Phonovoltaic. III. Electron-phonon coupling and figure of merit of graphene:BN

The phonovoltaic cell harvests optical phonons like a photovoltaic harvests photons, that is, a nonequilibrium (hot) population of optical phonons (at temperature $T_{p,\mathrm{O}}$) more energetic than the band gap produces electron-hole pairs in a $p\text{−}n$ junction, which separates these pairs to produce power. A phonovoltaic material requires an optical phonon mode more energetic than its band gap and much more energetic than the thermal energy $\left(E_{p,\mathrm{O}}>\mathrm{\Delta }{E}_{e,g}\gg k_{\mathrm{B}}T\right)$, which relaxes by generating electrons and power (at rate ${\dot{\gamma }}_{e\text{−}p}$) rather than acoustic phonons and heat (at rate ${\dot{\gamma }}_{p\text{−}p}$). Graphene (h-C) is the most promising material candidate: when its band gap is tuned to its optical phonon energy without greatly reducing the electron-phonon $\left(e\text{−}p\right)$ coupling, it reaches a substantial figure of merit $[Z_{\mathrm{pV}}=\mathrm{\Delta }{E}_{e,g}{\dot{\gamma }}_{e\text{−}p}/E_{p,\mathrm{O}}({\dot{\gamma }}_{e\text{−}p}+{\dot{\gamma }}_{p\text{−}p})\approx 0.8]$. A simple tight-binding (TB) model presented here predicts that lifting the sublattice symmetry of graphene in order to open a band gap proscribes the $e\text{−}p$ interaction at the band edge, such that ${\dot{\gamma }}_{e\text{−}p}\to 0$ as $\mathrm{\Delta }{E}_{e,g}\to E_{p,\mathrm{O}}$. However, ab initio (DFT-LDA) simulations of layered h-C/BN and substitutional h-C:BN show that the $e\text{−}p$ coupling remains substantial in these asymmetric crystals. Indeed, h-C:BN achieves a high figure of merit $\left(Z_{\mathrm{pV}}\approx 0.6\right)$. At 300 K and for a Carnot limit of $0.5(T_{p,\mathrm{O}}=600\text{K})$, a h-C:BN phonovoltaic can reach an efficiency of ${\eta }_{\mathrm{pV}}\approx 0.2$, double the thermoelectric efficiency $\left(ZT\approx 1\right)$ under similar conditions.

Figure 1

(a) The phonovoltaic (pV) cell and (b) its energy diagram, which shows the quasi-Fermi level $\left(E_{\mathrm{F}}\right)$, conduction and valence bands $(E_{e,c},E_{e,v})$, and the generation of an electron-hole pair. In a pV, a source excites a population of optical phonon modes more energetic than the band gap in a $p\text{−}n$ junction. This population relaxes by producing electrons (and power) or acoustic phonons (and heat).

Figure 2

The band gap and optical phonon energy of various semiconductors. An efficient pV requires $E_{p,\mathrm{O}}>\mathrm{\Delta }{E}_{e,g}\gg k_{\mathrm{B}}T$, as in tuned graphene. In typical materials at 300 K, however, $\mathrm{\Delta }{E}_{e,g}\gg E_{p,\mathrm{O}}$ or $E_{p,\mathrm{O}}\approx k_{\mathrm{B}}T$, as the strong bonds which enable energetic phonon modes also localize electrons and open a large band gap. The symmetry of group IV, $s{p}^2$ coordinated materials (e.g., graphene) enables them to overcome this trend. For traditional elemental and composite semiconductors, see Refs. [3, 4, 5]. For graphame (h-C:H), see Refs. [2, 6]. For materials with the $s{p}^1$ acetylene bond, graphdiyne, and BNyne, see Refs. [7, 8].

Figure 3

(a) The graphene sheet, its geometry, and Brillouin zone (BZ). In graphene, the $s{p}^2$ hybridized orbitals of carbon atoms form strong bonds with the three nearest neighbors in a hexagonal lattice (defined by two vectors $a_1$ and $a_2$ of length $a)$ with two symmetric, triangular sublattices, separated by ${\mathbf{\delta }}_1$. (b) Two h-C:BN structures. Boron and nitrogen atoms are distributed equally into both sublattices in the disordered h-C:BN structure [h-${\mathrm{C}}_{28}$(BN)${}_{2,\mathrm{d}}]$ and into separate sublattices in the ordered structure [h-${\mathrm{C}}_{48}$(BN)${}_{1,\mathrm{o}}]$. (c) h-C/BN bilayers. Real bilayers form Moire superlattices rather than an ideally stacked structure due to the small difference in the h-C and h-BN lattice constants.

Figure 4

The ratio of the interband generation rate to the total rate of optical phonon absorption by electrons in graphene. When the optical phonon energy is much larger than $k_{\mathrm{B}}T$ in an intrinsic semiconductor, the rate of interband generation events dominates the rate of intraband heating events until $\mathrm{\Delta }{E}_{e,g}\to E_{p,\mathrm{O}}$. As the optical phonon energy approaches $k_{\mathrm{B}}T$ or as the Fermi level approaches the conduction band edge $\left(E_{\mathrm{F}}\to E_{e,c}\right)$, the interband interactions dominate over a much smaller range of band gap energies. When $E_{p,\mathrm{O}}/k_{\mathrm{B}}T<5$, the intraband interactions always dominate.

