Tree-Level Electron-Photon Interactions in Graphene
Graphene’s low-energy electronic excitations obey a 2+1 dimensional Dirac Hamiltonian. After extending this Hamiltonian to include interactions with a quantized electromagnetic field, we calculate the amplitude associated with the simplest, tree-level Feynman diagram: the vertex connecting a photon with two electrons. This amplitude leads to analytic expressions for the 3D angular dependence of photon emission, the photon-mediated electron-hole recombination rate, and corrections to graphene’s opacity and dynamic conductivity for situations away from thermal equilibrium, as would occur in a graphene laser. We find that Ohmic dissipation in perfect graphene can be attributed to spontaneous emission.
pacs:78.67.Wj, 78.67.Ch, 13.40.Hq
Electron-photon interactions determine the opto-electronic properties of a material. The electrons in graphene, a single atomic layer of graphite, exhibit superlative electronic properties associated with their exotic Hamiltonian Neto et al. (2009); Das Sarma et al. (2010). In particular, a tight binding model Wallace (1947) of graphene produces a Hamiltonian that, for low energy excitations, is formally identical to a 2+1 dimensional Dirac equation for massless fermions Semenoff (1984), with the Fermi velocity and the sublattice state vector filling the roles of the speed of light and spin respectively. As part of an effort to understand how electron-hole recombination might limit the function of a graphene-based transistor, we use this Dirac Hamiltonian to calculate the amplitude for the electron-photon interaction diagrammed in Fig. 1. Rotating this diagram with respect to the time axis allows the consideration of both photon emission (i.e. recombination) and absorption rates, which we relate to graphene’s opacity and dynamic conductivity.
These measurable Novoselov et al. (2005); Nair et al. (2008); Mak et al. (2008); Kuzmenko et al. (2008) properties have been previously treated using semiclassical methods (where the electromagnetic field is not quantized) within the Kubo and Landauer formalisms Ando et al. (2002); Gusynin et al. (2006); Ryu et al. (2007); Ziegler (2007); Vildanov (2009) and perturbation theory Nair et al. (2008); Lewkowicz and Rosenstein (2009). Our fully quantum mechanical calculation reproduces results found previously, such as for the optical opacity Nair et al. (2008); Kuzmenko et al. (2008); Stauber et al. (2008) and Gusynin et al. (2006); Ziegler (2007); Ryu et al. (2007); Stauber et al. (2008); Vildanov (2009); Lewkowicz and Rosenstein (2009) for the zero-temperature conductivity. We extend these previous results to non-equilibrium situations (e.g. population inversion) and specify the full angular dependence of photon emission/absorption. Furthermore, we identify spontaneous emission as the mechanism of dissipation, present even in idealized graphene, that is usually left unspecified Ziegler (2007); Stauber et al. (2008); Vildanov (2009); Lewkowicz and Rosenstein (2009).
The carbon atoms in graphene form a two-dimensional honeycomb network with two inequivalent atomic sites per unit cell. In the simplest tight-binding description of graphene, an electronic energy is associated with each atomic site in the sheet, and an energy parametrizes the probability of an electron hopping from one site to its neighbor on the other sublattice. An operator creates a electron on the ‘A’ site in cell , with a corresponding destruction operator . With similar operators for the ‘B’ sites, the total Hamiltonian is
where runs over the sites in the sheet, and runs over the nearest neighbors of the site . Spin indices on the operators and the sums are understood. Fourier transforming the creation and annihilation operators (e.g. , where the are the wavevectors in the first Brillouin zone) allows the Hamiltonian (1) to be written,
There are two spin states per , and two mobile electrons per cell, so the first Brillouin zone (Fig. 2) is exactly filled in electrically neutral graphene at zero temperature. The energy origin is set at the energy of the highest occuppied states, which are those at the Brillouin zone corners Bena and Montambaux (2009). The label indexes the two inequivalent corners. For near a point the single-particle Hamiltonian is
where the momentum . With defined by the in-plane components of according to , the corresponding eigenvalue equation is
where the band index labels whether the energy is positive or negative (i.e. conduction or valence). Thus the Hamiltonian (3) produces a linear dispersion relation . The product gives the helicity eigenvalue for the state , where the helicity operator is defined as .
