Abstract
Seeking for thermoelectric (TE) materials with high figure of merit (or ZT), which can directly converts lowgrade wasted heat (400 to 500 K) into electricity, has been a big challenge. Inspired by the concept of multilayer thermionic devices, we propose and design a solidstate thermionic devices (as a power generator or a refrigerator) in using van der Waals (vdW) heterostructure sandwiched between two graphene electrodes, to achieve high energy conversion efficiency in the temperature range of 400 to 500 K. The vdW heterostructure is composed of suitable multiple layers of transition metal dichalcogenides (TMDs), such as MoS_{2}, MoSe_{2}, WS_{2} and WSe_{2}. From our calculations, WSe_{2} and MoSe_{2} are identified as two ideal TMDs (using the reported experimental material’s properties), which can harvest waste heat at 400 K with efficiencies about 7% to 8%. To our best knowledge, this design is the first in combining the advantages of graphene electrodes and TMDs to function as a thermionicbased device.
Introduction
The most common approach to harvest the waste heat to generate electricity is thermoelectrics (TE), which is based on the Seebeck effect (see Table 1). The performance of TEbased devices is characterized by the figure of merit (ZT), given by ref. 1
where α, T, κ_{l}, μ, n, and e are, respectively, the Seebeck coefficient, absolute mean temperature, lattice thermal conductivity, carrier mobility, carrier density and electron charge. Here, L is defined as the Lorenz number equal to 2.44 × 10^{−8} WΩ K^{−2}. This formula has recently been redefined to solve the inconsistence between theoretical predication and experimental measurement^{2}. Before the 1990s, the progress of improving ZT had been slow and the best TE material was Bi_{2}Te_{3} alloys with ZT ≈ 1.0 at 300 K^{3}. To increase ZT, many new approaches have been proposed^{4,5}, such as fabricating lowdimensional thermoelectric structures to increase large density of state, engineering the interface of materials to reduce the lattice thermal conductivity, and modulating dopants to increase carrier mobility. Subsequently, further improvements include ZT = 2.4 at 300 K for ptype Bi_{2}Te_{3}/Sb_{2}Te_{3} superlattice^{6}, and ZT = 3 at 550 K for Bidoped ntype PbSeTe/PbTe quantumdot superlattice^{7}. A prospective of nanostructured TE materials can be found in a review paper^{8}. For practical applications, other issue such as size, maintenance and fast response time must also be considered even if highefficiency TE materials are found ref. 1.
Recent interests in using twodimensional (2D) transition metal dichalcogenides (TMDs) as new TE materials have attracted extensive attention^{9,10} due to high Seebeck coefficient offered by these 2D TMDs: a bilayer MoS_{2} gives α^{2}σ = 8.5 mW/m/K^{2 }^{9}. If this MoS_{2}based TE material is able to realize the calculated thermal conductivity of κ ≈ 1.55 W/m/K^{11}, we will have ZT = 1.6, which corresponds to an efficiency of about 6.5% in harvesting waste heat at T_{h} (hot side) = 400 K and T_{c} (cold side) = 300 K [according to Eq. (1)].
For high temperature range, the more viable approach is based on thermionic energy convertor (TIC), which was first proposed by G. N. Hatsopoulos^{12}. Due to the high work function of the metallic electrode, however TIC is limited to hightemperature operation above 1500 K. A potential method to harvest waste energy at 900 K was recently proposed by using a suspended monolayer graphene as cathode to provide an efficiency of higher than 40%^{13}. This improvement is attributed to the new thermionic law given by J(ϕ, T, E_{F}) = A^{*} × T^{3} × exp[−(eϕ − E_{F})/k_{B}T], where A/cm^{2}/K^{3}, v_{f} is the Fermi velocity, E_{F} is the Fermi level, and ϕ is the barrier height at zero bias. Note that the new scaling has been compared well with a recent experiment^{14}. For the wasted heat generated in the industrial or domestic process, lowgrade heat (around 400 K to 500 K) is distributed more everywhere. developing an efficient approach remains a great challenge so far.
In this paper, we propose a highefficiency solidstate thermionic device by using van der Waals (vdW) heterostructure^{15} composed of 2D TMDs (MoS_{2}, MoSe_{2}, WS_{2}, and WSe_{2}) and graphene electrodes. By taking the advantage of the ultralow crossplane thermal conductance of the 2D materials and the new thermionic emission over the Schottky barrier (SB) contact between the graphene and 2D materials (tunable via gate voltage or chemical doping), we predict that it is possible to realize highefficiency power generation and refrigeration at the temperature of 300 K to 500 K, which may be better than (or at least comparable to) the traditional TE devices. Note that the concept of using multilayers or superlattices in the thermionic devices was first suggested by two groups (Shakouri and Mahan) in late 1990s^{16,17}. The performance of their proposed singlejunction thermionic device was predicted to be better than the TE device using the same lnGaAs/lnAlAs material^{18}. For simplicity, we will ignore the effect of nonconservation of lateral momentum in the thermionic emission^{18,19} in this paper. This is justified by the facts that Schottky barrier height is planar and homogenous at the interface between graphene and Transition metal dichalcogenide^{20}.
