Designing Two-Dimensional Dirac Heterointerfaces of Few-Layer

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Designing Two-Dimensional Dirac Heterointerfaces of Few-Layer Graphene and Tetradymite-Type SbTe for Thermoelectric Applications 2

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Woosun Jang, Jiwoo Lee, Chihun In, Hyunyong Choi, and Aloysius Soon ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.7b09805 • Publication Date (Web): 08 Nov 2017 Downloaded from http://pubs.acs.org on November 8, 2017

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ACS Applied Materials & Interfaces

Designing Two-Dimensional Dirac Heterointerfaces of Few-Layer Graphene and Tetradymite-Type Sb2Te3 for Thermoelectric Applications Woosun Jang,† Jiwoo Lee,† Chihun In,‡ Hyunyong Choi,‡ and Aloysius Soon∗,† †Department of Materials Science & Engineering, Yonsei University, Seoul 03722, Korea ‡School of Electrical & Electronic Engineering, Yonsei University, Seoul 03722, Korea E-mail: [email protected]

Abstract Despite the ubiquitous nature of the Peltier effect in low-dimensional thermoelectric devices, the influence of finite temperature on the electronic structure and transport in the Dirac heterointerfaces of few-layer graphene and layered tetradymite, Sb2 Te3 (which coincidently have excellent thermoelectric properties) are not well understood. In this work, using first-principles density-functional theory calculations, we investigate the detailed atomic and electronic structure of these Dirac heterointerfaces of graphene and Sb2 Te3 , and further re-examine the effect of finite temperature on the electronic band structures using a phenomenological temperature-broadening model based on Fermi-Dirac statistics. We then proceed to understand the underlying charge redistribution process in this Dirac heterointerfaces and through solving the Boltzmann transport equation, we present theoretical evidence of electron-hole asymmetry in its electrical conductivity as a consequence of this charge redistribution mechanism. We

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finally propose that the hexagonal-stacked Dirac heterointerfaces are useful as efficient p-n junction building blocks in the next-generation thermoelectric devices where the electron-hole asymmetry promotes thermoelectric transport by “hot” excited charge carriers.

Keywords Thermoelectrics; Graphene; Topological Materials; Heterointerfaces; Density-Functional Theory; Finite temperature Statistics; Fermi-Dirac Distribution

1

Introduction

After the successful exfoliation of graphene, many researchers have attempted to exploit the exotic electronic structure of this two-dimensional (2D) nanomaterial 1–3 by tuning and modulating the energy levels of its linear Dirac cone. The same can be said for another class of Dirac materials – the topological insulators (TIs) where their time-reversal symmetryprotected topological surface states exhibit a linear Dirac cone energy-momentum dispersion, and they are also extensively investigated for their equally fascinating quantum properties. 4–7 Graphene was first suggested as an almost ideal material for thermoelectric applications, given its high flexibility, carrier mobility and theoretical ZT value. 8,9 Unfortunately, pristine graphene alone could not be used independently in thermoelectric devices due to high thermal conductivity. 10 Recent studies have also demonstrated and discussed that the stacking sequence and thickness of multi-layered graphene can strongly affect its electronic structure in different unexpected ways. For instance, AA-stacked graphene shows two separated Dirac cones with an enlarged energy band gap, while the AB-stacked (or the so-called Bernalstacked) graphene displays parabolic bands rather than preserving its sharp linear Dirac cones. 11 Similarly, for the trilayer graphene system, the hexagonal- (AAA-), Bernal- (ABA), and rhombohedral- (ABC-) stacked graphene systems show very distinct electronic band 2 ACS Paragon Plus Environment

