Chapter Strain dependence of the heat transport properties of graphene nanoribbons Abstract: Using a combination of accurate density-functional theory and a nonequilibrium Green function’s method, we calculate the ballistic thermal conductance characteristics of tensile-strained armchair (AGNR) and zigzag (ZGNR) edge graphene nanoribbons, with widths between − 50 Å The optimized lateral lattice constants for AGNRs of different widths display a three-family behavior when the ribbons are grouped according to N modulo 3, where N represents the number of carbon atoms across the width of the ribbon Two lowest-frequency out-of-plane acoustic modes play a decisive role in increasing the thermal conductance of AGNR-N at low temperatures At high temperatures the effect of tensile strain is to reduce the thermal conductance of AGNR-N and ZGNR-N These results could be explained by the changes in force constants in the in-plane and out-of-plane directions with the application of strain This fundamental atomistic understanding of the heat transport in graphene nanoribbons paves a way to effect changes in their thermal properties via strain at various temperatures 6.1 Introduction Recently there is a surge in research activities on heat transport through nanostructures as evidenced by the emergence of a few review papers 22,232,233 The reasons for this change abound The first is related to heat management in nanoelectronic circuits, 234 since the miniaturization of electronic devices demands efficient dissipation of heat The second is related to the utilization of the thermoelectric effect 235 to harness heat in nanostructures that may help in alleviating the worldwide energy problem Graphene and its derivatives such as graphene nanoribbons (GNRs) are among the most promising materials in these respects Various experimental values for the heat 102 CHAPTER HEAT TRANSPORT PROPERTIES OF STRAINED GNRS 103 conductivity of graphene have been reported, e.g., 4840 − 5300 Wm−1 K−1 (Ref 26), 600 − 630 Wm−1 K−1 (Ref 236), and 1400 − 2500 Wm−1 K−1 (Ref 99) This points to the fact that high heat conductivity is expected for graphene (in stark contrast to, e.g., the heat conductivity of Ag which is only ∼ 430 Wm-1 K-1 at room temperature) The high thermal conductance of graphene has made it very popular for use as a filler in thermal interface materials 22 For example, the heat conductivity of epoxy resins was improved by 30 times upon addition of 25 vol% graphene additive, 237 and by 2.6 times when wt% of graphene was added to polystyrene 238 Graphene could also be potentially used as a thermoelectric material to generate thermoelectric power 22 The efficiency of thermoelectric materials can be quantified using the thermoelectric figure of merit ZT= S2 Ge T /(σe + σph ), where S is the Seebeck coefficient (also known as the thermopower), Ge is the electronic conductance, T is temperature and σe (σph ) is the electronic (thermal) conductance Graphene has a superior 20 electronic conductance Ge , and a large 239 theoretical value of S ∼ 30 mV/K Even though the experimental 240 values of S for graphene are more modest (40 − 80 µV/K) compared to that for the inorganic 235,241 thermoelectric materials (150 − 850 µV/K), graphene might still qualify as a good thermoelectric material if its high value of (σe + σph ) could be suppressed Although graphene is a semi-metal, its heat conduction is dominated by σph and not by σe due to the strong sp2 -hybridization that efficiently transmits heat through lattice vibrations 242 Various ways have been proposed to increase the phonon scattering centers in graphene, e.g., by increasing the disorder at graphene edges, 243 by introducing isotopes in graphene, 244 and by creating vacancy defects in graphene 245 GNRs with vacancy defects was predicted to have a ZT of up to 0.25 246 The experimental demonstrations of the excellent heat properties of graphene and GNRs have stimulated many theoretical works 23,25,247–253 It is known that applying strain to graphene induces changes to the electronic structure, 254 resistance, 255 Raman spectra, 256 and thermal conductivity 257 For GNRs, different theoretical approaches have been used to study the heat properties of both the unstrained and strained GNRs Thus CHAPTER HEAT TRANSPORT PROPERTIES OF STRAINED GNRS 104 far, molecular dynamics (MD) studies concluded that both compressive and tensile strains are detrimental to the heat conductivity of graphene 257 or GNRs 250,251,258 Wei et al 251 and Gunawardana et al 258 concluded that the conductivity of armchair edge GNRs is more sensitive toward strain than their zigzag edge