Having identified the eigenspinors of the single particle Hamiltonian , we can re-write the total Hamiltonian in terms of operators that create () and destroy () energy eigenstates,
where the sum is over near and () refers to the conduction (valence) band.
We introduce the electromagnetic field with a Peierls (minimal coupling) substitution , treating the new vector potential term Sakurai (1967) as a quantized perturbation in the full Hamiltonian ,
Here indexes the photon’s polarization states, is the normalization volume, is the relative permittivity, and . As is evident from the appearance of the speed of light (and not the Fermi velocity ) in this substitution, the electron-photon coupling implied follows from the local gauge invariance of the standard model Lagrangian, and is not related to the properties of the Hamiltonian (3) under gauge transformations.
The electron-photon interaction rate can be calculated using the standard arguments of Fermi’s Golden Rule, suitably modified to account for the system’s mixed dimensionality. The rate to go from an eigenstate of the unperturbed electronic Hamiltonian to a given final state is
In the position representation, the time-dependent solutions to the unperturbed electronic have the form
where and is the graphene area. Initially we consider processes that create a valence electron and a photon , while destroying a conduction electron . Then
Derived from the -extent of the carbon atomic orbitals, the normalized function is only appreciable within a few angstroms of the graphene plane. Since we are considering photons with optical or longer wavelengths , the integral over gives unity to excellent approximation. In atomic physics this step applies to all three spatial dimensions () and is known as the dipole approximation.
We square , and consider the interval to be short compared to the lifetime and long compared to the time scale set by the energy of the transition, i.e. . In this limit, with large area ,
where we have used the standard relations , , and , with the upper (lower) sign chosen for bosons (fermions). Thus we see that the recombination rate is proportional to the number of conduction electrons and the number of holes . The first and second parts of correspond to spontaneous and stimulated emission respectively Lasher and Stern (1964).
To evaluate the angular matrix element in (11), we define an orthonormal triple , , and that describes the photon and its possible polarizations. Summing over the possible polarizations of the created photon gives
As does not appear in , the component of the photon polarization along does not contribute to this matrix element. The integrals over the energy and momentum -functions in (11) can now be performed, with the result
Since is a small number Novoselov et al. (2005), several approximations are in order. To better than 1% accuracy and . The energy of the initial conduction electron is half that of the photon, and of the same magnitude but opposite sign of the final valence electron. The photon’s momentum is negligible in comparison to the electrons’; as a result and transitions are impossible in this low energy limit. To lowest order in the angular dependence of (13) is . Thus for small a conduction electron is slightly more likely to emit a photon opposite than along . Figure 3 shows various plots of the angular distribution in the small limit, which we will adopt henceforth.
When averaged over the possible momentum directions of the conduction electron, the emission or absorption of a photon depends on the polar angle from the normal to the graphene sheet like . Because this function falls off more slowly than the Lambertian function , a graphene sheet will appear progressively brighter (i.e. blacker) at angles away from normal incidence. At this level of analysis the angular matrix element (12), and thus the rate, is zero for the metallic nanotube case Jiang et al. (2004); Mecklenburg and Regan (2010). The interaction Hamiltonian contains only photons polarized along the nanotube axis, and such photons do not couple the initial and final electronic states.
The form of the matrix element (12) indicates that angular momentum conservation is enforced in an unusual way in this problem. In a more conventional condensed matter system a typical optical transition involves bands with different orbital angular momentum quantum numbers, and allows the possibility of a spin flip. For instance, in gallium arsenide interband transitions occur between orbitals with and symmetries Zutic et al. (2004). Here the transition is and there is no spin flip. Thus the usual sources of angular momentum for the photon do not contribute in graphene. The structure of the matrix element (12), which follows directly from the Hamiltonian (3) and the assumption of minimal coupling, implies that the pseudospin flip creates the angular momentum of the photon. We further explore this connection between pseudospin and angular momentum elsewhere Mecklenburg and Regan (2010).