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dispersions as compared to its single-layer counterpart. 12,13 It was only recently reported that so-called van der Waals epitaxial heterointerface structures (composed of graphene and metal chalcogenides) exhibit a dramatic improvement over the pristine components, owing to its very high thermoelectric power factor. 14,15 These van der Waals epitaxial heterointerface systems may indeed be a game changer in the field of thermoelectrics. Motivated by experimental advances in synthesis methods for the 2D Dirac multi-layered graphene-based heterojunctions, the graphene/TI heterostructures are intensively studied. 16–18 Owing to a small lattice mismatch (∼ 1 to 3 %) and the presence of heavy atomic species, TIs such as Sb2 Te3 and Bi2 Te3 are widely chosen as a the contact substrate for graphene. Much attention has also been given to the carrier transport behavior in these 2D heterostructures where the enhanced spin-orbit coupling (SOC) and proximity effects are reported to be beneficial. 19,20 Specifically, in epitaxially-grown graphene-Sb2 Te3 heterointerfaces, the presence of graphene aids the growth of Sb2 Te3 with superior crystallinity (where previously known structural defects are greatly minimized). 15 This high crystallinity of epitaxially-grown Sb2 Te3 films results in low defect concentration (and thus, low carrier concentration) which then explains the high Seebeck coefficient observed. 21 Specifically, the heterointerface of graphene-Sb2 Te3 boasts a 38 and 413 % increase in its Seebeck coefficient (when compared to pristine Sb2 Te3 and graphene, respectively), 15 while its thermoelectric power coefficient has a colossal increase of 256 and 111 % when compared to pristine Sb2 Te3 and other Sb2 Te3 -based thin films. 15 However, the direct measurements of the thermoelectrical properties in these graphene/TIbased systems are often technically challenging due to the natural occurrence of the Peltier effect which is caused by the temperature gradient between two current probes when an electrical current passes through the sample. Specifically, this temperature gradient set up via the Peltier effect in turn generates a Peltier voltage that results in a less accurate determination of the measured voltage. The Peltier voltage for a small amount of current is



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normally the same order of magnitude as the resistive voltage. 22–24 Moreover, other technical uncertainties, e.g. the exact positions of the probes and modes of contact, may further hinder a precise measurement of the electrical conductivity of these thermoelectric materials. 25,26 Given the ubiquitous nature of the Peltier effect, 27 it is somewhat unsettling to know that a complete microscopic (i.e. atomic-scale) understanding and description of these electrons with a finite temperature in low-dimensional thermoelectric nanomaterials is still very much lacking. However, it becomes immediately apparent that a rigorous theoretical treatment of these heavy-metal-containing heterointerfaces (e.g. SOC-inclusive electron-phonon coupling calculations from first-principles 28 ) can easily become computationally prohibitive. Thus, as a first step in this work, we will perform first-principles density-functional theory (DFT) calculations in conjunction with a phenomenological model to re-examine the finite temperature electronic band structures of few-layer graphene/Sb2 Te3 heterointerfaces.

2

Methods

All density-functional theory (DFT) calculations are performed using periodic boundary conditions, with the projector-augmented wave (PAW) 29,30 method as implemented in the Vienna Ab initio Simulation Package (VASP). 31–33 The generalized gradient approximation due to Perdew-Burke-Ernzerhof (PBE) 34 is used to treat the exchange-correlation energy, augmented with the Grimme’s D2 van der Waals correction (PBE+D2) scheme. 35 For the Brillouin zone integrations, a Γ-centered k-point grid of 12×12×1 is used for all calculations, with the exception of a much denser grid of 24 × 24 × 1 employed for solving the semiclassical Boltzmann transport equation within the constant relaxation time approximation, as implemented in the BoltzTraP program. 36 All DFT calculations are carefully tested for convergence with respect to the kinetic energy cutoff, k-point grid, vacuum region between repeated slabs, and substrate quintuple layer (QL) thickness, where the total energies and forces do not vary for more than 20 meV and 0.02 eV˚ A−1 . A planewave kinetic energy cutoff


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Figure 1: (Color online) (a) A bird’s eye view of the graphene/Sb2 Te3 heterointerface. (b) Top-view of the stacking sequence and geometry of the multi-layer (i.e. using layers A, B, and C) graphene. (c) Side-view of the bilayer graphene on 4 QL Sb2 Te3 (showing only 1 QL of Sb2 Te3 ). The vertical distances, labeled d1 and d2 represent the interlayer spacings between the adjacent graphene layers and the graphene-Sb2 Te3 interface, respectively. The C, Sb, and Te atoms are depicted as black, gray, and purple spheres, accordingly. of 400 eV and a vacuum separation of at least 20 ˚ A in the z-direction are employed for the slab calculations. Spin-orbit coupling (SOC) is considered only for the electronic structure analysis. We relax the atomic coordinates for all structures and since asymmetric slabs are used in this work, a dipole correction (along the z-direction) is employed for all slab calculations.