With the current advances in growing graphene and TMDs, the proposed vdW heterostructures such as Gr/TMDs/Gr (Gr is the monolayer graphene) can be assembled experimentally^{15,21,22,23,24,25,26}. For different 2D TMDs (MoS_{2}, MoSe_{2}, WS_{2} and WSe_{2}), the crossplane thermal conductivity κ was measured to be very low: κ = 0.05 W/m/K for disordered WSe_{2}^{27}, κ = 0.0084 to 0.3 W/m/K for MoS_{2}, WS_{2} and WSe_{2}^{28}, and κ = 0.085 W/m/K for 10 nmthickness of WSe_{2} synthesized via SeO exchange^{29}. Due to their low crossplane thermal conductivity, these 2D materials may seem to be good TE materials, but having very low electrical mobility^{30} will also offset the increment due to low thermal conductivity. Thus the thermionic emissionbased (or TIC) method may be a better approach than TE method, if the ballistic transport^{31} within the structure can be ensured by choosing a suitable thickness (to avoid collisions) and an optimal barrier height (to have highcurrent injection). Note that this ballistic assumption is very commonplace in the previous studies of thermionic emission based on the traditional Superlattices structure^{17,18,32}. In the conventional superlattices, the thickness is usually larger then 1 μm. Compared with the conventional suplattices structure, the van der waals heterostructure made of graphene and TMDs considered in our paper is far less than 1 μm in thickness of crossplane direction. Therefore the ballistic assumption in the crossplane is justified. Note that Gr/hBN/Gr heterostructures device^{33} has been fabricated to study the thermoelectric transport properties, but the measured low ZT calls for further effort to identify the most ideal sandwiched material and optimized parameters of devices to achieve the goal of highefficiency thermal energy conversion. In our work presented here, we theoretically identify two materials WSe_{2} and MoSe_{2}, together with employing Molecular dynamics simulations and ab initio calculations. we hope that these two identified candidates materials can further motivate experimentalist to explore the feasibility of achieving highefficiency thermal energy harvesting system based on thermionic emission mechanism in using other 2D materials assembled in a vdW heterostructure.
Figure 1 illustrates the proposed vdW heterostructurebased thermionic device with two monolayers graphene as top and bottom layers, and Nlayers of 2D TMDs materials of thickness d between two graphene layers. This configuration is similar to the recentlyreported Gr/phosphorene/Gr^{34} and Gr/hBN/Gr sandwich structure^{33}. The device can be either a power generator or refrigerator depending on the direction of the current flow. For power generation, electron flow is from the hot electrode at temperature T_{h} to the cold electrode at temperature T_{c}.
If a 2D material with suitable thickness d and low crossplane thermal conductivity κ is used as the vdW heterostruture, our calculations show that TIC device may offer higher efficiency as compared to the existing TE devices operating at lowgrade temperature (400 to 500 K). As a power generator, it can harvest waste heat at 400 K with about 8% efficiency using the reported experimental properties of the 2D materials.
For refrigeration at 260 K, the efficiency can be more than 40% of the Carnot efficiency. According to our model, the thickness of the layer must be in the intermediate range in order to reach high efficiency. For few layers, electron tunneling process will be dominant over the overbarrier thermionic emission which will reduce the efficiency^{35}. For larger values of d, it will induce a large crossplane thermal conductivity κ; and also makes the assumption of ballistic electron transport within the layers no longer valid.
For a given voltage of V, the electrical current density (J_{e}) and the thermal current density (J_{Q}) being transported across the electrodes are, respectively,
Here, ^{13} is the thermionic current density emitted from the cold graphene electrode at temperature T_{c} over an effective Schottky barrier height of ϕ′ formed at the interface between the vdW structure and the graphene electrode. Similarly, we have for the thermionic current density emitted from the hot graphene electrode at temperature T_{h}. Due to the unique properties of graphene in tuning its Fermi energy via bias voltage or chemical doping, the effective barrier is defined as ϕ′ = ϕ − E_{F}/e, where ϕ is the value at zero bias with an intrinsic Fermi level.