For states connected by the -functions in Eq. (11), the proportionality is general, and applies whether the ’s reflect equilibrium distributions or not. Non-thermal distributions are commonly handled by introducing a quasi-Fermi level that differs for electrons and holes Shockley and Read (1952). To simply illustrate the time scales relevant for photon-mediated electron-hole recombination, we consider a perfect population inversion, i.e. and . Integrating (13) over all directions of gives the rate for a conduction electron with energy to decay spontaneously () via photon emission,
where is the fine structure constant. This rate corresponds to a lifetime of about 3 ns for a 1 V conduction electron.
For thermal populations the averaged transition matrix element , where the density operators are given by and the trace is taken over the possible occupations: for the electrons, and for the photons Cohen-Tannoudji et al. (1977). Evaluating the trace gives Bose and Fermi distribution functions,
where we have allowed for a chemical potential . The second line of (Tree-Level Electron-Photon Interactions in Graphene) shows that recombination stimulated by the blackbody background becomes important for . At room temperature with a conduction state with energy V will be populated and decay with a characteristic lifetime of about 400 ns. For many practical purposes this rate is negligible, since, for instance, the second order (Auger) process gives picosecond lifetimes Rana (2007).
In contrast, the time-reverse of this recombination process, photon absorption, is observable practically by the unaided eye Kuzmenko et al. (2008); Nair et al. (2008). To analyze absorption we proceed as in the derivation of Eqs. (13–14), this time considering illumination normally incident on the graphene plane () at a rate . Then the promotion rate from the valence to the conduction band is
where we have included a factor of 4 for the valley and (normal) spin degeneracies. Discounting spontaneous emission into the illuminating beam, we take the net absorption rate to be the promotion rate minus the stimulated emission rate , which gives
With initial and Eq. (17) reproduces the result for the optical absorption of a graphene sheet Kuzmenko et al. (2008); Nair et al. (2008), and identifies spontaneous emission as the source of dissipation. For the absorption is negative, implying gain and the possibility of a graphene laser Lasher and Stern (1964); Ryzhii et al. (2007). As before, thermally averaging (17) replaces the ’s with Fermi functions, with the result that the absorption goes to zero for or .
We can relate the energy absorption rate implied by (17) to the conductivity by invoking Ohm’s Law, which implies that the power dissipated per unit area is . Here is the current density and is the electric field. Since the energy density of the electromagnetic field is , we have
which is at . This expression can be written
after thermal averaging, which is identical to the result found previously Stauber et al. (2008); Kuzmenko et al. (2008); Mak et al. (2008). Our calculation, like the previous ones, is not rigorous at , as the dc limit explicitly violates the assumption required to generate the energy -function in Eq. (11).
In conclusion, we have performed the first calculation of graphene’s optical properties with a quantized electromagnetic field. The calculation is fully quantum mechanical and free of thermodynamic assumptions until the final step, which allows the treatment of systems far from thermal equilibrium. Furthermore, the inherently three-dimensional formalism gives amplitudes as a function of photon polarization and propagation direction relative to the graphene plane. The dependence on electron and photon state occupation numbers follows directly from their fermionic and bosonic commutation relations. The former result in Pauli blocking, while the latter give terms that can be identified with spontaneous and stimulated emission. Spontaneous emission proves to be a source of dissipation present even in idealized graphene, with an implied violation of time-reversal symmetry whose introduction can be traced back to the use of Fermi’s Golden Rule. Stimulated emission from graphene could prove technologically useful, since an electronic population inversion would allow graphene to serve as the gain medium in a laser tunable over a broad band of frequencies.
This work was supported in part by the DARPA CERA program.
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