Maximum Thermoelectric Power Factor and Optimal Carrier Concentration of Bilayer Graphene at Various Temperatures

a Department of Electrical Engineering, Faculty of Engineering, Tokyo University of Science, Niijuku 6-3-1, Tokyo 125-8585, Japan b Department of Physics, Faculty of Science, Tokyo University of Science, Kagurazaka 1-3, Tokyo 162-8601, Japan c Research Institute for Science and Technology, Tokyo University of Science, Kagurazaka 1-3, Tokyo 162-8601, Japan † Corresponding author: takahiro@rs.tus.ac.jp

The thermoelectric response of bilayer graphene over a wide temperature range (0 < ≤ 400 K) was theoretically investigated using linear response theory combined with a Green's function technique. We found that the power factor for a fixed chemical potential exhibits a maximum at a certain . On the other hand, we found that the for a fixed exhibits a maximum ( max ) at an optimal [or optimal carrier concentration ( opt )]. In addition, we clarified the dependence of opt and max and explained the existence of opt in terms of the thermal excitation of electrons between the valence and conduction bands, which cannot be predicted by the conventional Mott formula.
In particular, bilayer graphene (BLG), which consists of two-layered graphene, is a new type of nanocarbon TE material whose TE performance can be controlled by applying an electric field normal to the BLG surface. In our previous study [24], we theoretically showed that the TE power factor ( ) for BLG can be enhanced by the applied electric field. We also determined the optimal chemical potential opt that gives the maximum power factor ( max ) for different electric fields at 300 K [24]. For example, the room-temperature max for BLG reaches approximately 7 mW K −2 m −1 in the absence of an applied electric field, and this theoretical prediction is in excellent agreement with the results of a recent TE experiment of BLG using electric double-layer transistors with ionic liquids [25]. However, the temperature dependences of max and opt (or corresponding carrier concentration) for BLG have not yet been elucidated even though they are important for designing BLG-based TE devices for operation at various environmental temperatures.
In the present study, we theoretically investigate the TE power factor for BLG over a wide temperature range (0 < ≤ 400 K) using linear response theory [26,27] combined with a Green's function technique.

A. Electronic states of bilayer graphene
The lattice structure of AB (Bernal)-stacked BLG [28] is shown in Figure 1. Because four carbon atoms are included in the unit cell, the Hamiltonian for this system can be described by a 4 × 4 π-orbital tight-binding Hamiltonian. However, to describe the low-energy physics in the temperature range 0 < ≤ 400 K, the Hamiltonian can be reduced to a 2 × 2 effective Hamiltonian given by [24, 29−31] is the effective mass for BLG, (≡ √3 0 2ℏ ⁄ ≈ 10 6 m s −1 ) is the group velocity for the monolayer graphene, (= | 1 | = | 2 | = 0.246 nm) is the lattice constant, ℏ is the Dirac constant, 0 (=3.033 eV) is the intralayer hopping integral between 1(2)A and 1(2)B, and 1 (= 0.390 eV) is the interlayer hopping integral between 1A and 2B [32]. The interlayer hopping integrals between 1B and 2A and between 1A(B) and 2A(B) are neglected because these effects are negligibly small in the low-energy region [33,34]. Here, = � , � is the wavenumber vector measured from the K point. By diagonalizing the 2 × 2 Hamiltonian in Eq. (1), we obtain the parabolic energy dispersions ± ( ) as Here, the origin of energy ( = 0 ) is chosen as the charge-neutral point (CNP) for the BLG and is defined as ≡ � 2 + 2 . Next, to account for the effects of disorder on the BLG, we introduce a retarded/advanced Green's function given by where is a 2 × 2 identity matrix and Σ R A ⁄ ( , ) is the retarded/advanced self-energy due to the disorder potential. Similar to the theoretical procedure described in our previous study [24], the k dependence of Σ R A ⁄ ( , ) is neglected; that is, Here, ( ) is the relaxation time, which depends on but not on k. The dependence of ( ) is given by where c , , and (0 < ≤ 1) are fitting parameters. In the reproduction of the experimental TE results for BLG in Ref. 25, these three parameters are determined as c = 60 fs, = 1.42 fs eV, and = 1.00, respectively. Similar behavior of ( ) has been experimentally confirmed elsewhere [35]. Using (∞) = c = 60 fs and the group velocity g~1 0 6 m s −1 in the high-| | region, the mean free path m is estimated as ~100 nm, which is consistent with the averaged grain size in the polycrystalline graphene reported in Ref. 36. On the other hand, m due to phonons is longer than that due to the grain boundaries in the region of ≤ 300 K except for the extremely low-carrier-concentration case [37]. Therefore, we assume that the electron scattering characterized by ( ) in Eq. (5) is mainly originated from the elastic scattering at the grain boundaries and the electron−phonon interaction can be neglected in this study. Under this assumption, the Green's functions can be straightforwardly rewritten as Here, the imaginary part of the square root in Eq. (7) is chosen to be positive. Using the retarded Green's function in Eq.
(6), we can calculate the density of states (DOS) ( ) per unit area of the BLG as which is independent of . Here, Ω is the system area and S (= 2) and V (= 2) are the spin and valley degrees of freedom, respectively. Interestingly, the DOS for BLG with disorder in Eq. (8) coincides with that without disorder obtained from the dispersions in Eq. (2). Using the DOS in Eq. (8), we express the net carrier concentration by Here, e ( , ) and h ( , ) are the electron and hole concentrations, defined as and is the Boltzmann constant, and is the chemical potential.

