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Thermoelectric Properties of a Monolayer Bismuth Long Cheng, Huijun Liu, Xiaojian Tan, Jie Zhang, Jie Wei, Hongyan Lv, Jing Shi, and Xinfeng Tang J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 23 Dec 2013 Downloaded from http://pubs.acs.org on December 26, 2013

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The Journal of Physical Chemistry

Thermoelectric Properties of a Monolayer Bismuth

Long Cheng1, Huijun Liu1,*, Xiaojian Tan1, Jie Zhang1, Jie Wei1, Hongyan Lv1, Jing Shi1, Xinfeng Tang2

1

Key Laboratory of Artificial Micro- and Nano-Structures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan 430072, China

2

State Key Laboratory of Advanced Technology for Materials Synthesis and Processing, Wuhan University of Technology, Wuhan 430070, China

*

Author to whom correspondence should be addressed. Electronic mail: [email protected]. 1

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Abstract The structural and electronic properties of a two-dimensional monolayer bismuth are studied using density functional calculations. It is found that the monolayer forms a stable low-buckled hexagonal structure which is reminiscent of silicene. The electronic transport properties of the monolayer bismuth are then evaluated by using Boltzmann theory with the relaxation time approximation. By fitting first-principles total energy calculations, a modified Morse potential is constructed, which is used to predicate the lattice thermal conductivity via equilibrium molecular dynamics simulations. The room temperature ZT value of a monolayer bismuth is estimated to be 2.1 and 2.4 for the n- and p-type doping, respectively. Moreover, the temperature dependence of ZT is investigated and a maximum value of 4.1 can be achieved at 500 K.

Keywords: density functional calculations; Boltzmann theory; molecular dynamics; thermoelectric properties

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I. INTRODUCTION Due to their novel properties and promising applications in power generation and cooling, thermoelectric materials have attracted much attention during the last few decades. Bismuth (Bi)-based compounds, such as bismuth telluride (Bi2Te3) and bismuth antimony (BiSb) alloys are known to exhibit good thermoelectric performance at both room and low temperature1, 2. The efficiency of a thermoelectric material is quantified by the dimensionless figure of merit ZT =

S 2σ T κe + κl

(1)

where S is the Seebeck coefficient, σ is the electrical conductivity, T is the absolute temperature, and κ e and κ l are the electronic and lattice thermal conductivity, respectively. Therefore, a high ZT value requires a large power factor ( S 2σ ) and/or a small thermal conductivity ( κ e + κ l ). Unfortunately, for traditional bulk thermoelectric materials, these transport coefficients are coupled with each other and it is extremely difficult to significantly improve their thermoelectric performance. In 1993, Hicks and Dresselhaus first proposed that by using low-dimensional systems or nanostructures, one could significantly enhance the ZT values of thermoelectric materials3, 4. This pioneering study has stimulated a large number of subsequent works, both experimentally and theoretically. In particular, the existence of two dimensional Bi-based layered structures and their favorable thermoelectric performance have been confirmed. Liao and Kuo5 prepared the Bi/Te bilayer thin film by sputter deposition and measured the Seebeck coefficient, the electric resistivity, and the thermal conductivity. The estimated ZT value is 1.43 at 200 ℃, which is higher than that of bulk Bi2Te3. Teweldebran et al.6,

7

prepared bismuth

telluride atomic quintuples and ultra-thin films by using graphene-like exfoliation technique. They found that the atomic thin films exhibit high electrical conductivity with unusual temperature dependence, and there is an increase of 40% in ZT value compared with that of bulk Bi2Te3. Using density functional theory and a Landauer approach, Zahid et al.8 predicted a ZT value of 7.15 for the Bi2Te3 quintuple layer (QL), which increases by a factor of 10 compared with that of bulk Bi2Te3. Maassen 3



