Systematic Enhancement of Thermoelectric Figure of Merit in Edge

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C: Energy Conversion and Storage; Energy and Charge Transport

Systematic Enhancement of Thermoelectric Figure of Merit in Edge-Engineered Nanoribbons Luke J. Wirth, and Amir Abbas Farajian J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b02132 • Publication Date (Web): 04 Apr 2018 Downloaded from http://pubs.acs.org on April 4, 2018

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The Journal of Physical Chemistry

Systematic Enhancement of Thermoelectric Figure of Merit in Edge-Engineered Nanoribbons Luke J. Wirth and Amir A. Farajian∗ Department of Mechanical and Materials Engineering, Wright State University, Dayton, Ohio 45435, USA E-mail: [email protected]

Abstract

tion occurs when a voltage is produced in the presence of a thermal gradient because of the process known as the Seebeck effect. Its efficiency is described by the figure of merit

Nanomaterials provide unique promise to thermoelectric energy conversion owing to possible phonon confinement and reduced thermal conductivity. These effects can, in particular, occur in nanoribbons upon edge-engineering. Here we study graphene, boron nitride, and silicene chevron nanoribbons (CNRs), because of their high edge-length to surface area ratio, to assess phonon boundary scattering effects on improving thermoelectric figure of merit (ZT ). Ab initio-based nonequilibrium Green’s function method is utilized to calculate quantum electronic and phononic thermal conductance, electrical conductance, and Seebeck coefficient. Our results show that, compared to straight nanoribbons, ZT in CNRs is systematically enhanced. Detailed contributions to CNRs’ ZT for different geometries and materials are analyzed, in particular, separation of electrical and electron-contributed thermal conductance versus chemical potential. Taking corresponding recent fabrications into account, edgeengineering of nanoribbons is shown to provide a possible strategy for achieving competitive thermoelectric energy conversion.

ZT =

σS 2 T , (κe + κp )

(1)

where σ is electrical conductance, S is the Seebeck coefficient or thermopower, κe is the electron-contributed thermal conductance, and κp is the phonon-contributed thermal conductance. Traditionally, thermoelectric devices have not been widespread because of their high costs and low efficiencies, aside from niche applications like spacecraft where these considerations are of low concern 1 . The primary obstacle to increasing their efficiency has been the interdependence of electrical and thermal conductance 2 . Nanomaterials offer unique promise to thermoelectric power generation, because their lowdimensional structures can confine phonons to suppress thermal transport while still allowing electronic transport, resulting in high ZT s 3,4 . These can be achieved even at room temperature, further enhancing the appeal of nanomaterials for thermoelectric power generation 5 . Knowledge about which of these materials hold especially high figures of merit is necessary for their utilization in solving global problems like energy generation and emissions reduction. Nanoribbons (NRs) have been a frequent target of thermoelectric research because manipulation of their phonon effects can lower

Introduction Thermoelectric power generation at the nanoscale is an emerging technology, made possible by the introduction of new nanomaterials and processing methods. The power genera-

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their thermal conductance, a factor that limits ZT 6,7 . Assessments of graphene are common, and its analogues boron nitride (BN) and silicene have also been studied due to their lower thermal conductances 8–11 . These lower conductances emerge through greater scattering. Silicene has particularly low conductances due to its buckled rather than perfectly planar structure, which strongly scatters its acoustic out-ofplane phonon modes relative to graphene 12,13 . Computations have found promising improvements in ZT at room temperature in different types of engineered graphene NRs 14–22 , BN NRs 23 , graphene-BN heterostructured NRs 24,25 , and silicene NRs 26–29 . Some of these find ZT ≥ 1, which is a benchmark value indicating high potential for thermoelectric applications. Various methodologies were employed to calculate electronic and thermal properties across these studies. Most comparable to our results are those from two fully quantum mechanical studies in structurally dislocated graphene chevron nanoribbons (CNRs) and narrow armchair silicene NRs, both of which found a ZT > 1 in at least one system 14,29 . In this paper, we use ab initio-based nonequilibrium Green’s function (NEGF) method to analyze how thermoelectric properties change in CNRs as material and width vary. CNRs are chosen owing to their edgeengineered geometry that can result in reduction of phonon transport because of high edge-length to surface-area ratio. It is known that phonon-boundary scattering is a limiting factor for phonon-contributed thermal conductance in narrow systems 7 . We consider graphene NRs along with BN and silicene ones, which were chosen for their low thermal conductances as mentioned above. Two widths . 1 nm are chosen, based on the observation that thermal conductance increases with increasing width in pseudo-one-dimensional graphene nanoribbons 30–32 . Chevron graphene nanoribbons were synthesized in 2010 by a bottomup approach based on etraphenyl-triphenylene monomers as precursors on a Au(111) substrate 33 . Smaller precursors could be used to produce ones with our geometries. Production may also be achieved through a top-down