counterparts Guo et al 250 used a slightly different approach than that used in Ref 251, which resulted in slightly different but essentially similar predictions We note that the MD method is well-suited for investigating heat conduction in the diffusive regime and at high temperatures However, in the ballistic regime and at low temperatures, intricate quantum mechanical effects come into play 253 Since the phonon mean free path in graphene is ∼ 775 nm at room temperature, 234 the heat conduction is ballistic for small-scale graphene nanodevices The phonon mean free path is reduced to ∼ 20 nm in presence of edge disorders 259,260 Zhai et al 261 addressed the thermal conductance of GNRs using a ballistic nonequilibrium Green’s function (NEGF) approach 262–264 They extracted the force constants of strained graphene via the elasticity theory and applied that to study strained GNRs They concluded that thermal conductance is enhanced with tensile strain, with an enhancement ratio of up to 17% and 36% for zigzag edge and armchair edge GNRs, respectively 261 However, we note that a large 19% strain applied in Ref 261 might put the applicability of the elasticity theory in the high strain regime to a severe test In this work, we investigate the thermal conductance characteristics of strained GNRs by using a combination of (1) density-functional theory (DFT) that accurately treats the atomic and electronic structures of sub-nanometer width GNRs, and (2) the NEGF method that has been extensively used to study the heat 262,264 and electron 265 transport through nanostructures Our results may shed light on how the heat conductivity of graphene-polymer composites could be affected under loading, and the possibility of using strain to tune the thermal conductance of GNRs to improve its ZT value 6.2 Models and methodology In this work, we calculate the thermal conductance of the zigzag (ZGNR-N) and armchair edge graphene nanoribbons (AGNR-N) as shown in Fig 6.1(a) using a combina- CHAPTER HEAT TRANSPORT PROPERTIES OF STRAINED GNRS (a) ZGNR-N 105 AGNR-N W W l0 l0 … … … N 4… N hydrogen carbon x l0 (b) Y X Z Figure 6.1: (a) The width W of the zigzag (ZGNR-N) and armchair (AGNR-N) edge graphene nanoribbons is controlled by N that represents the number of carbon atoms across the width of the ribbon Hydrogen atoms are attached to the edge carbon atoms to terminate the dangling bonds For very large W , the optimized length √ the of primitive cell along the edge direction should approach a0 for ZGNR-N and a0 for AGNR-N, where a0 is the lattice parameter of graphene The primitive unit cells are demarcated by dotted lines (b) A supercell comprising of nine primitive unit cells is constructed for the phonon calculations CHAPTER HEAT TRANSPORT PROPERTIES OF STRAINED GNRS 106 tion of first principles density-functional calculations and the ballistic nonequilibrium Green’s function (NEGF) method 262–264,266 The uniaxial strain imposed on the GNRs is described by the strain parameter ε = ( − )/ , where ( ) is the length (relaxed length) of the ribbon along the edges A tensile (compressive) strain corresponds to ε > (ε < 0) We note that the GNR edges have compressive edge stresses 27 that might cause the GNRs to buckle 267,268 that will lead to a decrease of thermal conductance 250,251,257 due to increased phonon-phonon scattering A proper treatment of buckled GNRs using DFT involves many atoms in a supercell and this demands extensive computing resources Therefore we limit this work to studying the effects of tensile strain on the flat GNRs The thermal conductance σ (T, ε) at temperature T and strain ε is calculated from the Landauer expression, ∞ σ (T, ε) = dν hνθ (ν) ∂ nB (ν, T ) ∂T (6.1) ehν/kT (ehν/kT − 1)2 (6.2) or equivalently, 269 σ (T, ε) = h2 kT ∞ dν ν θ (ν) where nB (ν, T ) = ehν/kT −1 is the Bose-Einstein distribution for frequency ν and temper- ature T , h (k) is the Planck (Boltzmann) constant The key quantity is the transmission function θ (ν) that may be calculated in general cases using the nonequilibrium Green’s function method 263 or by counting the number of phonon bands at frequency ν for quasi-one-dimensional periodic systems 269 We have used the latter approach to get θ (ν) due to its computational efficiency to treat the problem at hand (see Appendix A where we show that θ (ν) from the counting and NEGF methods are equivalent) It is interesting to note that at low temperatures T , only the very low-frequency modes contribute to thermal conductance Therefore θ (ν → 0) = Nm in Eq 6.2 may be taken out of the integral sign and this leads to a quantization 270,271 of the thermal