Letter pubs.acs.org/NanoLett
Enhanced Thermoelectric Efficiency via Orthogonal Electrical and Thermal Conductances in Phosphorene Ruixiang Fei,† Alireza Faghaninia,‡ Ryan Soklaski,† Jia-An Yan,§ Cynthia Lo,‡ and Li Yang*,† †
Department of Physics and Institute of Materials Science and Engineering, and ‡Department of Energy, Environmental, and Chemical Engineering, Washington University, St Louis, Missouri, United States § Department of Physics, Astronomy, and Geosciences, Towson University, Towson, Maryland, United States S Supporting Information *
ABSTRACT: Thermoelectric devices that utilize the Seebeck effect convert heat flow into electrical energy and are highly desirable for the development of portable, solid state, passively powered electronic systems. The conversion efficiencies of such devices are quantified by the dimensionless thermoelectric figure of merit (ZT), which is proportional to the ratio of a device’s electrical conductance to its thermal conductance. In this paper, a recently fabricated two-dimensional (2D) semiconductor called phosphorene (monolayer black phosphorus) is assessed for its thermoelectric capabilities. First-principles and model calculations reveal not only that phosphorene possesses a spatially anisotropic electrical conductance, but that its lattice thermal conductance exhibits a pronounced spatial-anisotropy as well. The prominent electrical and thermal conducting directions are orthogonal to one another, enhancing the ratio of these conductances. As a result, ZT may reach the criterion for commercial deployment along the armchair direction of phosphorene at T = 500 K and is close to 1 even at room temperature given moderate doping (∼2 × 1016 m−2 or 2 × 1012 cm−2). Ultimately, phosphorene hopefully stands out as an environmentally sound thermoelectric material with unprecedented qualities. Intrinsically, it is a mechanically flexible material that converts heat energy with high efficiency at low temperatures (∼300 K), one whose performance does not require any sophisticated engineering techniques. KEYWORDS: Black phosphorus, phosphorene, thermoelectric, anisotropic thermal conductance
T
possessing promising free-carrier mobility, it exhibits highly anisotropic electrical and optical responses.18−20 It is natural to check if the thermal conductance is also anisotropic. If this is the case, it is desirable for the preferred direction of electrical conductance in phosphorene to coincide with one of poor thermal conductance; such an alignment would significantly enhance phosphorene’s thermoelectric performance, as shown in Figure 1. In this paper, first-principles simulations and model calculations of monolayer phosphorene are used to study its electrical conductance, thermal conductance, Seebeck coefficient, and thermoelectric figure of merit (ZT). Excitingly, the electrical and thermal conductances are shown not only to be anisotropic but to exhibit respectively orthogonal preferred conducting directions, suggesting that a large (σ/κ) ratio can be achieved in this material. These results are illustrated in Figure 1. Additionally, the Seebeck coefficient is calculated using the Boltzmann transport equation (BTE).21 Within the reasonable estimation regime, ZT may ultimately reach 1 at room temperature and even higher values at T = 500 K using only moderate doping (∼2 × 1016 m−2 or 2 × 1012 cm−2). Therefore,
he task of designing practical and efficient thermoelectric materials is a decades-old problem that has spurred diverse and creative material engineering efforts for its purpose.1,2 The efficiency of thermoelectric materials is measured by its figure of merit, ZT = S2(σ/κ)T, where S is the Seebeck coefficient, σ is the electrical conductivity, κ is the thermal conductivity, and T is the temperature. The forefront of all-scale electronic and atomistic structural engineering techniques have served to enhance the (σ/κ) ratios of thirdgeneration thermoelectric materials, such as cubic AgPbmSbTe2+m, PbTe-SrTe, and In4Se3‑δ,3−5 achieving ZT values near 2 (around a 15% conversion efficiency) within a temperature range of 700 to 900 K.6 Alternatively, promising simple structures exhibit intrinsically low thermal conductances without requiring sophisticated structural engineering and achieve excellent thermoelectric performances.7,8 For example, SnSe crystal can reach a ZT of 2.6 at T = 923 K,8 although this value falls quickly for lower temperatures. Graphene-inspired 2D materials also possess unusual thermal properties in addition with optical and mechanical advantages,9−13 yet little progress has been made toward harnessing them for thermoelectric applications. Recently, a newly fabricated direct band gap semiconductor, few-layer black phosphorus (phosphorene),14−17 stands out as a possible intrinsically efficient thermoelectric material. Along with © XXXX American Chemical Society
Received: July 25, 2014 Revised: September 10, 2014
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Figure 1. Schematic of orthogonal electrical conductance and thermal conductance in monolayer phosphorene.