3 3.1

Results and discussion Atomic structures and interfacial energetics

To investigate the few-layer graphene/TI heterointerface structure, antimony telluride, Sb2 Te3 (where its PBE+D2 optimized a0 = 4.24 ˚ A) – one of the layered tetradymite (V2 VI3 ) – is se-



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lected as a representative TI substrate for graphene. In order to minimize the effect of lateral √ √ strain of graphene on Sb2 Te3 , a 3 × 3 R30◦ surface supercell of few-layer (i.e. monolayer, bilayer and trilayer) graphene is stacked on Sb2 Te3 (see Figure 1a). Here, the surface lattice constant, a0 of stacked graphene is 4.27 ˚ A, yielding only a small lattice mismatch of 0.7 % for the graphene/TI heterointerfaces considered in this work. Thus, we choose to fix the interface lattice parameter of the graphene/TI to that of the TI. For the single-layer graphene (so-called the “A” form), it has been determined to be the most energetically favored (and experimentally observed) interface structure where the outermost Te surface atom of Sb2 Te3 is located at the center of the hexagonal ring of carbon atoms in graphene (so-called P1 site). 17,19,37 Both the Bernal (AB) and hexagonal (AA) in the bilayer graphene 38–40 are then considered by building on top of the single-layer graphene. This is followed by examining the hexagonal (AAA), Bernal (ABA), and rhombohedral (ABC) forms of the trilayer graphene 41–43 stacked on the substrate (as shown in Figure 1b). To aid our discussion, we intuitively label our graphene/TI heterointerfaces as e.g. grABC /Sb2 Te3 for the rhombohedral trilayer graphene on Sb2 Te3 . The interlayer spacings between the adjacent graphene layers, and between the bottommost graphene and the uppermost surface Te atom of Sb2 Te3 are measured and reported as d1 and d2 , respectively (see Figure 1c). We find that although d2 remains almost constant (i.e. 3.46 ˚ A) for all the few-layer graphene/Sb2 Te3 heterointerfaces considered here, a larger d1 value of 3.50 ˚ A is found for the hexagonal bilayer (AA) and trilayer (AAA) graphene on Sb2 Te3 as compared to the smaller measured value of 3.25 ˚ A for the others. Our calculated structural parameters for the stacked graphene layers agree well with previous reports. 19,44,45 In order to determine the relative adsorption/adhesion strength of the few-layer graphene on Sb2 Te3 as a function of thickness and stacking sequence of the graphene, we calculate the work of adhesion, Wad normalized by the interface area, A, according to:

Wad =

) 1 ( slab i intf ESb2 Te3 + Egr − Egr i /Sb2 Te3 A 6

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,

(1)

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slab i intf where ESb , Egr , and Egr are the total DFT energies of the pristine 4 QL Sb2 Te3 slab, 2 Te3 i /Sb2 Te3

the free-standing few-layers graphene (with i denoting the different A, AA, AB, AAA, ABA, and ABC stacking sequence), and the combined few-layer graphene/Sb2 Te3 heterointerfaces, respectively. As shown in Table S1 in the Supporting Information, the value of Wad is found to be higher for the few-layer graphene on Sb2 Te3 as compared to that for the single-layered case. In particular, the Wad of the bilayer graphene/Sb2 Te3 systems are slightly higher than that of the trilayered ones (especially within the same stacking order). This may be rationalized by the weak repulsive interaction between the second and third graphene layers where net negative charges build up (see Figure 3c), and thus lowering the Wad of the trilayered systems. Taking the relative Wad for amongst the bilayer (and trilayer) graphene/Sb2 Te3 heterointerfaces, and the well-known weak van der Waals bonding between the graphene layers, we find only marginal differences (< 0.5 meV ˚ A−2 ) and may conclude that the stacking orders do not strongly affect the energetics of these heterointerfaces. We would like to further suggest that with such small energy differences, thermal contributions (even at room temperature and above) may well easily change the specific graphene stacking sequence for the bilayer and trilayer systems. Thus, in this work, we proceed to survey the specific electronic structure and properties of these few-layer graphene/Sb2 Te3 heterointerfaces with the inclusion of SOC and thermal effects.