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approach like using OH radicals as “chemical scissors,” which are capable of cutting graphene sheets, grown on arbitrary substrates, into any shape 34 . This second method could also be applied to silicene sheets grown on Ag 35 or BN sheets grown on Ni 36 . Our results show systematic improvement of thermoelectric figure of merit in edge-engineered CNRs relative to straight ones.

Methods We consider hydrogen-passivated CNRs of two different widths . 1 nm for each of graphene, BN, and silicene. One unit cell of each structure is shown in Fig. 1. To assess electronic and lattice vibrational properties based on ab initio methods, structure optimizations were performed by density functional theory (DFT) analyses employing the Becke, 3-parameter, Lee-Yang-Parr (B3LYP) method and 6-31G(d) basis set in gaussian 09 37–39 . Clusters containing five unit cells and 192 atoms each were optimized for the narrow ribbons, while sevenunit cell structures with 250 atoms were optimized for the wider ribbons. The greater widthto-length ratios of wider ribbons necessitated the use of longer structures so that their central unit cells could be extracted and repeated to effectively model infinite structures. blackFor each system, we have gaussian write molecular orbital coefficients and corresponding eigenvalues, which give us the Hamiltonian eigenvalues and eigenvectors. These, along with overlap and force constant matrices are obtained following optimization and input into our program tarabord 40–44 that calculates electron and phonon transmission coefficients (Te and Tp , respectively) as functions of energy, using ab initio-based non-equilibrium Green’s function (NEGF) method. The method mathematically constructs an infinite open system with a junction consisting of one unit cell and semi-infinite contact nanoribbons consisting of periodically repeated structures to the left and right. black Using the Hamiltonian (obtained from its eigenvalues and eigenvectors), overlap, and force constant matrices extracted


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Figure 1: Narrow (a) and wide (b) unit cells of graphene (i), boron nitride (ii), and silicene (iii) chevron nanoribbons (CNRs). These unit cells are repeated periodically and infinitely when ab initio-based non-equilibrium Green’s function (NEGF) method is applied for analysis. from cluster calculations, the NEGF method provides transport data for ribbons with periodic boundary conditions in transport direction. The electronic transmission is obtained using 45–47 Ge (E) = (zSJ − HJ − ΣeL − ΣeR )−1 ,

1 L1 (µ, T )2 κe (µ, T ) = L2 (µ, T ) − , T L0 (µ, T )

where σ is electrical conductance, e is the electron charge, µ is the chemical potential, T is the temperature, and Ln (µ, T ) is the Lorenz function:

(2)

and Te (E) = Tr[ΓeL Ge ΓeR G†e ],

1 L1 (µ, T ) , S(µ, T ) = |e|T L0 (µ, T )

∞

−∂fF D dE, ∂E −∞ (7) with h and fF D being the Planck constant and the Fermi-Dirac distribution function, respectively. Following geometry optimization, force constant calculations were also performed within gaussian 09 using the same method and basis set to obtain Hessian matrices containing the second derivative of energy with respect to position for each atom 39 . These matrices were used by our program 40,44 to calculate phonon transmission coefficients, Tp , as functions of frequency 50–52 , based on the NEGF method, using 2 Ln (µ, T ) = h