where e is the magnitude of the electron’s charge, μe and μh are the mobilities of the electrons and holes, respectively. n and p are the electron and hole densities obtained from the Dirac− Fermi distribution for a given Fermi level (chemical potential)22 (details in Supporting Information). The vertical thickness for monolayer, Lz, is taken to be 0.55 nm, which is the average interlayer distance of multilayer phosphorene.14,18,19 Because phosphorene is a nonpolar crystal, the carrier mobilities are calculated by first-principles Bardeen and Shockley’s theory30 of considering the coupling between free carriers and acoustic phonons for this nonpolar crystal26,27,31
monolayer phosphorene may stand out as a mechanically flexible thermoelectric material that can operate at room temperature and does not require extensive structural engineering. In present calculations, the atomic structure and electronic structure are calculated by density functional theory (DFT) with the Perdew, Burke, and Ernzerhof (PBE) functional.23 The Quantum Espresso computing package is used to solve the Kohn−Sham equations, producing the electronic band structure.24 The band gap is then refined to account for many-body interactions, using the GW approximation via BerkeleyGW.25 The electrical conductance is calculated using Bardeen and Shockley’s theory, including electron-acoustic phonon coupling. This method predicts reasonable mobility values26,27 and does not require the electron relaxation time, a quantity that is needed by most simulation packages. The phonon and thermal conductance calculations are performed in the framework of density functional perturbation theory (DFPT).28 Finally, the Seebeck coefficient is calculated using the BTE implemented in the BolzTrap package29 without needing to estimate the electron relaxation time. Anisotropic Electrical Conductance. The atomic structure of monolayer phosphorene is presented in Figure 2a. Each phosphorus atom is covalently bonded with its three neighboring phosphorus atoms to form a puckered 2D honeycomb structure. The electronic band structure of suspended monolayer phosphorene is presented in Figure 2b. The Γ-point hosts a 2.0 eV direct band gap. The band dispersion about the Γ-point is highly anisotropic; both the valence and conduction bands are significantly dispersive along the armchair lattice direction (Γ-Y) but are nearly flat along the perpendicular zigzag lattice direction (Γ-X). Accordingly, the effective masses of the free carriers in phosphorene, and thus its electrical conductance, are all highly anisotropic. Figure 2c,d details these anisotropies with an “8” shape; the effective mass along the armchair direction is about an order of magnitude smaller than that along the zigzag direction. The electrical conductance in monolayer phosphorene is σ=
μx _anisotropy =
(2)
Here m*e is the effective mass along the transport direction, and the density-of-state mass md is determined by md = (mex * mey * )1/2. The deformation potential constant E1x = ΔE/(Δlx/lx) is obtained by varying the lattice constant along the transport direction Δlx(lx is the lattice constant along the transport direction, here is x direction) and checking the change of band energy under the lattice compression and strain. The elastic module Cx_anisotropy is obtained via Cx_anisotropy (Δlx/lx)2/2 = (E − E0)/S0, where E − E0 is obtained by varying the lattice constant by small amount (Δlx/lx ∼ 0.5%) to obtain the change of the total energy, and S0 is the lattice area in the xy plane. Unlike in widely used transport-simulation packages, such as BTE,29 the approach discussed above does not require a freecarrier lifetime, which is unknown for phosphorene. This calculation proves to produce surprisingly good results when comparing with available experiments. At room temperature, the calculated electron and hole mobilities along the armchair direction are 1800 and 2300 cm2/v·s, respectively. Recent experimental results support this simple model calculation, finding that the holes are the more mobile of the carriers and reporting the hole mobility around 1000 cm2/v·s.14 In practice, an improved sample quality will bolster the measured mobility and yield results that are in better agreement with the calculated ones. The electrical conductance is presented in Figure 2e. It depends exponentially on the Fermi level, because of the nearly constant mobility of the free carriers in the Drude model. In accordance with the above discussion of the effective masses,
(μe en + μ h ep) Lz
eh3Cx _anisotropy (2π )3 kBTme*md E12x
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Figure 2. (a) Atomic structure of monolayer phosphorene. (b) GW-calculated electronic band structure of suspended monolayer phosphorene. (c,d) Effective mass of electrons and holes according to spatial directions. (e) Electrical conductance of monolayer phosphorene by the Drude model. The scale of conductance is a logarithmic scale. The minus sign of the Fermi level means hole p-doping and the plus sign means electron n-doping.