Figure 5

(a) The band structure $\left({\epsilon }_{\pm }\right)$ and (b) fraction of the electron wave function composed by atomic orbitals centered on atoms in one of the two sublattices $\left(|{\beta }_i^{\pm }|/\sum |{\beta }_i^{\pm }|\right)$ and resulting overlap $\left(I_{{\mathit{k}}_e,{{\mathit{k}}_e}^{\prime }}^{\pm }\right)$ of the valence (-) and conduction(+) bands with (—) and without () sublattice symmetry. In a symmetric crystal like graphene, no gap exists and the valence- and conduction-band wave functions overlap. In a nonsymmetric crystal like h-BN, a band gap opens, the valence and conduction bands collapse into atomic orbitals centered on different sublattices, and the overlap between them vanishes at the band edge, proscribing the $e\text{−}p$ interaction.

Figure 6

The band gap of h-C:BN and h-C/BN for variations in the BN concentration. The band gap tends to increase as the concentration increases, particularly when the B and N atoms are substituted into different sublattices (ordered h-C:${\mathrm{BN}}_o)$, as previously predicted [23]. Conversely, the band gap widens slowly with the disordered substitution of BN onto random sublattices (h-C:${\mathrm{BN}}_d)$. The experimental results [24] show that the gap widens even more quickly than in h-C:${\mathrm{BN}}_o$. Regardless, a few of the simulated crystals meet the primary pV condition: $E_{p,\mathrm{O}}>\mathrm{\Delta }{E}_{e,g}$.

Figure 7

The electronic structure of (a) h-C and h-BN, (b) h-C/BN, and (c) h-${\mathrm{C}}_{48}({\mathrm{BN})}_{1,o}$. The band structure of h-C/BN strongly resembles the superposition of the h-C and h-BN band structures. The band structure of h-${\mathrm{C}}_{48}({\mathrm{BN})}_{1,o}$, although complicated by the band folding, resembles that of h-C. However, the $\pi$ bands are less energetic and have some dispersion throughout the BZ. Note the narrow band gap formed in h-C:BN and h-C/BN and the large band gap of h-BN.

Figure 8

The phonon density of states $\left(D_p\right)$ for h-C:BN and h-C/BN. The density of states in h-C/BN is nearly the superposition of that in h-C and h-BN. While the density of states in h-C:BN strongly resembles that of h-C, the h-C:BN $D_p$ calculation uses a single ab initio dynamical matrix (at ${\mathit{k}}_p=\mathrm{\Gamma })$. Therefore the $D_p$ is not as well resolved.

Figure 9

Ab initio (a) interband $e\text{−}p$ coupling and (b) $e\text{−}p$ scattering rate in h-C, h-BN, h-C/BN, and h-C:BN for the zone-center, LO phonon mode. The BZ and high-symmetry points are shown, along with the ring for which energy is conserved $\left({\delta }_{\mathrm{E}}\right)$. Also shown are two TB models: TB 1, which uses a scaled overlap integral [Eq. (26)] as the $e\text{−}p$ matrix element, and TB 2, which assumes the matrix element is independent of the band gap. The scattering rate is approximately halved when h-C is doped with BN. However, the $e\text{−}p$ coupling remains strong at the K and $K^{\prime }$ points, regardless of the asymmetry and band gap. Therefore ${\dot{\gamma }}_{e\text{−}p}$ is nearly independent of $\mathrm{\Delta }{E}_{e,g}$ until $\mathrm{\Delta }{E}_{e,g}\simeq E_{p,\mathrm{O}}$, at which point the number of states available for interband transitions vanishes. The TB 2 model predicts this trend.

Figure 10

(a) The $p\text{−}p$ coupling in the first BZ, (b) down-conversion pathways, (c) down-converted phonon energy distribution, and (d) down-conversion rates in h-C and h-C/BN. (a), (b), and (c) combine to show the momentum and energy of the acoustic phonon modes produced during down-conversion of the LO phonon. These results show that the $p\text{−}p$ coupling is nearly unaffected by the disruption of the sublattice symmetry, such that the down-conversion rate of h-C/BN approximately equals that in graphene.

Figure 11

(a) Spectral function and (b) defect scattering rate for h-C:BN. While the rate of scattering is negligible at a low BN concentration, it increases substantially with the BN concentration. However, only a small BN concentration is required to tune the band gap to the optical phonon energy [24].

Figure 12

(a) The optical phonon scattering rates of h-C, h-BN, h-C/BN, and h-C:BN and (b) resulting h-C:BN performance metrics. The TB 2 model (Sec. 5c1) is used to model the ${\dot{\gamma }}_{e\text{−}p}$ trends and interpolate performance between the ab initio h-C:BN results. While the incorporation of BN in h-C disrupts its strong $e\text{−}p$ coupling, ${\dot{\gamma }}_{e\text{−}p}$ remains much faster than ${\dot{\gamma }}_{p\text{−}p}$ until $\mathrm{\Delta }{E}_{e,g}\to E_{p,\mathrm{O}}$. Moreover, the defect scattering rate remains negligible in comparison to both ${\dot{\gamma }}_{e\text{−}p}$ and ${\dot{\gamma }}_{p\text{−}p}$. Thus the ${\dot{\gamma }}_{e\text{−}p}^{*}$ in h-C:BN remains near 0.75 and $Z_{\mathrm{pV}}$ nearly reaches 0.7 at $\mathrm{\Delta }{E}_{e,g}=181$ meV. As the nonequilibrium between optical phonon population and cell increase $\left({\eta }_{\mathrm{C}}\to 1\right)$, the fill factor of the cell increases, and the efficiency $\left({\eta }_{p}V\right)$ approaches the figure of merit $\left(Z_{\mathrm{pV}}\right)$ times the Carnot limit $\left({\eta }_{\mathrm{C}}\right)$, as shown using the color gradation (Sec. 1a).