(3)

where Ge is the total electronic Green’s function projected onto junction unit cell, z is the complex energy, SJ and HJ are the overlap and Hamiltonian matrices corresponding to the junction unit cell, ΣeL and ΣeR are the electronic self-energies of the semi-infinite parts to the left and right sides of junction with ΓeL and ΓeR defined based on their imaginary parts, and Te is the electronic transmission coefficient 40 . We use Te to calculate the three electronic properties that factor into ZT (Eq. 1) 48,49 : σ(µ, T ) = e2 L0 (µ, T ),

(6)

Z

Te (E)(E − µ)n

(4) Gp (ω) = [(ω + iη)2 I − KC − ΣpL − ΣpR ]−1 , (8) (5) and

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Tp (ω) =

Tr[ΓpL Gp ΓpR G†p ],

cantly smaller κp relative to graphene and BN when the integral in Equation 10 is evaluated: a smaller energy range where non-zero phonon transmission coefficients occur and relatively smaller coefficients at given energies within that range. The smaller range owes primarily to system material and is physically explained by Slack’s theory, which states that lower thermal conductance values are found in structures with greater atomic masses, weaker bonds, and greater anharmonicity 64 . Silicon atoms have the greatest mass of those considered here, the sp2 -sp3 hybridization of silicene 65 causes it to have weaker bonds, and its buckled structure introduces anharmonicity into its structure, so these criteria are satisfied. The buckle heights in these silicene ribbons are in the range of 0.410.42 ˚ A, in good agreement with previous experiments and simulations 66 . Nonzero electronic transmission coefficients also span a smaller energy range in silicene compared to graphene and BN (Fig. 2 panels (a)), but this does not have the same impact on electrical conductance σ. The reason is that within the energy range with significant contribution to the integrals in Equation 7, dictated by the derivative of the Fermi-Dirac distribution, nonzero electronic transmission exists for all systems. However, the phonon energy spans in Fig. 2 panels (b) are much smaller compared to the electronic ones in panels (a). Therefore silicene’s zero phonon transmission within the energy range with significant contribution in Equation 10, dictated by the derivative of the Bose-Einstein distribution, causes smaller κp for silicene compared to graphene and BN. Suppression of phonon transmission coefficients Tp (ω) at particular energies corresponds to reduction of the number of phonon bands that span those energies. Within the ballistic transmission mode considered here, the greatest contributions to overall conductance will come from phonon bands that span larger energy ranges 67 . Because of strict two-dimensional geometries of graphene and BN systems, the out-of-plane flexural acoustic and optical modes (ZA and ZO) are decoupled from in-plane vibration modes, and the corresponding phonon bands can cross. In silicene and other buck-

(9)

where ω is phonon frequency, η is a small real number, I is the identity matrix, KC is the junction Hessian, and the rest of the functions are the phonon counterparts of the electron ones used in Equations 2 and 3. Tp was then used to calculate the phonon contribution to thermal transport via the Landauer formula, an effective means to evaluate ballistic phonon contribution to thermal conductance, κp , in quasi-one-dimensional systems 53,54 : ~ κp (T ) = 2π

Z 0

∞

∂fBE (ω) Tp (ω)ωdω, ∂T

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(10)

where fBE (ω) is the Bose-Einstein distribution function.

Results and Discussion Our electron and phonon transmission coefficients are plotted for each of our systems in Fig. 2. According to the acoustic sum rules, nanowires generally have four acoustic phonon modes at zero energy because of translational and rotational symmetries 55–59 . The necessity to impose a cutoff radius to include a finite number of nearest neighboring atoms in ab initio calculations of force constants, however, can break the general rule 57,60 and cause the phonon transmission coefficient near zero energy to reduce below four. Depending on the methodology, four or less acoustic phonon modes are reported near zero energy in studies on phonon transmission in chevron nanoribbons 19,22 and straight nanoribbons made of graphene 32,61 and other materials like boron 62 and black phosphorus 63 , as well as other quasi-1D systems 59 . The aforementioned deviation from the acoustic sum rules is restricted to energies near zero. From Fig. 2b it is observed that in our results this energy range is small compared to the whole energy range of nonzero phonon transmissions. The deviation therefore does not affect our overall results and conclusions. Two factors cause silicene to have a signifi-