conductance according to σ (T, ε) = ∞ k2 T u2 eu h Nm du (eu −1)2 2 k = Nm π 3h T CHAPTER HEAT TRANSPORT PROPERTIES OF STRAINED GNRS 107 We perform density-functional theory (DFT) calculations using the SIESTA package 151 The local-density approximation is used for the exchange-correlation functional Doubleζ basis sets and Troullier-Martins pseudopotentials are used for the C and H atoms We use a vacuum separation of 15 Å in the y and z directions consistent with the convention adopted in Fig 6.1(b) The mesh cutoff is 400 Ry The atomic positions are relaxed using the conjugate gradient algorithm with a force tolerance criterion of 10−3 eV/Å As was demonstrated in Refs 30 and 27, spin-polarization effects are particularly important for ZGNRs Therefore we perform spin-polarized (nonspin-polarized) calculations for ZGNR-N (AGNR-N) Phonon dispersion relations of GNRs are calculated using the supercell method 167,272,273 To minimize interactions from the distant periodic images of a displaced atom from its equilibrium position, a supercell of nine primitive cells sufficient for this purpose is used 253 We displace the ith atom in a primitive cell from its equilibrium position by ±δiα = ±0.015 Å and evaluate the forces acting on the jth atom in the supercell Fjβ (±δiα ) using the Hellmann-Feynman theorem α and β denote the Cartesian directions We then use a finite central-difference scheme to evaluate the matrix elements of the force constant matrix K, where Kiα, jβ = ∂ 2E ∂ riα ∂ r jβ =− Fjβ (+δiα )−Fjβ (−δiα ) 2δiα To reduce the number of static DFT calculations, we exploit the space group operations of AGNR-N and ZGNR-N so that only atoms in the inequivalent positions are displaced The forces 167 or interatomic force constants on the equivalent atoms are deduced from that of the inequivalent atoms (see Sec 3.3.1.1 for detailed discussion on how to transform the force constants) AGNR-N with N = 2(p + 1) and N = 2p + belong to the space group number 51 and 47, respectively, where p ≥ is a positive integer; while ZGNR-N with N = 2p and N = 2p + belong to the space group number 47 and 51, respectively The unstrained (strained) graphene belongs to the space group number 191 (65) CHAPTER HEAT TRANSPORT PROPERTIES OF STRAINED GNRS 108 2.50 AGNR-N, N=3p AGNR-N, N=3p+1 AGNR-N, N=3p+2 ZGNR-N a0 (Å) 2.49 2.48 2.47 10 20 30 GNR width, W (Å) 40 50 Figure 6.2: The approach of the optimized lateral lattice parameter a0 for AGNR-N (N = to 41) and ZGNR-N (N = to 25) toward a0 , the optimized lattice parameter of graphene (denoted by a horizontal dash line) as the width W of the ribbon increases A three-family behavior is observed for AGNR-N 6.3 Results and discussion 6.3.1 Optimized lattice parameter of GNRs Since this work concerns the effect of strain ε = ( − )/ we first need to obtain the optimized length (or equivalently, width W ) We obtain 0 on the thermal conductance, (see Fig 6.1) of GNR-N for different N for each N by performing atomic relaxation of GNRs with different ribbon lengths in the x direction (see Fig 6.1) The total energies of the relaxed structures are then fitted to a polynomial function to obtain the optimized ribbon length For ease of comparison between AGNR-N and ZGNR- N, the optimized lateral lattice parameter a0 is calculated according to a0 = a0 = √0 and for AGNR-N and ZGNR-N respectively From Fig 6.2, we find that while a0 for ZGNR-N monotonically increases toward a0 = 2.471 Å (the optimized lattice parameter of graphene) with increasing W , a0 of AGNR-N monotonically decreases toward a0 with an observation that AGNR-N exhibits a three-family behavior for a0 , i.e., the convergence of a0 is systematic when the AGNR-N are grouped according to N modulo We note that other three-family behaviors for AGNR-N have also been found for the electronic band gap, 30 edge energy, 27,274 and the LO/TO splitting 275 The CHAPTER HEAT TRANSPORT PROPERTIES OF STRAINED GNRS 109 three-family behavior for a0 of the AGNR-N ribbons may be understood using the concepts of aromaticity and resonance bond theory 11 Wassmann et al argued that AGNRs can be classified into three different families depending on the number of equivalent Clar’s structures that can be constructed for each AGNR-N 274 Clar’s structures must contain the maximum number of aromatic π-sextets which can be accommodated by the structure In Fig 6.3, we show examples of the equivalent Clar’s structure that can be constructed for AGNR-N belonging to the three different families, and the bond lengths for the optimized structures For AGNR-N where N = 3p and p is an integer, only one Clar’s structure can be constructed; for N = 3p + 1, there are two equivalent Clar’s structures, and for N = 3p + 2, more than two equivalent Clar’s structures can be constructed Since the C–C resonance bond is shorter than the C–C single bond, the N = 3p structures will have longer bond lengths – which results in a larger a0 – as compared to the N = 3p + and N = 3p + structures In contrast, for ZGNR-N with unpaired spins at the edges, more than two equivalent Clar’s structure can be drawn for any N 274 For width W ∼ 50 Å, the value of a0 differs from that of the bulk graphene by less than 0.1% (0.02%) for AGNR-N (ZGNR-N) 6.3.2 Thermal conductance of unstrained GNRs Using the optimized and atomic coordinates for GNR-N, we perform the phonon dispersion calculation (see Fig 6.6 for typical results) and subsequently obtain the thermal conductance by the counting method 269 Fig 6.4(a) shows the thermal conductance σ (T, ε) at T = 300 K and ε = 0.00 for GNRs as a function of W We find that ZGNR-N have higher conductances as compared to AGNR-N with comparable W This is due to the fact that the phonon dispersions of ZGNR are more dispersive (a single phonon branch is more dispersive if it covers a larger frequency range) compared to that of AGNRs, 253 thus increasing the thermal conductance through a change in the transmission function θ (ν) Since the dispersiveness is related to the gradient dω/dk, which is the phonon velocity, 270 we may also say that ZGNRs have higher phonon velocities than CHAPTER HEAT TRANSPORT PROPERTIES OF STRAINED GNRS 110 AGNR-6, N=3p 1.111 1.387 1.415 1.451 1.425 1.435 1.439 1.435 1.425 1.451 1.415 1.388 AGNR-4, N=3p+1 1.112 1.382 1.427 1.447 1.428 1.447 1.427 1.382 AGNR-5, N=3p+2 1.111 1.383 1.423 1.438 1.433 1.441 1.433 1.438 1.423 1.383 Figure 6.3: The three families of AGNR-N: the N = 3p family has only one possible Clar’s structure, the N = 3p + family has two Clar’s structures and the N = 3p + family has more than two Clar’s structures The bond lengths for each optimized AGNR are also shown The longest C–C bonds in each structure are highlighted in red CHAPTER HEAT TRANSPORT PROPERTIES OF STRAINED GNRS AGNR-N ZGNR-N 2.00 Armchair (b) Zigzag 0 1000 2000 -1 Frequency, ν (cm ) (c) 0.1 1.50 No of states σ(300 K,0.00) (nW/K) (a) θ(ν) 2.50 1.00 111 AGNR-11 AGNR-10* 0.05 (d) ZGNR-7 0.05 ZGNR-6* 10 12 14 GNR width, W (Å) 0 2000 -1 Frequency, ν (cm ) Figure 6.4: (a) Thermal conductance σ (T = 300 K, ε = 0.00) of AGNR-N (N = to 11) and ZGNR-N (N = to 7) The slopes of the dash lines are deduced from the thermal conductance of bulk graphene at 300 K in the armchair and zigzag directions (b) Average transmission function for bulk graphene along the armchair and zigzag directions (c) Phonon densities of states (DOS) of AGNR-11 and AGNR-10∗, which is the sum of the DOS of AGNR-10 and the DOS of bulk graphene (d) Phonon DOS of ZGNR-7 and ZGNR-6∗, which is the sum of the DOS of ZGNR-6 and the DOS of bulk graphene AGNRs, resulting in higher thermal conductance While bulk graphene is a fully π-resonant structure with equal C–C bond lengths between the C atoms, the presence of edges in the GNRs limits the extent of the πresonance Hence not all of the C–C bond lengths are equivalent as explained in Fig 6.3 Therefore, we expect the thermal conductance of GNRs to be different from bulk graphene due to this edge effect In Fig 6.4(b), we show the average transmission function θ (ν) for the bulk graphene in the armchair and zigzag directions The σ (300 K, 0.00) for bulk graphene in the armchair (zigzag) direction is 0.18 nW/K (0.32 nW/K) As W → ∞, the edge effect of GNRs should converge to some finite value, and hence the conductance increase from AGNR-(N − 1) to AGNR-N should approach the conductance value of bulk graphene in the armchair direction The expected conductance slopes for AGNRs and ZGNRs are shown in Fig 6.4(a) Fig 6.1 shows that for the largest W investigated in this study — AGNR-11 and ZGNR-7 — the conductance slope for 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N=3p+2 1.111 1 .38 3 1.4 23 1. 438 1. 433 1.441 1. 433 1. 438 1.4 23 1 .38 3 Figure 6 .3: The three families of AGNR-N: the N = 3p family has only one possible Clar’s structure, the N = 3p + family has... intrinsic strength of monolayer graphene Science, 32 1, 38 5? ?38 8 (2008) [14] Moser, J., Barreiro, A., and Bachtold, A Current-induced cleaning of graphene Appl Phys Lett., 91, 1 635 13? ??1 635 13? ? ?3 (2007) [15]... PROPERTIES OF STRAINED GNRS 110 AGNR-6, N=3p 1.111 1 .38 7 1.415 1.451 1.425 1. 435 1. 439 1. 435 1.425 1.451 1.415 1 .38 8 AGNR-4, N=3p+1 1.112 1 .38 2 1.427 1.447 1.428 1.447 1.427 1 .38 2 AGNR-5, N=3p+2 1.111