3.2

Electronic band structures of few-layer graphene and Sb2 Te3

Before considering the electronic structure of these few-layer graphene/Sb2 Te3 heterointerfaces, we will discuss briefly on the electronic structure of the TI substrate, Sb2 Te3 as well as free-standing few-layer graphene. From our previous work 46 and other reports, 19,47 the band structure (and topology) of the layered tetradymite (V2 VI3 ) family is known to show a strong correlation not only on its stacking sequence but also on its number of QL where the zero-gap topologically non7 ACS Paragon Plus Environment


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trivial surface states will only emerge when this number exceeds a certain critical value. The fundamental band gap of Sb2 Te3 is reported to be around 0.14 ∼ 0.28 eV. 48 With the inclusion of SOC effects, Sb2 Te3 shows a drastic decrease in the band gap energy as its number of QL increases, and forms a non-trivial topological surface state at Γ-point when the thickness of film became thicker than its topological critical thickness. Specifically, this minimum critical QL number for Sb2 Te3 is reported to be around 4 to 5. 19,46,47 Hence, we have opted to use a minimum 4 QL Sb2 Te3 model in this work as the substrate for stacked graphene to investigate the influence of Sb2 Te3 ’s non-trivial band topology on graphene. The electronic band structure (with SOC) of 4 QL Sb2 Te3 is plotted in Fig. S1 of the Supporting Information. Likewise, the electronic structure of free-standing few-layer graphene is known to show a strong dependency on both the number of atomic layers as well as its stacking sequence. 12,13 Other structural modifications e.g. morphological defects and mechanical strains have also been noted to exert an influence on its electronic properties. 12,13 As are clearly shown in our calculated band structures of free-standing few-layer graphene structures (in Supporting Information, Figs. S1 and S2), and in agreement with other reports, 12,13 the hexagonal stacked graphene, i.e. grA , grAA , and grAAA display their Dirac cones (as a multiple of the number of stacked layers) near the Fermi level. A micro-gap opening is also found for grAA (30 meV) and grAAA (10 meV), within the PBE+D2 approximation. This is in contrast to that found for the Bernal and rhombohedral graphene where clear parabolic bands are found near the Fermi level, with the known exception for grABA which possesses both the parabolic and linear energy dispersions. 12,13

3.3

Phenomenological temperature-broadening model

Now, we turn our attention to the electronic structure of these few-layer graphene/Sb2 Te3 heterointerfaces, and how finite temperature effects may potentially alter their interface band structures. 8 ACS Paragon Plus Environment

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Figure 2: (Color online) Temperature-broadened electronic band structure of various few-layer graphene/Sb2 Te3 heterointerfaces: (left to right) grA /Sb2 Te3 , grAA /Sb2 Te3 , grAB /Sb2 Te3 , grAAA /Sb2 Te3 , grABA /Sb2 Te3 , and grABC /Sb2 Te3 , with a finite temperature of 0 K (upper panel), 300 K (middle panel), and 600 K (bottom panel) presented (with the SOC effect included). The vertical energy scale is set to 0 at the Fermi level, ϵf . The effect of finite temperature is approximated and applied using a convolution between a Fermi-Dirac smearing function and a normal distribution function (cf. Equations 2 to 4). To consider thermoelectric transport in the graphene/Sb2 Te3 heterointerfaces, a rigorous theoretical approach to account for all electron-phonon interactions from first-principles 28 is generally desired but computationally demanding. Instead, we approach this problem by phenomenologically modelling the temperature effect via a statistical treatment of temperaturebroadened electron occupancies in our band structure calculations at specific temperatures. 49 According to the Fermi-Dirac statistics, an elevated temperature results in a broadened distribution of electrons. To include the effect of temperature in our band structure calculations, we employ a Fermi-Dirac smearing function (which mimics the grand-canonical



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extension to DFT due to Mermin 50 ) to approximate this broadening effect, 51,52 and use a Gaussian normal distribution to represent weight in momentum-energy space. We then use a convolution between the two distribution functions to obtain a temperature-broadened projected electronic band structure. Specifically, once the DFT eigenvalues in the ground-state projected band structures are determined, we apply this temperature broadening in two steps. We first project the band occupancies via a normal distribution, Γ along the energy axis, E using ) ( 1 E2 Γ = √ exp − 2 , 2σ σ 2π

(2)

where our broadening parameter, σ is defined as the product of the Boltzmann constant and absolute temperature (i.e. kB T ). In a similar fashion to the previous work using the Gaussian smearing function, 49 we adopt the smearing function based on the Fermi-Dirac distribution 50 with the following equation,

( f

E−ϵ kB T

) (

= exp

1 E−ϵ kB T

)

,

(3)

+1

where ϵ denotes the eigenvalues. Then we perform the final weighted-projection upon the energy axis for each k-point with

ξ(E, T ) = ωf ⊗ Γ .

(4)

Here, the temperature-dependent broadening of the band occupancy is reflected by the convolution of the Fermi-Dirac smearing and normal distribution functions (i.e. f and Γ, respectively) multiplied by the calculated weight ω. In our finite temperature approach, the broadening parameter kB T is about 0.05 eV at 600 K in accordance with the variance range of the explicit phonon-induced band structure calculations. 49 As shown in Figure 2, the 0 K electronic band structures for grA /Sb2 Te3 , grAA /Sb2 Te3 ,


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grAB /Sb2 Te3 , grAAA /Sb2 Te3 , grABA /Sb2 Te3 , and grABC /Sb2 Te3 are plotted in the top panel, while that at 300 and 600 K are displayed in the middle and bottom panels of Figure 2, respectively. At the outset, we first examine the electronic band structures of these heterointerfaces without temperature effects (as in the upper panel of Figure 2). To discuss and compare the changes in the electronic band structure of these heterointerfaces with respect to the free-standing graphene and Sb2 Te3 , we project the computed wavefunctions at each k-point of the band structure onto a set of orthogonalized pseudo-atomic orbitals (as determined within the PAW potential used). This allows us to trace (and color-code) the contributions coming from either the graphene layer(s) or the Sb2 Te3 substrate (see Figure S3 of Supporting Information). Depending on the coupling strength of electronic states between the graphene layer(s) and Sb2 Te3 , the computed wavefunctions may be strongly delocalized (system-dependent) and thus assigning such contributions unambiguously can be challenging and may not always be apparent. To complicate matters, the inclusion of SOC effects splits the bands further, and hinders a straightforward comparison. Thus, to assist our discussions below, we have chosen to focus on a schematic description of the electronic band structure of these heterointerfaces (in the lower panel of Figure S3 in the Supporting Information). We find that the overall band dispersions for the few-layer graphene is generally maintained when compared to their free-standing counterparts. They experience a small band gap opening and the splitting of degenerate states near the Fermi level (ϵf ) due to the proximity effect as discussed by others, 19,20,53 as well as – especially for the hexagonal stacked graphene layer(s) – a displacement of the Dirac cones to lower energy (i.e. n-doped). Upon injecting the electrons from Sb2 Te3 to the graphene layers, the topological surface states of Sb2 Te3 begin to interact with Dirac cones of graphene, displaying several signatures of SOC effects in graphene’s band structure. Beside the opening of the band gap energy, we find the fourfold degenerate states (and their multiple) in grA /Sb2 Te3 (and other bilayer