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led structures, buckling breaks the mirror symmetry and therefore ZA and ZO modes can hybridize with in-plane modes 68 . This results in avoided crossings, creation of phonon band gap, and smaller energy span of phonon bands. Planar geometries therefore support bands that span greater energies relative to buckled geometries. For instance, the in-plane longitudinal acoustic (LA) band spans approximately 80% of the phonon spectrum in graphene, while it spans less than 25% of that in silicene 68 . Comparing phonon dispersion curves between graphene 67 and silicene 27 nanoribbons confirms that this phenomena is carried over to the quasi-1D case. Thermoelectric properties were calculated for each system at temperatures of 300 and 600 K as functions of chemical potential. For pristine and unbiased systems, the intrinsic chemical potential was taken to be at the center of the electronic band gaps present in all six systems. The band gaps are presented in Table 1. For conventional armchair and zigzag nanoribons of graphene 69 , BN 70 , and silicene 71 , the band gaps generally decrease with increasing width. Armchair nanoribbons of all types show band gap fluctuations as well, causing nanoribbons with particular width indices to have small band gaps even at small widths 69–71 . Despite the fact that chevron nanoribbons cannot be identified as conventional armchair or zigzag type, the band gaps presented in Table 1 are comparable to those reported previously with similar widths. In particular, band gaps increase from silicene to graphene to BN consistent with previous studies. For doped and/or biased systems, the chemical potential can deviate from the center of the gaps. In the following discussions, the intrinsic chemical potential is shifted to zero. Figure 3 demonstrates how thermoelectric properties change with respect to chemical potential in our silicene systems at both temperatures. While this sample has a high concentration of ZT peaks over a small energy range, it displays a feature that is generally seen in all systems at both temperatures: ZT has many peaks along ranges where the Seebeck coefficient is relatively large while κe is small relative to κp . Changes in thermoelectric

properties of our graphene and BN systems are presented in Figs. S1 and S2 in Supporting Information, respectively. From Fig. 3 we notice that silicene displays asymmetries in its ZT behavior when it is prather than n-doped. Higher ZT s are generally found at negative chemical potentials, though more small peaks are present at positive ones. Owing to its small band gap and low phonon thermal conductance, ZT s near or exceeding 1 are found for both widths at both temperatures within ±1 eV of the center of the gap. This would be beneficial as small changes in chemical potential are possible through moderate doping or bias. Silicene CNRs also show more frequent Seebeck coefficient peaks per unit energy, resulting in more ZT peaks relative to graphene and BN (not shown in Fig. 3). The ZT peaks in silicene are generally higher than those in other systems due to relatively low κp . Despite the explicit dependence of ZT on temperature in Eq. 1, the ZT peaks at 600 K are less than double the corresponding 300 K ones, owing to reduction of Seebeck coefficient and increase of κp . Therefore identifying materials and system geometries with high thermoelectric performance at room temperature is an approach that should be preferred over simply increasing temperature to maximize ZT . High Seebeck coefficients within the band gap that do not coincide with nonzero electrical conductance are not relevant for thermoelectric purposes. However, the narrowing of the band gap at high temperatures does mean that the ZT peak nearest the band gap will become taller and wider as a result. For graphene CNRs, our results show that the effects of changing chemical potential are largely similar at 300 and 600 K, and are fairly symmetrical on both sides of the band gap. They are almost completely symmetrical in our wide nanoribbon. In both narrow and wide systems, almost no variation is seen in electrical conductance between temperatures. However, the peaks surrounding Seebeck coefficient sign changes are more pronounced at 300 K than at 600 K, and κe peaks are seen to approximately double with temperature, with


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Figure 2: (a) Electron transmission coefficients and (b) phonon transmission coefficients with respect to energy in (i) graphene, (ii) boron nitride, and (iii) silicene chevron nanoribbons. Centers of the energy gaps (intrinsic chemical potentials) are shifted to 0 eV in (a) panels.