B. Thermoelectric effects in bilayer graphene
In linear response theory, the Seebeck coefficient is defined as the electric field induced by a finite temperature gradient under the condition of no electrical current density, which leads to .
Similarly to in Eq. (12), the power factor PF (≡ 11 2 ) is also expressed in terms of 11  .
Thus, we can easily obtain and PF after we obtain 11 and 12 . Here, Onsager's coefficients 11 and 12 are called the electrical conductivity and the thermoelectrical conductivity, respectively. For the present system (i.e., BLG with a disorder potential), 11 and 12 can be expressed as Sommerfeld-Bethe relations [15−17, 24, 38, 39]: and Here, is the elementary charge and ( ) is the spectral conductivity (i.e., the electrical conductivity at T = 0). Using the retarded/advanced Green's function in Eq. (6) and the velocity matrix ( ) ( = ℏ −1 ( )/ ) in the x-direction along the temperature gradient of the BLG, the spectral conductivity can be calculated as Here, is the system volume and (≈ 7 Å) is the thickness of the BLG.

A. Spectral conductivity
As already mentioned in Sec. II.A, we chose = 1.42 fs eV, c = 60 fs, and = 1.00 in Eq. (5), which reproduce the experimental results at = 300 K in Ref. 25. Figure 2 shows ( ) for ( ) = 60 fs + 1.42/ | | (black solid curve) with constant (= 60 fs) (black dashed curve). In the limit of | | → c ⁄ , the solid curve approaches the dashed curve; by contrast, in the present region of | | 1 ≲ ⁄ 0.8 ( c | | ≲ 18.7 fs eV) shown in Figure 2, the influence of = 1.42 fs eV on ( ) is not negligible. In the weak-disorder region of | | ( ) ⁄ = 2( c | | + ) ℏ ⁄ ≫ 1 , which means that | | is much larger than the characteristic energy Note that, in the present case of = 1.42 fs eV, the whole region can be regarded as the weak-disorder region of | | ( ) ⁄ ≫ 1, where ( ) can be well approximated by Eq. (18), as shown by the red dotted line in Figure 2.

