Magnetic-field-free thermoelectronic power conversion based on graphene and related two-dimensional materials

For the exploration of the outer solar system, thermoelectric generators powered by radioactive decay processes are commonly used with great success.¹ The optimization of thermal and electrical conductivities in thermoelectric materials is the key for the realization of efficient thermoelectric conversion.² Suitable materials are therefore systematically sought and optimized.³ As shown by Schlichter,⁴ vacuum may be used in converters as thermoelectric active component rather than a conventional material. Indeed, due to the absence of phonons or kinetic heat transport, vacuum is characterized by a low thermal conductivity. With ballistic charge transport, it may at the same time feature a high electric conductivity. Thermionic power generators in which electrons are thermally emitted from a hot electrode into vacuum to condense on a second, colder electrode, have therefore been investigated intensively over many decades for efficient energy conversion. Indeed, thermionic generators have been used with success to power two satellites.^5,6 Thermionic generators can, in principle, be highly energy efficient⁷ because fundamental loss channels of the conversion process are moderate and the Carnot efficiency as well as the Chambadal-Novikov efficiency,^8,9 the efficiency for device operation at maximum power output, are large.

These generators face the problem, however, that the space charge of the electrons ejected into the interelectrode space tends to block the emission of further electrons. To realize viable devices, this so called space-charge problem has to be overcome without introducing significant secondary losses. Today, several approaches are pursued to solve this problem: (a) separating the electrodes by a few micrometers only, which is sufficiently small to be comparable to the space charge screening length;^10–12 (b) neutralizing the space charge by injecting positive ions such as Cs⁺ into the space charge cloud;¹³ (c) using positively charged gate electrodes or grids to remove the space charges.¹⁴ In the past, Cs⁺ injection had been the concept of choice for the space charge mitigation. While ion injection based converters were successfully used in the satellites, the injection of ions into the space charge requires energy and almost halved the converter's efficiency, however.¹³ 

In all approaches that use gate electrodes to remove the space charges in thermionic generators, an electric field generated by a positive gate potential accelerates the electrons and thereby removes the space charge. This principle is explained with the band diagram shown in Fig. 1. The task of the gate electrode is to generate a potential trough typically 5–10 V deep, while absorbing as few electrons and as little heat as possible. The positive bias of the gate, however, tends to pull the electrons into the gate. This electron attraction has to be avoided, because gate currents dramatically reduce the conversion efficiency, as the resulting power loss scales with the gate voltage. In the devices explored in the past, gate currents have reduced the efficiency to lesser values that the gate-based thermionic converters have been dismissed as impractical⁷ and have never been realized.

In the thermoelectronic generators,¹⁴ the gate-current problem was solved by suppressing electron absorption in the gate by using microstructured gates together with a magnetic field of order 0.5 T applied in the emitter-collector direction. Such thermoelectronic converters are potentially highly efficient and work with thermionic, photo-enhanced thermionic, and photoeffect-based electron emission. The magnetic field forces the electrons onto cycloidal paths around the magnetic field lines. The electrons are therefore not deflected into the gate wires, which keeps the gate currents small and the collector currents proportional to the geometrical transparency of the gate.¹⁴ Magnets, however, add mass and complexity to the converters. The geometrical transparency we here define as the ratio of the areas of the patterned holes to the complete area of the grid.

The need for a magnetic field could be eliminated if the gates consisted of an electrically conducting material that is transparent to electrons with energies of several eV. All materials used in the past for gates absorb electrons well. As was recently found, however, under operation conditions viable for thermoelectronic generators, graphene has an electron absorption of only ∼40% for electrons with energies in the relevant energy range of 2–40 eV crossing the graphene perpendicular to the plane.¹⁵ Graphene or related two-dimensional materials therefore offer the possibility to realize electron-transparent gates for thermoelectronic energy conversion. Without magnets, such generators have very little mass, even more if operated in the vacuum of outer space. Their mass is determined only by the packaging and by the thermal and electric connections. Therefore, these generators also require only limited material resources.

Even for an electron transparency of 60%, however, gate losses are large. They even exceed the output power generated by the collector current. For practical applications, gate transparencies well above 90% are desirable. To enhance the electron transparency, the gates therefore need to be optimized, for example, by perforating them with holes¹⁶ as sketched in Fig. 2. Nanomechanical devices patterned from single graphene sheets demonstrate the astonishing resilience¹⁷ of graphene, indicating that such perforated single graphene sheets are indeed suitable for thermoelectronic generators. The thermal robustness of graphene has been demonstrated in Ref. 15 where the transparency of low energy electrons was measured in a triode setup similar to the one discussed here.

With holes, the electric fields generated by the graphene gates are inhomogeneous. The induced curvature of the electric field enhances the gate current if transverse electric field components deflect electrons into the graphene. Additional complications arise because the patterned graphene has to be supported by a carrier grid, as we will discuss below. To clarify to which extent the electric field inhomogeneities of patterned graphene enhance the gate currents, we have numerically calculated the electron paths in thermoelectronic converters with patterned gates using the IBSimu software package.^18,19 This code self-consistently solves the Poisson and Vlasov equations, taking space charge into account. The distribution of the electrostatic potential was modeled in two-dimensional and three-dimensional spaces. In these calculations, grids with pitch sizes ranging from 10 μm to 200 μm in three dimensions and from 10 μm to 160 μm in two dimensions were analyzed.²⁰ The work functions of emitter, gate, and collector were set to be the same, without the loss of generality.²¹ 

Fig. 3 shows the results of the two-dimensional calculations, displaying the equipotential lines and electron paths of thermoelectronic converters with emitter–collector distances of 500 μm. In each panel, a red trace highlights the trajectory of a typical electron. With decreasing distance between two grid pillars, the homogeneity of the electric field is enhanced, resulting in a strong increase of the transparency. The geometrical transparency of the grids used in this simulation was fixed to 80%. Three-dimensional simulations of the gate transparency are shown in Fig. 4. In this figure, the pitch size was varied while either keeping the grid thickness constant or scaling it congruently to the pitch size for the grid beams to maintain a square cross-section. Both curves show a steep increase of transparency for smaller pitches, almost approaching the geometrical transparency for a pitch of 1/50 of the emitter-collector distance.

Even with its superior resilience, graphene requires a support grid to be held in place over larger areas, as it will also be attracted by the emitter electrostatically. This case is analyzed in Fig. 5. The equipotential lines are curved only on the backside of the grid. Arriving electrons are accelerated by an almost homogeneous electric field. The rather bright background color of Fig. 5 resulting from parallel electron trajectories provides evidence for the high transparency of this grid configuration. The calculations reveal that a high transparency is achieved if two conditions are met: First, the pitch of the hole pattern has to be much smaller than the emitter-collector distance, to avoid bending of the electric field lines. Because of the scaling properties of the electric field, the results shown in Figs. 3 and 4 scale down to the length scale at which the atomic character of the grid is relevant. Therefore, these results are also applicable to two-dimensional materials such as graphene. We conclude that at emitter–collector distances of tens of micrometers, pitch sizes below 1 μm are required. Second, the work function of the carrier grid has to match the work function of graphene, ∼4.5 eV.²² A work function difference between the two materials would induce transverse components to the electron trajectories on the downstream side of the grid stack. Therefore, carbon-based carrier grids, or grids that are coated with a carbon layer are desired.

We conclude that holes with diameters d < 10 μm (for the simulation geometry) patterned into graphene enhance the effective electron transmission of the graphene grid proportional to the geometrical transparency of the hole-pattern, so that the transparency of the gate is given by the graphene transparency times the geometrical transparency. For example, a graphene sheet patterned into a square lattice with hole diameters of 1 μm and a periodicity of 1.1 μm has an absorption coefficient of 7%. For sufficiently small holes, the transparency is affected neither by the shape of the holes nor by their arrangement.

In addition to the absorption of electrons in the graphene, excitations of the graphene may cause inelastic scattering of the traversing electrons, thereby providing a second loss channel for the energy conversion process. In the relevant energy range, loss channels due to interband excitations and plasmons exist.^23,24 The losses do not affect electrons passing through the holes. Therefore, nano-patterning of holes minimizes electron absorption as well as inelastic energy losses in the gate. Electrons that have experienced a small energy loss may still reach the collector if the collector voltage is correspondingly reduced (with an equivalent reduction of the output voltage, however). The measured data of the graphene transparency in a triode setup given in Ref. 15 include these losses and they are therefore taken into account in our efficiency considerations below.

With this understanding, we propose a magnetic-field-free thermoelectronic converter that uses nanopatterned graphene or other nanopatterned two-dimensional materials as the gate electrode as illustrated in Fig. 2. Electrons are thermionically evaporated or are excited by photoelectronic processes from the hot emitter, are accelerated by a nano-patterned two-dimensional gate electrode, then decelerated as they approach the collector and return the part of their kinetic energy to the gate field they previously gained it from.

In addition to the gate, also the emitter and the collector may be fabricated from foils, or even from two-dimensional materials. Indeed, graphene has already been proposed as an emitter.²⁵ Also, as shown recently,²⁶ forests of carbon or other nanotubes grown on foils are especially promising material systems for electron emitters, as they are lightweight and can easily be locally heated by light irradiation to temperatures around 2000 °C due to the enormous heat trapping that is characteristic for these 1D-materials.²⁷ 

The efficiencies of thermoelectronic converters can in principle exceed 40%,¹⁴ with power densities of several W/cm². If two-dimensional materials are used, the mass of the key components such as emitter, gate, and collector is indeed minute. With an ultimate limit of the mass of 10⁻⁹ kg/cm², an upper limit of the specific power density of 10¹⁰ W/kg is obtained. A lighter generator seems hard to conceive, as any generator has to include a hot and a cold electrode, which cannot be thinner than a monoatomic layer. In practice, materials need to be mounted on a carrier such as a silicon wafer, possibly using spacer foils, and be equipped with contacts and thermal shielding.

In terrestrial applications, vacuum encapsulation needs to be provided. For this, there are at least two approaches: vacuum encapsulation within the wafer,²⁸ as seems preferable for small devices, or vacuum packaging of large wafer assemblies, e.g., for large-scale generators to be used in coal combustion plants or house-heating units. For wafer-based converters, the wafer mass is important for determining the weight. For 10 W/cm² to be generated on a 0.5 mm thick wafer, the upper limit to the specific power density is of 10⁵ W/kg. This limit of the specific power density not only applies to large generators but also valid for small converters with an output power of, say, less than a Watt. In contrast to energy converters such as jet engines or gas turbines, very high specific power densities may therefore be achieved by small and lightweight devices. While, of course, the limits cannot be reached in practice, these estimations nevertheless suggest that thermoelectronic converters are interesting candidates for power generators with very high specific power densities.

Questions remain to be answered and problems need to be solved before such converters can be built. For example, adsorbate formation on the graphene needs to be limited to keep the electron transparency at desirable levels. Further, the mounting of the graphene has to ensure acceptable values of the mechanical and thermal stresses to which the graphene is exposed. Yet, we also point out that the principle of using patterned 2D materials to create ambipolar, homogeneous electric fields may find applications in vacuum-electronic devices other than thermoelectronic converters, such as electron guns, vacuum tubes, or nanoelectronic devices based on the electron propagation across vacuum gaps.

The authors acknowledge T. H. Geballe, T. Pan, P. Herlinger, and J. H. Smet for helpful discussions as well as for the support by the DFG (Leibniz Grant).