Table 1: Electronic Band Gaps [eV] of Narrow and Wide Graphene, BN, and Silicene Chevron Nanoribbons. System Narrow Wide

Graphene BN Silicene 2.77 6.39 1.05 1.44 6.20 0.71


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Figure 3: Thermoelectric properties of narrow and wide chevron silicene nanoribbons near the band gap, at 300 and 600 K. Intrinsic chemical potential, i.e., that of pristine and unbiased systems, is at the center of the gap and shifted to zero. slightly higher increases at negative chemical potentials. There are p-type and n-type ZT peaks having nearly equivalent values in both graphene systems, which may enhance the flexibility of graphene CNRs in thermoelectric applications. As for BN CNRs, the thermoelectric properties vary somewhat more as chemical potential changes, relative to those of graphene. ZT has a more asymmetric nature in BN on each side of the band gap: There are no peaks > 0.2 in any BN system between -6 and -4 eV, but four peaks are observed in narrow BN from 4 to 6 eV. At these high chemical potential levels the narrow BN CNR actually performs slightly better at 300 K, owing to a combination of Seebeck coefficient sign changes and very low κe , both of which are much less pronounced in the wide BN CNR. Electrical conductance shows little variation with temperature for BN CNRs, albeit greater differences than in graphene at different temperatures. In general, high ZT s are found in systems with low κp , which changes with temperature but not chemical potential, and thus serves as a limiting factor in our figure of merit calculations. Maximum ZT values within ±5 eV of

the band-gap center, together with corresponding thermoelectric characteristics, are shown in Table 2 for all systems at 300 K. Such shifts of chemical potential from zero can be achieved, e.g., by applying a gate bias. In devices based on graphene nanoribbons, gate biases on the order of 10 V 72 and 20 V 73 have been used. As the thickness of oxide layer is much smaller than the lateral dimension of highly doped silicon wafer used as gate, the electric potential at the surface of the oxide layer, that is at the location of the nanoribbon device, can be considered to be that of an infinite, charged plane held at constant potential. Assuming that the silicon oxide layer completely covers the gate surface and that the oxide thickness is enough to act as bulk in screening electric potential, the potential at the surface of the oxide will be V0 /r , where V0 is the potential at the gate surface and r is the relative permittivity of the oxide (for silicon oxide r ' 3.9). The presence of silicon oxide layer therefore converts ∼ 10 − 20 V gate bias to ∼ 2.5 − 5 V at the location of the nanoribbon. This estimate is consistent with the result of a previous study on electrostatic lever arm of carbon nanotube field effect transistors 74 .


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Table 2: Thermoelectric Properties of Narrow and Wide Graphene, BN, and Silicene Chevron Nanoribbons at 300 K.a System Narrow graphene Narrow BN Narrow silicene Wide graphene Wide BN Wide silicene

µ [eV] max ZT −1.98 0.77 2.00 0.77 −3.65 0.84 3.87 0.97 −1.08 1.91 1.41 1.08 −0.70 0.34 0.69 1.05 −3.27 0.38 3.94 1.02 −4.64 2.56 −0.32 1.02 0.32 1.02

σ[µS] 29.5 32.3 23.5 33.3 28.2 14.8 20.1 14.6 19.6 42.6 57.1 26.5 26.5

S [µV/K] 232 −223 −236 −219 −236 240 −201 232 −204 −233 247 −217 217

κe [nW/K] κp [nW/K] 0.055 0.560 0.061 0.037 0.432 0.063 0.053 0.193 0.044 0.039 0.682 0.027 0.038 0.600 0.083 0.093 0.051 0.315 0.051

a

The listed properties are at chemical potentials where maximum ZT values occur within ±5 eV from the center of the band gap (set at zero). Two p-type maxima are given for wide silicene; the larger one is both remarkably high and far from gap center, unlike that of any other system.

As mentioned before, CNRs have high edgelength to surface-area ratios that can reduce κp due to phonon-boundary scattering. In order to assess how κp in our graphene CNRs compares to that in non-chevron ribbons, we can compare with very thin zigzag edged nanoribbons (ZNR) that make up the straight sections of our CNRs. Tan, et al. used a similar quantum mechanical approach to find κp values of approximately 0.8 and 1.1 in 2- and 3-atom wide ZNR, respectively 32 . Our corresponding values from Table 2 suggest a reduction in κp of more than 30% based on chevron shape. An alternative approach is to compare with slightly wider armchair-edged nanoribbons (ANR) that our CNRs could be carved out of. Tan, et al. report κp of ∼0.8 and, by extrapolation, ∼1.0 nW/K for 6- and 7-atom wide ANRs, and we again see a > 30% reduction in κp for each case. For silicene systems, Yang, et al. find κp values of approximately 0.4 in 3-ZNR and 6- and 7ANRs; the values in our narrow and wide edgeengineered CNRs are about 50 and 25% lesser in magnitude 29 . To our knowledge, no fully quantum mechanical study has been conducted on phonon-contributed conductance in BN NRs of