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and Lundstrom9 considered the Bi2Te3 films ranging in thickness from 1 to 6 QLs and found that one QL has an electronic ZT ( ZTe ) that is 5 times greater than that of bulk Bi2Te3. All of the above studies are focused on the bismuth telluride layers and films which are originated from the bulk Bi2Te3. The thermoelectric properties of a bismuth layer are less known, however. A bismuth layer can be viewed as exfoliation from bulk Bi, with a similar crystal structure as bulk Bi2Te3. It was previously found that bulk Bi has a maximum ZT value of 1.44,10 which is considerably larger than the one for bulk Bi2Te3. It is thus reasonable to expect that the two-dimensional monolayer Bi may exhibit even better thermoelectric performance, which is the motivation of our current work. In this paper, the first-principles calculation is combined with Boltzmann theory to study the electronic transport properties of Bi monolayer. The equilibrium molecular dynamics (EMD) method is then used to investigate the phonon transport. Our theoretical calculations indicate that the ZT value of a monolayer Bi can be significantly enhanced at appropriate operating temperature and doping level (or carrier concentration).

II. COMPUTATIONAL DETAILS The structure optimization and energy band calculations are performed by using the plane-wave pseudopotential formulation within the framework of density function theory (DFT). The code is implemented in the Vienna ab initio simulation package (VASP)

11 ,

12 , 13

. The exchange correlation energy is in the form of

Perdew-Burke-Ernzerhof14 with generalized gradient approximation (GGA). As Bi is a heavy element, the spin-orbit (SO) coupling is explicitly considered in our calculations. The monolayer Bi is modeled by using a hexagonal cell containing two atoms, and the length of c vector is set to be 15 Å so that the monolayer and its periodic images can be treated as independent entities. The energy cutoff is set as 210 eV, and the Brillouin zone is sampled with 9×9×1 Monkhorst meshes. Optimal atom positions are determined until the magnitude of the force acting on each Bi atom

4


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becomes less than 0.05 eV/Å. Based on the calculated band structure, the electronic transport coefficients can be derived using the Boltzmann theory15 and relaxation time approximation. In this approach, the Seebeck coefficient S is independent of the relaxation time τ , while the electrical conductivity can only be determined with respect to the relaxation time. That is, what we actually calculated is σ τ . The relaxation time τ can be estimated by fitting the available experimental data. The electronic thermal conductivity is calculated according to the Wiedemann-Franz law 16 . Note that we have done convergence tests and found that 3000 k points is needed to get reliable results. The lattice thermal conductivity is calculated by performing MD simulations, where an accurate interatomic potential is needed. Based on the potential model proposed by Huang17 and Qiu18 for bulk and low-dimensional Bi2Te3, we construct a modified Morse potential which contains the two-body bond and three-body angle potentials U = U two + U three ,

(2)

Here the two-body potential is in the form

{

2

}

− a r −r U two = De 1 − e ( 0 )  − 1 ,

(3)

where De is the depth of the potential well, a is the measure of bond elasticity, r is the interatomic separation, and r0 is the equilibrium bond distance. The three-body angle potential is given by U three =

1 2 k ( cos θ − cos θ 0 ) , 2

(4)

where k is the force constant, θ is the bond angle, and θ 0 is the equilibrium bond angle. These potential parameters ( De , r0 , a , k , θ 0 ) can be determined by fitting the energy surface from first-principles calculations, which is performed using the GULP code19. In the MD simulations, we use a rectangular simulation box and the temperatures considered range from 200 K to 500 K, with an interval of 50 K. The time step is 0.5 fs. The Verlet leapfrog algorithm is adopted for the calculation, while 5



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Nose-Hoover thermostat is used to control the temperature. The system is first simulated in an NVT (constant number of atoms, volume and temperature) ensemble for 600 ps to ensure it reaches equilibrium at the desired temperature, and then it is switched into an NVE (constant number of atoms, volume and energy) ensemble for 600 ps to arrive at equilibrium. At each temperature point, 30 ns heat current data are obtained for the calculation of the heat current autocorrelation function. The thermal conductivity is then obtained by using the Green-Kubo auto correlation decay method20,

21, 22

, where an integration time of 50 ps and a 30×30×1 unit cell containing

3600 atoms is needed to obtain converged results.