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and trilayer graphene in this work) begin to split under the influence of the SOC effect, producing Rashba-type split states 54–56 in both the valence and conduction bands. These observations are very much in-line with other DFT+SOC calculations 18,19 and tight-binding models, 20 where the TI substrate invokes these SOC-induced changes to the overall band structure of graphene layer(s). With regards to the energy bands of Sb2 Te3 , we find a general up-shifting of the bands near ϵf (i.e. p-doped) and also the classic signature of Rashba SOC interactions, e.g. in the cases of grAA /Sb2 Te3 and grAAA /Sb2 Te3 where Dirac and Kramers points are clearly illustrated near ϵf . 57 Returning to Figure 2, we will now turn to the effect of finite temperature on the electronic band structures of these heterointerfaces. As outlined above, we do this phenomenologically via a statistical treatment of temperature-broadened DFT eigenvalues in our band structure calculations at both 300 and 600 K. Here, we find that with the inclusion of finite temperature effects, the DFT eigenvalues are broadened and smoothed out, causing many of the SOCeffects mentioned above to be no longer that apparent. However, we do observe that these temperature-broadened band structures (especially obvious for the ones modeled at room temperature of 300 K in the middle panel) retain the overall dispersion and characteristics of the 0 K ones. This is especially evident for the Dirac cones of the graphene layer(s) where the linear dispersions are hardly affected (albeit slightly down-shifted to lower energies). Overall, from Figure 2, for all temperatures, we find that the overall band structures of these few-layer graphene/Sb2 Te3 heterointerfaces strongly suggest a charge redistribution across the vdW layer between graphene layer(s) and Sb2 Te3 , resulting in a p-doped Sb2 Te3 and n-doped graphene heterojunction, particularly for the hexagonal-stacked systems. For a better understanding of this charge redistribution, we next proceed to calculate and examine the difference electron density between the graphene layer(s) and Sb2 Te3 , and further correlate how this charge redistribution might assist in the thermoelectric properties of these heterointerfaces.


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Figure 3: (Color online) Planar-averaged ∆ρavg (z) (where ∆ρ determined from Equation 3) as projected along the z-axis for (a) grA /Sb2 Te3 , (b) grAA /Sb2 Te3 , and (c) grAAA /Sb2 Te3 , respectively. The relative z-positions ∫ of C, Sb, and Te atoms are indicated alongside these plots. (d) Integrated planar-averaged ∆ρavg (z) as projected along the z-axis for grA /Sb2 Te3 , grAA /Sb2 Te3 , and grAAA /Sb2 Te3 , and the insert shows the volumetric ∆ρ for grA /Sb2 Te3 . ∫ Small vertical arrows are placed where the ∆ρavg (z) goes to zero for each system, and the vdW gap between graphene layer(s) and Sb2 Te3 is highlighted in pale green. The isosurface level used for ∆ρ in the insert is ±1.25 × 10−4 e/bohr3 , and the accumulation and depletion of electron densities are indicated by red- and blue-colored regions, accordingly.

3.4

Interfacial charge redistribution between few-layer graphene and Sb2 Te3

To examine the nature of bonding and charge redistribution between the few-layer graphene and Sb2 Te3 , we calculate the difference electron density (∆ρ), where ∆ρ = ρgri /Sb2 Te3 − ρgri − ρSb2 Te3

.

(5)

Here, ρgri /Sb2 Te3 is the total electron density of the combined few-layer graphene/Sb2 Te3 heterointerfaces from which the electron density of both clean Sb2 Te3 substrate slab, ρSb2 Te3 and that of the free-standing few-layers graphene, ρgri are subtracted while keeping their respective atomic positions to be the ones of the corresponding geometry-relaxed system. Furthermore, to obtain the planar-averaged difference electron densities (∆ρavg (z), electron densities are projected in one dimension along the out-of-plane direction (i.e. z-direction))



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to rationalize the charge redistribution between the few-layer graphene and Sb2 Te3 . Sub∫ sequently, we also perform an integration of ∆ρavg (z) (i.e. ∆ρavg (z)) to further analyze this charge redistribution at the heterointerface. We will focus on the hexagonal-stacked grA /Sb2 Te3 , grAA /Sb2 Te3 , and grAAA /Sb2 Te3 to discuss on the trends in charge redistribution involving the Dirac cones of graphene. The calculated ∆ρ and ∆ρavg (z) (and its integrated form) for the hexagonal stacked graphene layer(s) are presented in Figure 3. The most evident feature of charge redistribution in these heterointerfaces is the strong asymmetric accumulation/depletion of electrons near the van der Waals gap, i.e. the vertical void between the graphene layer(s) and Sb2 Te3 . ∫ This is also clearly shown by the highest peak in the ∆ρavg (z) for each heterointerface in Figure 3d. In the case of the graphene layer that is closest to Sb2 Te3 , the charges are very asymmetrically polarized (i.e. the small gain in electrons by the C atoms (colored in red) sandwiched in between the losses in electrons (colored in blue) on both surfaces of this graphene layer). It is easy to deduce that this asymmetric polarization of charges for the closest graphene layer is due to the proximity effect of this graphene layer to the surface of Sb2 Te3 where an internal electrostatic field is generated by the potential gradient near the graphene/Sb2 Te3 interface. 19 This polarizing field is also known to account for the giant SOC interaction induced in the graphene in contact. 19 In passing, this is in contrast to the case where graphene is sandwiched between two TI layers where symmetric polarization is reported. 58 Turning to their differences, by looking at the region just after the vdw gap (highlighted ∫ in pale green) in Figure 3d, the ∆ρavg (z) differs for all three hexagonal stacked heterointerfaces, suggesting that the electronic conductivity and thermoelectric transport for the few-layer graphene on Sb2 Te3 may indeed be quite unlike that of the monolayer. To provide a perspective of how the thermoelectric properties of these few-layer graphene may change in the presence of Sb2 Te3 , we solve the Boltzmann transport equation (BTE) (via the BoltzTraP program 36 ) and present our results for both the pristine free-standing hexag-