As already pointed out by previous work^{13,36}, traditional vacuum thermionic converter cannot operate near room temperature due to the high vacuum work function (around 4.51 eV) of graphene. To operate at room temperature, the work function is required to be below 0.34 eV^{36}, which can be attainable in the graphene/TMDs heterostructure. In Fig. 2, we present the verification (in comparison with experimental results) of the revised thermionic emission law^{13} is valid in describing the electron flow over the Schottky barrier of graphene/TMDs contact for graphene/MoSe_{2} (44 layers) contact^{24} with a reported barrier height of ϕ′ = 0.2 V (best fit). The calculated value of ln(J/T ^{3}) has an excellent agreement with experiment^{24} from 1000/T (about T = 250 to 300 K) as shown in Fig. 2a. The agreement with experiment indicates that our thermionic emission model can be good enough to describe the carrier’s transport across the Vdw heterostructure.
A prior firstprinciples calculation^{37} has also indicated that the conduction band edge of MoSe_{2} will reduce with increasing layer number (black line with black symbols in Fig. 2b). As a result, the Schottky barrier height decreases from 0.614 to 0.38 volt as the layer number increases from one layer to eight layers (red line with symbols) as shown in Fig. 2b. The trend of change in Schottky barrier height with increasing layer number is consistent with previous experiment^{24}, which reports that the Schottky barrier height between graphene and MoSe_{2} becomes saturable with larger than 50 MoSe_{2} layers. Based on the experiment facts and firstprinciple calculation, it is reasonable to believe that the Schottky barrier height can be reduced down to around 0.2 volt when the layer number increases up to 44 layers. For completeness, the band structure of other contacts between graphene and one layer of TMDs materials (MoS_{2}, WS_{2} and WSe_{2}) can be found in the Supplementary materials (see S3).
In Eq. (3), the 3k_{B}T term measures the average heat energy per emitted electron, which is obtained through the internal energy of electron in graphene associated with one degree of freedom, , with Ξ(k) being the partition function. In the last term of Eq. (3), R is the thermal resistance including all the contributions due to interface (between graphene and TMDs, and different layers within TMDs), barrier layers and electrode. We will only consider thermal conductance due to TMDs and limit our study to multilayers TMDs, as the molecular dynamics simulation (see discussions section) shows that the contribution from the interface between graphene and TMDs and electrode is small, as compared to the resistance due to TMDs itself. For simplicity and a conservative estimation, we use R = d/κ in Eq. (3). Including other effects will increase the efficiency predicted in this paper. By defining the average temperature as T = (T_{h} + T_{c})/2, and the temperature difference as δT = T_{h} − T_{c}, we calculate J_{e} as the thermionic emitted current density at the mean temperature T. In the limit of and , Eqs (2) and (3) become
where α = eϕ′/k_{B}T, , γ = (T_{R}/T)^{3}e^{α}, and . Here, we have introduced a temperaturelike parameter T_{R} to characterize the performance of the device (together with the effective barrier height, ϕ′), where T_{R} is proportional to (κ/d)^{1/3} or inversely proportional to the barrier resistance, R^{−1/3}. Numerically, we have T_{R} [K] = 4666 × (κ/d)^{1/3} for κ [W/m/K] and d [nm].
The experimental verification of our model proposed in this work closely depends on the design of Gr/TMDs/Gr. In this paragraph, we will briefly state that our design of Gr/TMDs/Gr can be easily realized by using the current capability in fabrication of vdW heterostructure by many different research groups^{22,23,24}. For example, Gr/WSe_{2}/Gr had been fabricated ranging from d = 2.2 nm (3 layers of WSe_{2}) to 40 nm^{22}. Similarly, we have d = 50 nm and d = 9 to 50 nm, respectively, for Gr/MoS_{2}/Gr^{23}, and for Gr/MoSe_{2}/Gr^{24}. The combination of more than one type of TMDs materials is also possible, such as Gr/MoS_{2}/WSe_{2}/Gr^{38}. Thus we have used some of reported experimental parameters to illustrate the performance of our design in harvesting waste heat at 400 K as shown below.
Results
Refrigerator
For the device to operate as a refrigerator, it require J_{Q} > 0, which poses a condition of in Eq. (5). The cooling efficiency is calculated by η = J_{Q}/(J_{e} × V) and its maximum efficiency (η_{max}) is obtained by taking first derivative with respect to V, which gives
The variables α and β are functions of ϕ′ and T_{R}. The term T/δT is approximately regarded as the Carnot efficiency. In Fig. 3, we plot the maximum cooling efficiency η_{max} (in terms of the Carnot efficiency) at T_{c} = 260 K as a function of effection barrier height ϕ′ = 0 to 0.5 volt for various T_{R} = 500 K down to 10 K at T_{h} = 300 K (solid lines) and 350 K (dashed lines). It is clear that the cooling efficiency increases with smaller T_{R} as expected. For a given T_{R}, the maximal efficiency can be achieved by tuning the effective barrier height ