Enhancement of gaps in thin graphitic films for heterostructure formation

There are a large number of atomically thin graphitic films with a structure similar to that of graphene. These films have a spread of band gaps relating to their ionicity and, also, to the substrate on which they are grown. Such films could have a range of applications in digital electronics, where graphene is difficult to use. I use the dynamical cluster approximation to show how electron-phonon coupling between film and substrate can enhance these gaps in a way that depends on the range and strength of the coupling. It is found that one of the driving factors in this effect is a charge density wave instability for electrons on a honeycomb lattice that can open a gap in monolayer graphene. The enhancement at intermediate coupling is sufficiently large that spatially varying substrates and superstrates could be used to create heterostructures in thin graphitic films with position-dependent electron-phonon coupling and gaps, leading to advanced electronic components.

Figure 1

Dynamical cluster approximation (DCA) subzones for cluster sizes of up to $N_C=100$. Axes show the $x$ and $y$ components of the momentum, $k_x$ and $k_y$. Within each subzone, the self-energy is taken to be momentum independent. This allows Green functions to be calculated in the thermodynamic limit, and convergence properties are particularly good as $N_C$ is increased. Note that the only symmetry taken into account is translation in $\mathbf{k}$ space.

Figure 2

Feynman diagrams showing the perturbation theory used in the work presented here. (a) Hartree diagram. For symmetry-broken states, this cannot be absorbed into the chemical potential and is the main contributor to modification of the gap. (b) The Fock potential is responsible for the frequency dependence of the self-energy. (c) Dyson equation for the phonon propagator. This renormalizes the phonon frequency, which can lead to further enhancement of the band gap.

Figure 3

Values of ${\lambda }_{XY}\left({\mathbf{K}}_i\right)$ plotted for zones that are centered on the high-symmetry directions. The Fröhlich interaction used has an average $\lambda$ value of 1, so due to the inhomogeneity of the interaction across the Brillouin zone, the interaction is $>$1 in unit cells close to the $\Gamma$ point and $<$1 elsewhere. The peak is suppressed for small clusters due to averaging across DCA subzones but reaches the peak value for cluster sizes of order 144. The difference between ${\lambda }_{AA}\left(\mathbf{K}\right)$ and ${\lambda }_{AB}\left(\mathbf{K}\right)$ at $\Gamma$ is important for the effective interaction strength in the Hartree diagram.

Figure 4

Comparison of the Hartree and Fock parts of the self-energy for the case of the Holstein interaction, $\Sigma ={\Sigma }^{\prime }+i{\Sigma }^{\prime \prime }$, vs the Matsubara frequency in a nine-site cluster. The largest element in the self-energy matrix is the real part of the on-diagonal Hartree term, which is momentum independent, and it is this value that defines the size of the gap. Left panels: Individual Hartree and Fock contributions to the gap. Right panels: Momentum dependence of the total self-energy. ${\Sigma }_{AA}$ is essentially momentum independent (so in the upper panel, curves for points K and $\Gamma$ lie directly under those for point M), and ${\Sigma }_{AB}^{\prime }$ has a weak momentum dependence, so the points can only be differentiated under a high magnification. ${\Sigma }_{AB}^{\prime \prime }$ is very small. Primed points are related to unprimed ones by inversion around the origin. (N.B.: The Fock term is still the most important contribution in some cases; for example, the inverse mass depends on derivatives of the self-energy, so this will be given by the Fock term.) The points marked M and M${}^{\prime }$ represent zones that border on point M(but where the DCA subzone center in the $N_C=9$ cluster is offset slightly from point M so that the values are distinct). $T=0.02t$, $\Delta =0.1t$, $\Omega =0.01t$, and $\lambda =2$.

Figure 5

Comparison of the Hartree and Fock parts of the self-energy for a Fröhlich interaction, $\Sigma ={\Sigma }^{\prime }+i{\Sigma }^{\prime \prime }$, vs Matsubara frequency. On-diagonal contributions can be seen at the top, and off-diagonal ones at the bottom. Again, the largest contribution to the self-energy matrix is the real part of the on-diagonal Hartree term, which defines the gap and is momentum independent. As in the Holstein case, the Hartree contibution to the off-diagonal self-energy is necessarily 0, and the Fock diagram is the largest off-diagonal contribution, although it is still about one-fourth the magnitude of the on-diagonal Hartree term. The Fock term has a significantly bigger momentum dependence than in the Holstein case. (N.B.: The contributions to ${\Sigma }_{AB}^{\prime \prime }$ for points K, K${}^{\prime }$, and $\Gamma$ are all 0 so cannot be distinguished from each other.) Here, $T=0.02t$, $\Delta =0.1t$, $\Omega =0.01t$, $\lambda =2$, and $N_C=9$. The Fock contributions become relatively smaller compared to the Hartree term as the cluster size increases.