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Figure 4: (Color online) Calculated electrical conductivity (σ/τ , measured in 1/Ω m s) of (a) free-standing pristine graphene layer(s) and (b) few-layer graphene/Sb2 Te3 heterointerface structures within the relaxation time approximation. Both free-standing and interface structures with grA , grAA , and grAAA are depicted in black, orange, and red lines, respectively. (c) Schematic diagram to illustrate a possible thermoelectric device structure, consisting of Sb2 Te3 /graphene p-n heterointerface structure. Blue, red, and gray boxes denotes graphene, Sb2 Te3 , and heat absorber/rejector components, accordingly. onal stacked graphene and grA /Sb2 Te3 , grAA /Sb2 Te3 , and grAAA /Sb2 Te3 , respectively, in Figures 4a and 4b.

3.5

Electron-hole asymmetry in electrical conductivity simulations

We first begin by briefly outlining how we calculate the electrical conductivity of both freestanding and supported graphene layer(s). By solving the semi-classical Boltzmann transport 15 ACS Paragon Plus Environment


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theory within the constant relaxation time approximation, the group velocities of the charge carriers are first obtained from the electronic band structure via the relation υ(i, k) =

1 ℏ

▽k

ϵi,k , where the i, k, and ℏ denotes the band index, wave vector, and reduced Planck constant, respectively. Within the relaxation time approximation, the transport distribution function tensor is then calculated with the equation σ αβ (ϵ) =

1 Σ τ N i,k

υα (i, k)υβ (i, k)

δ(ϵ−ϵi,k ) , dϵ

where α and β

are tensor indices. Based on the transport distribution tensor, the electrical conductivity ∫ e) tensor can be calculated from the equation σαβ (T, µ) = Ω1 σ αβ (ϵ)[− δf0 (T,ϵ,µ ]dϵ where Ω, δϵ µe , and f0 denotes the unit cell volume, the electron chemical potential of carriers, and the Fermi-Dirac distribution function of the carriers, respectively. Here, we can find some interesting physical insights from solving the BTE. In Figure 4a, the calculated electrical conductivity for the pristine free-standing graphene layer(s) is close to perfectly symmetric about the zero value of the electron chemical potential, µe , i.e. having no electron-hole asymmetry. As we go from a monolayer to the bilayer and then to the trilayer for the hexagonal stacked graphene, we see a rather dramatic symmetrical increase in the σ value by 2 to 3 orders of magnitude (i.e. between 1018 to 1020 (Ω m s)−1 , noting that it is plotted in the log scale). This tells us that the (thermo)electric transport behavior of pristine few-layer graphene should be very simliar in both the electron and hole regime, 59–61 while we will then see that this is very much a different picture for the few-layer graphene/Sb2 Te3 heterointerfaces. Referring to Figure 4b, here we plot the same electrical conductivity for the few-layer graphene/Sb2 Te3 heterointerfaces but on a slightly different scale range. We first notice that the electrical conductivity behavior of heterointerfaces are very asymmetric about the zero value of µe , and these conductivities are way much higher than their free-standing counterparts (∼ 1020 (Ω m s)−1 and above). In pristine few-layer graphene structures, it is clear from their band structures that the conduction and valence bands are symmetric (as reflected in the Dirac cones in Figures. S1 and S2). This corroborates with the symmetric