comparison with the lattice thermal transport will be discussed later. Given this approximation, the thermal conductivity κ of phosphorene at a finite temperature T can be considered to be comprised of the phonon mode contributions: κ = ∑λKλ, where the sum is taken over all phonon branches. The contribution from each mode is given by36,37
the conductance is spatially anisotropic (Figure 2c,d); the conductance along the armchair direction is about an order of magnitude larger than it is along the zigzag direction, across all doping levels. For a p-doping density of 2 × 1016 m−2, the conductance along the armchair and zigzag directions is calculated to be 1.33 × 106 and 1.04 × 105 S/m, respectively. Anisotropic Thermal Conductance. The thermal conduction in most semiconductors is mediated chiefly by lattice vibrations (phonons).32−34 Free-carrier contributions of monolayer phosphorene have been studied recently35 and the
κλ = C
1 LxLyLz
∑ τλ(q ⃗)Cph(ωλ)[vλ(q ⃗)·n]̂ 2 λ,q
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Figure 3. (a) Phonon dispersion and DOS of monolayer phosphorene. (b) Thermal conductance of monolayer phosphorene along the armchair and zigzag directions, respectively. The solid symbols are estimation based on the long-wave relaxation time (60 ps) derived from experimental data of bulk black phosphorus.30 Different curves are using different long-wave relaxation time to cover the possible range in monolayer phosphorene.
where τλ is the phonon relaxation time, Cph(ωλ) is the specific heat contribution of the mode, and vλ(q⃗) is the phonon sound velocity with wave vector. n̂ is the unit vector that points along the thermal gradient ∇T. The phonon dispersion of monolayer phosphorene, as calculated by first-principles DFPT calculations, is detailed in Figure 3a, which is consistent with previous result.38 Optical phonons have smaller sound speed vλ(q⃗)) and much smaller contributions to the heat capacity Cph(ωλ) (details can be found in the Supporting Information). Therefore, according to eq 3 we neglect those optical-phonon contributions to the thermal conductance, which is similar to the simplification employed in the case of graphene.28 Two in-plane acoustic modes, the transverse acoustic mode (TA) and the longitudinal acoustic mode (LA), all exhibit linear dispersions, while the off-plane acoustic mode (ZA) exhibits a parabolic dispersion, which is in accordance with phosphorene’s 2D geometry. The LA phonon stands out for exhibiting an anisotropic speed of sound. The lattice structure of phosphorene decides that the sound speed of the LA branch is larger along the zigzag direction than it is along the armchair direction. The longitudinal vibrations can easily distort the puckered armchair structure, whereas the zigzag lattice is more robust and more difficult to excite. Therefore, the LA branch’s speed of sound along the zigzag direction is 8397 m/s, which is almost twice the speed along the armchair direction 4246 m/s. According to eq 3, the thermal conductance is proportional to the squares of these speeds. Phosphorene’s preferred direction of thermal conduction will thus be enhanced along the zigzag lattice direction, orthogonal to the preferred direction of electrical conduction. To quantify the thermal conductance, the relaxation time (τλ) of a specific phonon mode (λ) is calculated by considering nonlinear lattice vibrations that are chiefly mediated by phonon−phonon scattering in the lowest-order approximation37,39 is given by
τλ =
ωm 1 Mν 2 ω 2 2γλ2 kBT
