ERRATUM: A bottom-up route to enhance thermoelectric figures of merit in graphene nanoribbons

We propose a hybrid nano-structuring scheme for tailoring thermal and thermoelectric transport properties of graphene nanoribbons. Geometrical structuring and isotope cluster engineering are the elements that constitute the proposed scheme. Using first-principles based force constants and Hamiltonians, we show that the thermal conductance of graphene nanoribbons can be reduced by 98.8% at room temperature and the thermoelectric figure of merit, ZT, can be as high as 3.25 at T = 800 K. The proposed scheme relies on a recently developed bottom-up fabrication method, which is proven to be feasible for synthesizing graphene nanoribbons with an atomic precision.


DFTB Calculations of Electrons and Phonons
In this work, electronic Hamiltonians and overlap matrices as well as the interatomic force constants are obtained using the density functional tight binding (DFTB) method as implemented in the DFTB+ sofware package 1 . The advantage of the DFTB approach, when compared to genuine density functional methods, is its accuracy and efficiency in the calculation of systems with large unit cells. As shown in Figure 1a, the calculated phonon dispersions of 2D graphene by using finite displacement scheme 2 and the experimental results (see Ref. 3) shows the accuracy of DFTB in describing the vibrational properties of sp2 carbon structures. The electronic bands are also well decribed with DFTB (see Figure 1).
In Figure 2, we show the electronic bands, DOS (a, c) and ballistic electron transmission spectra (b, d) for s-GNR (a, b) and c-GNR (c, d). Likewise, phonon dispersions and transmission spectra of s-GNR and c-GNR are plotted in Figure 3. In Figure 3a and 3c phonon dispersions and ballistic transmissions are shown except dispersionless high frequency C-H modes lying above 3000 cm −1 . Figures 3b and 3d are close-ups to lower energy modes that dominate vibrational heat transport. The fundamental effect of chevron geometry is the formation of mini-bands in electron and phonon dispersions, which results in a subsequent increase of van Hove singularities in the densities of states and narrowing of ballistic transmission plateaus. As a result the electronic structure of c-GNR becomes more compatible with Mahan-Sofo criteria and phonon transmission is substantially reduced due to the fragmented dispersions and geometry induced energy gaps.

Nonequilibrium Green's Functions (NEGF)
The partitioning scheme is employed in transport calculations, where the system is divided into three regions as the central disordered region (C) and the left and right pristine reservoirs (L and R) of the same material. Having obtained the Hamiltonian H and the overlap matrix S from DFTB simulations, the retarded GF is defined as where ε = E + iδ and δ being an infinitesimal real positive number. Using the identity the retarded GF of the central region is expressed as with being the self energies due to coupling to the reservoir states. Here, G r,0 el,LL/RR = (εS LL/RR − H LL/RR ) −1 are the surface GF of the semi-infinite free reservoirs and they are calculated using the iterative scheme as explained in Ref. 4. Having obtained the broadening functions Γ L/R = i(Σ r L/R − Σ a L/R ), the lesser and greater self energies are defined as by using Fermi-Dirac distribution function f FD and chemical potentials µ L/R . Correspondingly, the lesser and greater GFs are and the advanced GF G a el is obtained by setting ε = E − iδ in Equations 3 and 4. Electrical current is written as and the transmission spectrum in the coherent limit is obtained by setting µ L = 1 and µ R = 0 with The scatterings in the central region are taken into account through implementation of the recursion algorithm as explained in Section 1.4

Atomistic Green's Functions
Phonon transport properties are calculated using the AGF method, which is an efficient tool for addressing phonon scatterings at boundaries, interfaces and disordered systems, especially in reduced dimensions. The system is parti-tioned as the central region and the reservoirs as is done for electrons, with the central part being isotopically disordered, and left and right reservoirs are semi-infinite pristine GNRs. Calculating the elements of the force constant matrix K in the harmonic approximation, AGF is defined in terms of the dynamical matrix where M is the diagonal matrix of corresponding atomic masses and ½ is the identity matrix.
Similar to the case of electronic GFs, the GF of the central region is with Σ L/R = D CL/CR G ph,LL/RR D LC/RC being the self energies due to reservoirs. Calculating the broadening of vibrational modes as the transmission spectrum is obtained from In both electronic and phononic calculations, the bottleneck is the computation of GF of the central region, which are carried out using the recursion scheme.