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electrical conductivity plots (in Figure 4a) where µe relates to the Fermi level in our DFT band structure calculations. From our finite temperature DFT band structure calculations, we find that the hexagonal-stacked heterostructures are mildly n-doped, hence left-shifting the electrical conductivity plots (in Figure 4b). This also results in the observed asymmetry (in part, due to the charge redistribution). Returning back to our earlier discussion of thermoelectric relations in low-dimensional nanomaterials, recent studies on the interfacial charge transfer to graphene from electrode contacts 62,63 have further supported that electronhole asymmetry (as deduced from experimentally determined current/voltage (I/V ) curves) is beneficial for setting up an efficient thermoelectric p-n junction. Particularly, for the AA-graphene/Sb2 Te3 heterointerface, as shown in Figure 4b, its lower electrical conductivity may be accounted for by the difference in polarization of the graphene layers (as shown in Figure 3b). Here, we find the polarization results in a clear charge redistribution of the first and second graphene layers whereby an overall depletion (− + −, colored in blue/red/blue) and accumulation (+ − +, colored in red/blue/red) charge layer is built up, respectively. We suggest that this could confine the carriers between the electrostatically attractive field setup by these charge layers. In contrast, in Figures 3a and 3c, the redistribution is of the same nature (i.e. in the order − + − colored in blue/red/blue, and thus lacking this electrostatic attraction between the layers) and hence reflecting a higher electrical conductivity in A- and AAA-stacked heterointerfaces. Concluding, we would like to propose that the hexagonal stacked few-layer graphene/Sb2 Te3 heterointerfaces can be used as p-n junction building blocks to construct an efficient thermoelectric device (as conceptually illustrated in Figure 4c) that exploit the electron-hole asymmetry by enhancing the carrier transport with “hot” excited charge carriers.



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4

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Summary

To summarize, using first-principles DFT calculations, we survey and investigate various graphene/Sb2 Te3 heterointerfaces in different stacking sequences. Using the DFT(+D2) optimized atomic structures, we calculate the electronic band structures of these graphene/Sb2 Te3 Dirac heterointerfaces, including the effects of SOC. We find that most essential topological band features are well reproduced and using a phenomenological temperature-broadening model based on the Fermi-Dirac statistics, we re-examine the effect of finite temperature on the electronic band structures. Consequently, we also find that the DFT eigenvalues are broadened and smoothed out with increasing temperature, obscuring many of the spin-orbit interactions. Nonetheless, it is evident that the Dirac cones of the graphene layer(s) are hardly affected (albeit slightly down-shifted to lower energies). From the calculated band structures, we identify the underlying charge redistribution process by analyzing the difference electron densities to find large polarization effects at the interface, driving the charge redistribution of electrons between Sb2 Te3 and graphene. When solving the Boltzmann transport equation for these heterointerfaces, we have also found a strong case of electron-hole asymmetry in its electrical conductivity as a signature of enhanced thermoelectric properties. In closing, we offer these graphene/Sb2 Te3 Dirac heterointerfaces as useful p-n junction building blocks for the next-generation thermoelectric devices where the carrier transport may be enhanced by “hot” excited charge carriers.

Acknowledgement We acknowledge that this work is supported from the Basic Research Laboratory (BRL) Program by the National Research Foundation (NRF) of Korea (Grant No. 2016R1A4A1012929). Computational resources have been provided by the Korea Institute of Science and Technology Information (KISTI) supercomputing center (KSC-2017-C3-0007) and the Australian National Computational Infrastruc18 ACS Paragon Plus Environment

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ture (NCI).

Supporting Information Available Supporting Information. Optimized structural parameters for each graphene/Sb2 Te3 interface system; Calculated electronic band structures of pristine Sb2 Te3 and free-standing few-layer graphene systems; Projected electronic band structure of various few-layer graphene/Sb2 Te3 heterointerfaces (with and without the inclusion of spin-orbit coupling effect). This material is available free of charge via the Internet at http://pubs.acs.org/.

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Designing Two-Dimensional Dirac Heterointerfaces of Few-Layer

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