in which M is the atomic mass, ωm is the Debye frequency, kB is Boltzmann constant, and γλ is the Grüneisen parameter, which is given by the negative logarithmic derivative of the phonon frequency with respect to the change in volume.36,37,39 The average velocity ν for a 2D phonon comprised of the LA and TA modes is approximated by 2/ν2 = 1/νLA2 + 1/νTA2 in the long wavelength limit.37,39 An outstanding issue arises when considering only the above two-phonon scattering mechanism: eq 4 predicts diverging acoustic phonon lifetimes at the long-wave limit. It is necessary to include three-phonon Umklapp scatterings or scattering with rough boundaries when ω approaches zero. In practice, a truncation scheme is typically implemented in which the long wavelength lifetime is estimated from experimental data. Because there is no such data for monolayer phosphorene, the relevant thermal conductance value used in the present calculation is obtained from data regarding bulk black phosphorus (60 ps).40 Additional details are provided in the Supporting Information. It is assumed that the van der Waals interactions between phosphorene layers are weak and thus will not dramatically change the long wavelength phonon scatterings within a single layer. This is a reasonable approximation. For instance, the measured thermal conductivity of suspended monolayer hexagonal boron nitride (h-BN) is only about 1.4 times greater than that of its few-layer structure versions.41 For the sake of being systematic, we choose a wide range of long-wave lifetimes to account for a broad window of possible relaxation times (45−150 ps); this approach is widely used in theoretical studies of thermoelectric materials42,43 and it is found that the final results are robust in this regard. Given the above calculations and approximations, the lattice thermal conductances are plotted versus temperature in Figure 3b. The multiple curves, corresponding to different long-wave lifetimes, are plotted. The thermal conductance is consistently larger along the zigzag direction than it is along the armchair direction. That is, the preferred direction for thermal conductance in monolayer phosphorene lies perpendicular to the direction of preferred electrical conductance, as presented in Figure 1.
(4) D
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Figure 4. (a,b) Seebeck coefficient according to the doping density, calculated by the BTE at T = 300 and 500 K, respectively. (c,d) The thermoelectric figure of merit according to the doping density under T = 300 and 500 K, respectively. Different long-wave relaxation times of phonons are included. The unit of the doping density is m−2.
Seebeck Coefficient. The Seebeck coefficient (S) of monolayer phosphorene is calculated using BTE (see the Supporting Information for details). Figure 4a,b shows the S for monolayer phosphorene at room temperature and 500 K with different free carrier doping densities. The coefficient varies dramatically with doping density, indicating that optimizing the density is crucial for achieving efficient thermoelectric performance. Unlike many of the other features of phosphorene, the Seebeck coefficient is nearly isotropic. These calculated values are comparable with typical thermoelectric materials8 and are in accord with recent studies of phosphorene.35 Thermoelectric Figure of Merit (ZT). Figure 4 c depicts the thermoelectric figure of merit as a function of free-carrier density along the zigzag and armchair lattice directions, respectively, at room temperature. As expected, given the above observations and discussions, ZT exhibits a strong spatial anisotropy in monolayer phosphorene. The armchair direction fosters a larger ZT than does the zigzag direction. Given the orthogonal preferred directions of the electrical and thermal conductances, the values of ZT along the armchair direction can be enhanced well above the figures of merit of typical materials.3−8 A doping density of ∼2 × 1016 m−2 yields a figure of merit well above 1 for reasonable relaxation time (60 ps). Even with the most conservative estimation of the phonon relaxation, ZT is still around 0.6 (Figure 4c), which is extremely attractive for applications. For a higher temperature (T = 500 K), the figure of merit is even better as shown in Figure 4 d; it is larger than 1.7 and can even reach 2.5 under the same doping density. Our calculated phosphorene’s ZT is promising for applications. According to the Carnot efficiency for heat conversion, phosphorene based thermoelectric devices are poised to reach an energy conversion efficiency of 15−20%,32