Recursion Scheme
Electron and phonon GFs of GNRs as long as 5 µm, consisting of more than 300,000 atoms, are not possible to compute with direct inversion of the matrices of the central region. The recursion scheme explained below is an efficient and numerically exact way of handling such large matrices. Here, we summarize the decimation of electronic GFs using nonorthogonal basis sets. Its implementation to phonons is straightforward.
Similarly the retarded GF is written as Starting from the second cell, the effective Hamiltonians and the corresponding G (n) of the (n + 1)st recursion step are with n = 0, . . . , N − 3. Repeating the decimation N − 2 times, one arrives at the equivalent GF for the central region which consists of only G 11 , G 1N , G N 1 , and G N N . For phonons, one sets ε = (ω 2 + iδ), S = ½ and H → D.

Thermoelectric Coefficients
Thermoelectric coefficients are defined in the linear response regime using the integral functions with n being integers. 5 Electric conductance in the linear response regime can be expressed as G(µ, T ) = e 2 L 0 while the Seebeck coefficient and the electronic contribution to heat conductance are and the thermoelectric figure of merit is 2

Phonon Mean-Free-Paths
In Figures 4 and 5 phonon mean free paths ℓ ph with different isotopes and densities are compared for s-GNR and c-GNR, respectively. We consider d = 10% and 50% of 14 C isotopes with atomic and precursor distributions. The details of ℓ ph (ω) are mainly due to the phonon DOS of the structures. Highly oscillating behavior of ℓ ph for c-GNRs in the entire spectrum and vanishing ℓ ph values at certain energies are rooted in the increased number of singularities in DOS and the energy gaps. For both types of GNRs, precursor distributions yield longer ℓ ph at higher frequencies for a given isotope density and vice versa at lower frequencies (see the insets of Figure 5). Since it is mainly the low frequency modes which contribute thermal conduction, high frequency phonons having longer ℓ ph with precursor distribution do not affect the resulting κ ph , but reduced ℓ ph of low frequency phonons does. Comparison of low frequency ℓ ph of s-GNR and c-GNR at a given d shows that c-GNR has shorter ℓ ph by approximately one order of magnitude in the average.

Thermoelectric Coefficients
Investigation of the ballistic regime is a prerequisite for a detailed study of the effects of nano-structuring on thermoelectric transport. In this section we present thermoelectric properties of c-GNR within the ballistic assumption for both electrons and phonons; also considering the case when phonons are scattered by isotopic clusters while the electrons stay ballistic. Finally the details of TE coefficients are presented when scatterings are included for both electrons and phonons. Ballistic Electrons and Phonons− The effect of geometry on TE transport in c-GNR is evident in the ballistic limit for both electrons and phonons. The power factor P is as high as 1 µW/K 2 and κ ph /A is lower by more than a factor of 3 compared to s-GNR, yielding significantly high ZT values ( Figure 6) (see also Ref. 6). Phonons being dominant in κ at µ where ZT is maximized, points to the possibility of significantly higher ZT upon phonon engineering, i.e. inclusion of heavy precursors.
Ballistic Electrons and Non-ballistic Phonons− The merit of isotope engineering is best visible when ballistic electron assumption is kept and phonons are scattered from heavy precursors with d = 50%. Since G, S and P remain the same with those in Figure 6, only κ and ZT are shown in Figure 7. Vast reduction of κ at the charge neutrality point is purely due to the suppression of phonons. It is clear that, thermal transport is dominated by electrons for the whole range of µ where P is appreciably high (see also Figure 6c). As a result, ZT values higher than 3 are calculated for all T = 300 K, 500 K and 800 K for devices of lengths L =430 nm, 140 nm and 75 nm. We note that, these L correspond to the optimum system lengths when electron scatterings are included (cf. Figure 8). If all L are set to 5 µm and ballistic electron limit is considered, the ZT values can be as high as 4.9, 7.2 and 11 for T = 300 K, 500 K and 800 K, respectively.
Anderson Disorder− In Figure 8, electrical conductance G, Seebeck coefficient S, power factor P , thermal conductance per cross section κ/A and ZT are plotted at T =300, 500 and 800 K for optimal c-GNR lengths that yield the highest ZT . Optimal L are due to the trade off between increasing thermal resistance and decreasing power factor with L. ZT is enhanced by a factor of 4 compared to the ballistic limit without isotope engineering and Anderson disorder, while it is reduced by 32% when compared to the ballistic electron limit.    ) and Anderson disorder w A = √ 12k B T at T = 300 K (blue), 500 K (green) and 800 K (red) are plotted for optimum system lengths, L =0.43 µm, L =0.14 µm, and L =0.07 µm.