B. Electrical conductivity and thermoelectrical conductivity
Using ( ) in Figure 2, the electrical conductivity 11 and the thermoelectrical conductivity 12 can be obtained from Eqs. (14) and (15), respectively. In the present study, we use three values of where the minus (plus) sign corresponds to the case of > 0 ( < 0). Notably, Eq. (20) is independent of , as shown by the dashed line in Figure 3(b). As increases, 12 deviates downward from the 2 behavior at B | | ⁄ 0.1 because of cancellation of the contributions from conduction electrons and valence holes.

C. Seebeck coefficient and power factor
Substituting the 11 data in Figure 3(a) and the 12 data in Figure 3(b) into Eqs. (12) and (13), we obtained the Seebeck coefficient and the power factor . Figure 4(a) represents the dependence of for BLG for / 1 = −0.1 (black solid curve), −0.3 (blue solid curve), and −0.5 (red solid curve). In the low-T region, S increases linearly with increasing T irrespective of . Such behavior can be expressed by the well-known Mott formula [40] as which can be derived by substituting Eqs. (19) and (20) into Eq. (12). Here, the plus (minus) sign is for the case of > 0 ( < 0). As shown in Figure 4(a), for the three values deviates downward from the prediction of Mott formula in Eq. (21), which is shown by the dashed lines at | | ~ 0.1 ⁄ , corresponding to the temperatures where 11 in Figure 3(a) and 12 in Figure 3(b) deviate from Eqs. (19) and (20), respectively. As increases, for 1 ⁄ = −0.1 eventually begins to decrease at ~ 260 K. That is, 2, and ~77.0 μV K −1 , respectively. We emphasize that the appearance of the peak in cannot be predicted by the conventional Mott formula. Figure 4(b) displays the dependence of the for BLG for 1 ⁄ = −0.1 (black solid curve), −0.3 (blue solid curve), and −0.5 (red solid curve). The extremely low-T exhibits 2 behavior, which can be expressed by Eq. (22), which is derived from Mott formula as and is displayed by the dashed curves. Similarly to , when increases, shows a decrease from the 2 curve at | | ~ 0.1 ⁄ and the peak. The peaks in are observed at ~ 330, ~900, and ~1240 K for From another perspective, exhibits a maximum value (≡ max ) for the optimal chemical potential (≡ opt ) at fixed . Figure 5(a) shows the dependence of the for BLG for = 150 K (black curve), 250 K (blue curve), and 350 K (red curve). As can be seen, max depends on and appears at opt 1 ⁄ −0.12, −0.19, and −0.25 for = 150, 250, and 350 K, respectively, as indicated by the arrows. Next, we show opt [or the optimal carrier concentration opt ( ≡ Δ � opt , � = � opt , � − ℎ � opt , � in Eq. (9))] and max as functions of T. Figure 5(b) shows the T dependence of opt and the corresponding opt for the BLG. max is shown in the inset as a function of . It is evident that max and opt monotonically increase with increasing and eventually reach ~11 mW K −2 m −1 and ~3.3 ×10 12 cm −2 at = 400 K, respectively. Therefore, to maximize the for BLG in the region 0 < ≤ 400 K, carrier doping should be performed to the concentration of ~3.3 × 10 12 cm −2 .

IV. CONCLUSION
We investigated the TE properties of BLG over a wide range (0 < ≤ 400 K) on the basis of linear response theory combined with a Green's function technique. and for fixed show a peak at certain values because of  thermal excitation of electrons between the valence and conduction bands, which cannot be predicted by the conventional Mott formula. Moreover, the values for a fixed have maximum values ( max ) at the optimal chemical potentials ( opt ), and we determined opt and the corresponding carrier concentration opt as functions of . As increases, opt monotonically increases, eventually reaching ~3.3 × 10 12 cm −2 at = 400 K, where max is ~11 mW K −2 m −1 . On the basis of these findings, we expect that the for BLG is optimized by optimal carrier doping using electric double-layer transistors with ionic liquids, as reported in Ref. 25.