surpassing the criterion for commercial deployment. Outstandingly, unlike most thermoelectric materials whose ZT are meager at lower temperatures, the ZT of phosphorene is still substantial at room temperature, as shown in Figure 4c. This feature, plus phosphorene’s environment-friendly composition and exceptional mechanical elastic properties, distinguishes it as an exceptional candidate for thermoelectric applications at room temperature. It is imperative to consider factors that may in practice affect the value of ZT in phosphorene. First, the role of doping free carriers as conduits of thermal conduction has not been considered. Recent studies of electron-induced thermal conductance shows that a p-doping density of 6 × 1016 m−2 would lead to a thermal conductance around 1.5 W/K·m.35 This is an order of magnitude smaller than the predicted conductance along the armchair direction (11.8 W/K·m), as seen in Figure 3b. Therefore, the contributions of the dopants to the thermal conductance will not significantly modify ZT. Second, the calculations and simulations pertain to suspended monolayer phosphorene. Realistically, defects will act to reduce the conductance and diminish the anisotropies. To this end, modulation doping of 2D structures based on charge transfer44,45 may act to diminish the effect of scattering defects and preserve the high ZT. Third, the band gap of phosphorene may vary across different environments. This band gap effect is discussed in the Supporting Information. Ultimately it does not significantly impact our conclusions. Fourth, we must emphasize that the thermoelectric performance of realistic phosphorene results from the culmination of many complicated physical processes; here, we can only focus on the most important intrinsic contributions and provide only conservative predictions of the material’s properties in an effort to account for these necessary simplifications. Finally, we estimate the E
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(3) Hsu, K. F.; Loo, S.; Guo, F.; Chen, W.; Dyck, J. S.; Uher, C.; Hogan, T.; Polychroniadis, E. K.; Kanatzidis, M. G. Cubic AgPbmSbTe2+m: bulk thermoelectric materials with high figure of merit. Science 2004, 303, 818−821. (4) Biswas, K.; He, J.; Blum, I. D.; Wu, C. I.; Hogan, T. P.; Seidman, D. N.; Dravid, V. P.; Kanatzidis, M. G. High-performance bulk thermoelectrics with all-scale hierarchical architectures. Nature 2012, 489, 414−418. (5) Rhyee, J. S.; Lee, K. H.; Lee, S. M.; Cho, E.; Kim, S. I.; Lee, E.; Kwon, Y. S.; Shim, J. H.; Kotliar, G. Peierls distortion as a route to high thermoelectric performance in In4Se3‑δ crystals. Nature 2009, 459, 965−968. (6) Zhao, L. D.; Dravid, V. P.; Kanatzidis, M. G. The panoscopic approach to high performance thermoelectrics. Energy Environ. Sci. 2014, 7, 251−268. (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. (8) Zhao, L. D.; Lo, S. H.; Zhang, Y.; Sun, H.; Tan, G.; Uher, C.; Wolverton, C.; Dravid, V. P.; Kanatzidis, M. G. Ultralow thermal conductivity and high thermoelectric figure of merit in SnSe crystals. Nature 2014, 508, 373−377. (9) Balandin, A. A. Thermal properties of graphene and nanostructured carbon materials. Nat. Mater. 2011, 10, 569−581. (10) Balandin, A. A.; Ghosh, S.; Bao, W.; Calizo, I.; Teweldebrhan, D.; Miao, F.; Lau, C. N. Superior Thermal Conductivity of SingleLayer Graphene. Nano Lett. 2008, 8, 902−907. (11) Ghosh, S.; Bao, W.; Nika, D. L.; Subrina, S.; Pokatilov, E. P.; Lau, C. N.; Balandin, A. A. Dimensional crossover of thermal transport in few-layer graphene. Nat. Mater. 2010, 9, 555−558. (12) Jo, I.; Pettes, M. T.; Kim, J.; Watanabe, K.; Taniguchi, T.; Yao, Z.; Shi, L. Thermal Conductivity and Phonon Transport in Suspended Few-Layer Hexagonal Boron Nitride. Nano Lett. 2013, 13, 550−554. (13) Sreeprasad, T. S.; Nguyen, P.; Kim, N.; Berry, V. Controlled, Defect-Guided, Metal-Nanoparticle Incorporation onto MoS2 via Chemical and Microwave Routes: Electrical, Thermal, and Structural Properties. Nano Lett. 2013, 13, 4434−4441. (14) Li, L.; Yu, Y.; Ye, G. J.; Ge, Q.; Ou, X.; Wu, H.; Feng, D.; Chen, X. H.; Zhang, Y. Black phosphorus field-effect transistors. Nat. Nanotechnol. 