these sizes. Comparing our κp values to those obtained from BN NRs using the tight-binding method suggests a similar trend, although they are not directly comparable due to the different methodologies used 75 . The results that we present here show that by considering smaller widths and a variety of materials, it is possible to achieve systematic enhancement of figure of merit based on accurate ab initio calculations. Some general observations and comparisons of ZT components are as follows: In all systems, we see σ as a function of chemical potential remain fairly constant with regard to temperature, while κp invariably rises. Electrical conductance changes little in graphene of either width at different temperatures, while in BN and silicene it tends to fluctuate more at low temperatures, especially in our wider structures. The Seebeck coefficient varies less with energy at higher temperatures and widths across systems. Width suppresses S more than temperature in graphene and BN, while temperature has a greater impact than width in silicene. Our graphene ribbons show very few ZT peaks for more than ±3 eV from the center of the band gap because their S rarely vary


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when away from the gap, even at energies where the ratio of σ to κe is relatively high. Their possession of the highest κp of any of our systems makes this ratio additionally unfavorable. Our narrow BN ribbon is expected to perform slightly better, and our narrow silicene ribbon exceeds these by more than a factor of two. In BN CNRs, we observe that our wider system generally has large ranges of high κe with few reductions to the order of magnitude of its κp , causing it to have figures of merit consistently lower than its narrow counterpart. A notable exception is near +4 eV, where low κe in the wider system result in ZT peaks that do not exist in the narrow ribbon. These ZT peaks do not occur right at the edge of the band gap but are also not far from it, which is a unique feature among the three wide CNRs that we considered. The variation of width in silicene produces two features not seen in other systems: We notice that relatively large Seebeck coefficient peaks ±10 eV from the center of the band gap in narrow silicene are, especially at negative chemical potentials, often fairly closely aligned with Seebeck magnitude in wider silicene NRs; this behavior is uncommon in graphene and BN. Full-width at half-maximum (FWHM) values of ZT peaks versus chemical potential indicate sensitivity of the peaks to small changes in doing and/or bias. FWHM values for all systems are generally about 0.06-0.09 eV at 300 K and 0.15-0.19 eV at 600 K. Peaks with ZT ≥ 1 tend to have FWHM values in the upper limits of these ranges. This suggests that FWHM values are generally more dependent on temperature than system width or material. Furthermore, when chemical potential may be prone to fluctuations, maintaining high ZT s will be more feasible at high temperatures. Phonon contributions to thermal conductance may contain anharmonic effects at higher temperatures. An ab initio study has found that anharmonic effects become significant at a strain of approximately 0.1 in BN nanoribbons 76 . Previous studies on graphene 77 , BN 78 , and edge-passivated silicene 27 have shown that variations of lattice parameters are much smaller than 10% for 0 < T < 600 K. Fur-

