and F. M. Peeters, Phys. Rev. B 87, 125415 (2013). [55] A. Molina-Sánchez and L. Wirtz, Phys. Rev. B 84, 155413 (2011). [56] B. S. El-Mahalawy and B. L. Evans, J. Appl. Crystallogr. 9, 403 (1976).

035422-5