2014, 9, 372−377. (15) Liu, H.; Neal, A. T.; Zhu, Z.; Luo, Z.; Xu, X.; Tománek, D.; Ye, P. D. Phosphorene: An Unexplored 2D Semiconductor with a High Hole Mobility. ACS Nano 2014, 8, 4033−4041. (16) Reich, E. S. Phosphorene excites materials scientists. Nature 2014, 506, 19−19. (17) Koenig, S. P.; Doganov, R. A.; Schmidt, H.; Neto, A. C.; Oezyilmaz, B. Electric field effect in ultrathin black phosphorus. Appl. Phys. Lett. 2014, 104, 103106. (18) Xia, F.; Wang, H.; Jia, Y. Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics. Nat. Commun. 2014, 5, 4458. (19) Fei, R.; Yang, L. Strain-Engineering the Anisotropic Electrical Conductance of Few-Layer Black Phosphorus. Nano Lett. 2014, 14, 2884−2889. (20) Tran, V.; Soklaski, R.; Liang, Y.; Yang, L. Layer-Controlled Band Gap and anisotropic excitons in few-layer black phosphorus. Phys. Rev. B 2014, 89, 235319. (21) Allen, P. B. Boltzmann Theory and Resistivity of Metals. In Quantum Theory of Real Materials; Chelikowsky, J. R., Louie, S. G., Eds.; Kluwer: Boston, 1996; pp 219−250. (22) Ashcroft, N. W., Mermin, N. D. Solid State Physics; Harcourt, Inc.: Orlando, FL, 1976. (23) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (24) Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; et al. QUANTUM ESPRESSO: a modular and opensource software project for quantum simulations of materials. J. Phys.: Condens. Matter 2009, 21, 395502.
thermoelectric performance of monolayer phosphorene at room temperature and 500 K, respectively. Although bulk black phosphorus is stable up to 700 K, the monolayer structure is not necessarily stable at such a high temperature. Upon heating, phosphorene may experience a phase transition to other possible allotropies.46 Substrates and encapsulations are important extrinsic factors that may affect the thermoelectric performance of phosphorene as they will impact both its thermal and electric conductances. For example, the phonon and carrier scattering between phosphorene and a substrate can change its conductance by an order of magnitude. Moreover, as shown in other 2D materials, encapsulating high-K dielectrics can effectively enhance the screening and reduce CI scattering.47 Therefore, state-of-the-art experiments will be needed to probe the robustness of phosphorene’s thermoelectric character in the presence of substrates and encapsulated environments. Conclusion. Monolayer phosphorene was assessed for its intrinsic thermoelectric capabilities. Accordingly, the electrical conductance, thermal conductance, and Seebeck coefficient of were calculated and used to determine ZT. It was found that the electrical and thermal conductances exhibit strong spatial anisotropies such that their respective preferred directions of conductance are orthogonal to one another, resulting in an anisotropic thermoelectric figure of merit that is large along the armchair direction. Promising thermoelectric figures of merit can be achieved at relatively low temperatures using moderate doping. Monolayer phosphorene is thus an attractive candidate for energy applications based on inexpensive and flexible thermoelectric materials that may operate near room temperature.
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ASSOCIATED CONTENT
S Supporting Information *
Calculation details of the anisotropic electrical conductance, anisotropic lattice thermal conductance, Seebeck coefficient, and the GW-corrected band gap impact on thermoelectric efficiency. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We acknowledge fruitful discussions with Vy Tran and Anders Carlsson. R.F., R.S., and L.Y. are supported by the National Science Foundation Grant DMR-1207141. J.-A.Y. is supported by the Faculty Development and Research Committee of Towson University (OSPR No. 140269) and by the FCSM Fisher General Endowment Fund of Towson University. The computational resources have been provided by the Lonestar of Teragrid at the Texas Advanced Computing Center (TACC).
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