thermore, we previously showed that relative bond-length variations in lithiated silicene do not reach or exceed 10% unless the temperature is 900 K or higher 79 . Therefore we do not expect significant anharmonic effects in our systems as we consider them at or below 600 K. Figure 4 displays our systems’ maximum ZT values when they are p- and n-doped. One feasible method of doping would be introducing physisorbed dopants which would not change the electronic properties of the system in the way that covalently bonded dopants would, e.g., using tetracyanoquinodimethane (TCNQ; p-type) or tetrathiafulvalene (TTF; n-type) for silicene 28 . Identifying particular adsorbtion candidates require special care, as we previously found that absorption of a single CO molecule on a pristine silicene NR significantly alters its quantum conduction 80 . In the CNRs considered here, ZT magnitudes are most commonly similar regardless of type of doping. However, wide BN CNR has a high ZT at 300 K when it is n-doped, a peculiarity that could lend it appeal over a system like narrow graphene, which otherwise generally performs better. Silicene displays some interesting variations, with its narrow system having approximately the same p-type maximum ZT at 300 and 600 K, while displaying the more expected behavior of increasing ZT with temperature when n-doped. Wide silicene has the typical relatively high ZT near the edge of its band gap, but also has a ZT equal to 2.56 at -4.64 eV. This is due to notably high σ and Seebeck coefficient, greater than those seen at any other system’s max ZT within ±5 eV. Besides its magnitude, this value is unusual in that it occurs in both a wide structure and at at lower temperature. It should be noted that by considering an even wider range like ±10 eV, we do see relatively large ZT peaks emerge far from the gap in other systems. In this expanded range, silicene consistently outperforms the other two systems, and our very highest ZT value is 2.70 in narrow silicene at 600 K and -9.72 eV. Another consideration regarding practical applications is the sign of the Seebeck coefficient, which is irrelevant for the calculation of fig-


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Figure 4: Variation in maximum thermoelectric figures of merit under (a) p-type doping and (b) n-type doping with respect to temperature, for our narrow and wide graphene, BN, and silicene chevron nanoribbons, within ±5 eV from the intrinsic chemical potential (centered in the band gap, and set at zero). Maximum ZT peak occurs for p-type wide silicene and happens far from the gap, unlike other peaks that occur near the band gap edge. ure of merit but is related to the directions of thermal and electric current. Sample Seebeck coefficients at 300 K are provided in Table 2. Seebeck coefficient signs associated with maximum ZT at one temperature will not necessarily be the same at another because temperature causes these peaks to shift, emerge, or diminish. A material-specific feature that can affect ZT is application of an external electric field; which can widen the band gap of a silicene NR, but not a graphene one 81–83 . This could shift the ZT peaks away from the energy where one would otherwise expect to see them. Figure 5(a) displays how a typical high ZT peak, specifically the peak at -0.3 eV in wide silicene CNR at 300 K, emerges as a result of its components’ behavior. Each plotted component is normalized based on its behavior from -0.5 to -0.1 eV, where 1 is its maximum value across that range (these values can be found in Table 2). One factor contributing to the presence of a ZT peak is proximity to a Seebeck coefficient peak, regardless of whether S is increasing or decreasing, positive or negative. The second, more interesting, factor is proximity to chemical potentials at which electrical and electron-contributed thermal conductance

shift away from one another. These two generally increase or decrease along with each other, but some differences can be observed, such as greater variance at a peak in σ relative to the corresponding peak in κe , or the apparent lag effect seen in Fig. 5(a). When this happens, a ZT peak will emerge at the intersection of normalized σ and S 2 , the two terms that directly enhance the figure of merit. Higher ZT peaks will occur at chemical potentials where this lag behavior occurs closer to a Seebeck coefficient peak. These effects are material- and geometrydependent, therefore providing a possibility for engineering thermoelectric applications. Phonon contribution to thermal conductance depends only on temperature, as shown in Fig. 5(b). This limits the maximum ZT that can emerge when its other components interact as was just described. As Table 2 indicates, ZT peaks consistently occur at energies where electron-contributed thermal conductance is approximately one order of magnitude less than the phonon contribution. As we see in Fig. 3(d), the rarity of this occurring limits the number of high ZT s that can occur across an energy range. Figure 5(b) clearly shows that choice of either width or material


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Figure 5: (a) Normalized values of the thermoelectric figure of merit and its components near the peak at -0.3 eV in wide silicene CNR at 300 K, where 0 eV is the system’s intrinsic chemical potential. (b) Phonon-contributed thermal conductance (κp ) for each system as a function of temperature. can be the factor that results in a lower κp . Silicene has lower κp than any graphene or BN system, and a narrow CNR of either graphene or BN will have a lower κp than a wide one. Here we focused on the nanoribbons’ materials and geometries for improving thermoelectric performance. In experimental situations, other system factors could affect overall device performance and therefore would need to be carefully understood. The basic idea is to determine how much change is induced in electrical and thermal currents through the materials under experimental conditions. Prior studies have considered some of these effects: A continuum bulk model applied to thin film thermoelectric materials deposited on a substrate concluded that intrinsic ZT values are reduced more in presence of a thick conducting substrate compared to a thin insulating one 84 . Another study using classical atomistic simulations showed that thermal transport in black phosphorene supported by a substrate was reduced compared to a suspended one 85 . In this case ZT can increase in the presence of a substrate as it effectively acts as a phonon heat sink, assuming that possible electron tunneling across the van der Waals distance results in (less significant) reduction in both electrical and electron-contributed thermal currents. Considering an interface of metal electrodes attached to quantum dots, it was concluded that adjust-