Figure 6

(a) Gap enhancement ${\Delta }^{\prime }/\Delta$ vs electron-phonon coupling $\lambda$ for a Holstein interaction with $\Delta /t=0.1$, comparing results from the dynamical mean-field theory (corresponding to $N_C=1$) and the DCA. There are only small corrections due to momentum dependence. The inclusion of the phonon self energy (polarization bubble, PB) is responsible for a significant increase in the gap. (b) Gap enhancement for the long-range Fröhlich interaction, $R_{\mathrm{sc}}=2a$. (c) Gap enhancement for the long-range Fröhlich interaction. Here $T=0.02t$ and $\Omega =0.01t$. The initial increase in the gap as $N_C$ increases arises because the long-range interaction is not homogeneous across the Brillouin zone, so the effective value of $\lambda (\mathbf{Q}=0)$ (which is relevant to the Hartree diagram) is larger than the average $\lambda$ in all cluster sizes except $N_C=1$. The enhancement is essentially converged for cluster sizes of $N_c=144$. (N.B.: Since $\Delta =0.1t$, the gap enhancement is 10 times larger than the gap, ${\Delta }^{\prime }$, which can be read from the real part of the self-energy at a high Matsubara frequency.)

Figure 7

Comparisons of the phonon self-energy for Holstein and Fröhlich interactions with various $\lambda$ values. As the interaction strength increases, the phonon self-energy decreases due to the gap at the Fermi energy, which reduces the value of the Green function at low Matsubara frequencies. The self-energy is only weakly momentum dependent, but it is possible to discern the variation across the Brillouin zone.

Figure 8

Effective coupling in the Holstein model for a range of couplings, $D_{XY}(Q,0)$. The effective interaction becomes larger more rapidly than $\lambda$, and the form changes from momentum independent to weakly momentum dependent.

Figure 9

Effective coupling in the Fröhlich model for a range of couplings, $D_{XY}(Q,0)$. This property is strongly momentum dependent, and the momentum dependence becomes slightly larger as coupling increases. It is the effective coupling that leads to nonlocal position space variations that require large clusters to treat.

Figure 10

Corrections taking account of the finite depth of the bulk substrate. The traces show the effect of interactions between electrons in the monolayer and vibrations in the surface atoms only ($N_z=1$) and interactions with vibrations in three and six layers of bulk substrate, respectively. (N.B.: Electrons only hop in the monolayer, and there is no hopping into the bulk of the substrate. The effect of the bulk of the substrate is to reduce the enhanced gap by around 20%.)

Figure 11

Effect of taking ${\lambda }_A\ne {\lambda }_B$ when $R_{\mathrm{sc}}=2a$ and $N_C=36$. Since electrons on different sublattices are contained in orbitals of different atoms, the interaction between electrons and phonons may depend on the sublattice. The enhancement is mainly determined by the largest of ${\lambda }_A$ or ${\lambda }_B$, with small changes to the overall enhancement as the other coupling is varied.

Figure 12

(a) Spontaneous symmetry breaking (gap generation) in a graphene monolayer. The $y$ axis shows the spontaneously formed gap ${\Delta }^{\prime }$; the $x$ axis, the electron-phonon coupling $\lambda$. Here $T=0.02t$, $\Omega =0.01t$, $N_C=1$, and $\Delta =0$. (b) Effect of cluster size on the CDW state formed from a Fröhlich interaction with $R_{\mathrm{sc}}=a$. (N.B.: A single point is calculated for a cluster size of $N_C=225$ to confirm convergence.)

Figure 13

Schematic showing the possible use of two substrates with different electron-phonon coupling, $\lambda$, with thin graphitic films to make a heterostructure in a single thin film of a graphitic material. This could be manufactured by laying down an interface between the two substrate materials before cleaving perpendicular to the interface and then depositing the thin film. The film above the substrate with the largest $\lambda$ would have a bigger gap, leading to a heterostructure within the interface region.