III. RESULTS AND DISCUSSIONS We begin with the structural properties of a monolayer Bi. Fig. 1 shows the ball-and-stick model of bulk Bi and monolayer Bi. The former has a rhombohedral —

crystal structure that belongs to the space group D35d ( R 3 m) 23. Along the z direction, bulk Bi exhibits a layered structure as shown in Fig. 1(a). The layer-layer interactions are very weak and are usually in the form of the van der Waals force, which makes it possible to exfoliate ultrathin layer from the bulk, as has been done in graphite24 and bulk Bi2Te36, 7. In the fully relaxed monolayer Bi shown in Fig. 1(b), each Bi atom is covalently bonded to three neighboring Bi atoms with identical bond lengths of 3.05 Å and bond angles of 90.8°, both are a little smaller than those found in bulk Bi (the bond lengths and bond angles are 3.07 Å and 93.8°, respectively). The bucking distance, labeled as ∆ in Fig. 1(c), is 1.73 Å for monolayer Bi and 1.65 Å for bulk Bi. These values are consistent with previously reported calculations 25 ,

26 , 27

. The

formation energy of a Bi monolayer is calculated to be −1.97 eV/atom if the chemical potential of Bi is set at that of a Bi atom. To confirm the structure stability of the monolayer Bi, we calculated the phonon spectrum by using the density function perturbation theory (DFPT), which is implemented in the QUANTUM-ESPRESSO package28. As can be seen from Fig. 2(a), there is no imaginary frequency in the phonon dispersion relations, which indicates that the monolayer Bi is indeed stable.

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As another evidence, we have performed ab-initio molecular dynamics (MD) on the monolayer. The starting temperature is 300 K and a microcanonical ensemble has been used. The MD runs for 2.5 ps with a time step of 0.5 fs. Fig. 2(b) shows the nearest Bi-Bi distance of the monolayer in the whole MD running time. We see there is only slight ﬂuctuation around the equilibrium bond length of 3.05 Å, which suggests that the monolayer Bi is rather stable. We next consider the electronic properties. It is well known that bulk Bi is a semimetal with the valance band and conduction band overlapped at the centre of the hexagonal face in the Brillouin zone29. However, we see from Fig. 3 that monolayer Bi is semiconducting with a direct band gap of 0.55 eV at the Γ point. The opening of a band gap can be attributed to the well-known quantum confinement effect. If the SO coupling is explicitly included in the calculations, the valance band maximum (VBM) is a little away from the Γ point while the conduction band minimum (CBM) remains at the Γ point, and the band gap becomes 0.50 eV. As shown in Fig. 3, the difference in the band structure with and without SO coupling is obvious and will definitely affect the electronic transport properties. To get reliable results, the SO coupling is explicitly considered throughout this work. As mentioned before, the electronic transport coefficients of monolayer Bi can be derived from the energy band structure by using the Boltzmann theory and the relaxation time approximation. The accurate evaluation of the relaxation time depends on the detailed scattering mechanism. Here we use a simple approach in which the relaxation time τ is estimated by fitting the experimentally measured electrical conductivity or resistivity30. Unfortunately, such experimental data is not available for monolayer Bi, and as an alternative, we use the corresponding bulk value. This is a reliable approximation since monolayer Bi has very similar covalent bonding compared to that of the bulk structure, as confirmed by our calculated differential charge densities. Such an approach has been widely used in previous works modeling the electronic transport of nanoscale systems31,

32, 33

. The fitted relaxation time in the

temperature region from 100 K to 300 K is summarized in Table I. As the electron scattering is more frequent at high temperatures, we find that the relaxation time τ 7



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decreases with increasing temperature and can be fitted as

τ × 1013 = 62.9 exp(−T / 71.8) + 0.56 .

(5)

We assume that the relaxation time of a monolayer Bi is independent of carrier concentration since it does not vary much around the Fermi level. Fig. 4 shows the calculated room temperature Seebeck coefficient S, the electrical conductivity σ , the power factor S 2σ , and the electronic thermal conductivity κ e as a function of chemical potential. Within the rigid band approach34, the chemical potential indicates the doping level (or carrier concentration) of the system. In the case of n-type doping, the Fermi level ( µ =0) shifts up corresponding to µ >0; for p-type doping, the Fermi level moves down and µ

Thermoelectric Properties of a Monolayer Bismuth - The Journal of

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