ing energy levels in quantum dots relative to the Fermi level of electrodes can tune ZT 86 . A similar effect was reported for semiconducting electrodes attached to single-molecule junctions 87 . Electron-phonon interaction, which is not considered here, may cause the Seebeck coefficient to diminish and shift towards the band gap, resulting in decreased ZT , particularly in internally disordered systems 88 . This shift would also change the chemical potential that maximizes ZT . On the other hand, edge disorder is known to decrease phonon-contributed thermal conductance 44 , so potential trade-offs between this effect and any induced electronphonon scattering would need to be analyzed.

Conclusions We used ab initio-based non-equilibrium Green’s function method to study the thermoelectric functionality of edge-engineered chevron nanoribbons with two different widths, each constructed out of graphene, BN, or silicene. We found that all six systems had thermoelectric figure of merit (ZT ) peaks distributed near the edges of their band gaps. The magnitudes of these peaks were usually greater in narrow systems, and were consistently highest in silicene. Even greater ZT values can be found far from the band gap.


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Factors that affect ZT are identified and analyzed, in particular, separation of electrical and electron-contributed thermal conductance versus chemical potential as well as lower electron-contributed thermal conductance than the phonon contribution. These effects are material- and geometry-dependent, therefore providing possibilities for engineered thermoelectric applications. While choosing silicene over the other two materials always resulted in a higher ZT , either width or material could be the key property when choosing between graphene and BN. Our highest near-band gap ZT of 1.91 in narrow silicene at 300 K suggests that it holds particular promise for thermoelectric applications. Geometry engineering and material selection are shown to provide possibility for systematic ZT enhancement in nanomaterials that could be instrumental in future competitive thermoelectric devices.

(5) Venkatasubramanian, R.; Siivola, E.; Colpitts, T.; O’Quinn, B. Thin-Film Thermoelectric Devices with High RoomTemperature Figures of Merit. Nature 2001, 413, 597–602.

Acknowledgement L. J. W. acknowledges financial support through a first-year assistantship grant from the Ph.D. in Engineering program at Wright State University. Computational resources were generously provided by the Ohio Supercomputer Center. Supporting Information. Figures like Fig. 3 for graphene and BN.

(8) Balandin, A. A.; Ghosh, S.; Bao, W.; Calizo, I.; Teweldebrhan, D.; Miao, F.; Lau, C. N. Superior Thermal Conductivity of Single-Layer Graphene. Nano Lett. 2008, 8, 902–907.

(6) Ryu, H. J.; Aksamija, Z.; Paskiewicz, D. M.; Scott, S. A.; Lagally, M. G.; Knezevic, I.; Eriksson, M. A. Quantitative Determination of Contributions to the Thermoelectric Power Factor in Si Nanostructures. Phys. Rev. Lett. 2010, 105, 256601. (7) Hochbaum, A. I.; Chen, R.; Delgado, R. D.; Liang, W.; Garnett, E. C.; Najarian, M.; Majumdar, A.; Yang, P. Enhanced Thermoelectric Performance of Rough Silicon Nanowires. Nature 2008, 451, 163–167.

(9) Ghosh, S.; Calizo, I.; Teweldebrhan, D.; Pokalitov, E. P.; Nika, D. L.; Balandin, A. A.; Bao, W.; Miao, F.; Lau, C. N. Extremely High Thermal Conductivity of Graphene: Prospects for Thermal Management Applications in Nanoelectronic Circuits. Appl. Phys. Lett. 2008, 92, 151911.

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Systematic Enhancement of Thermoelectric Figure of Merit in Edge

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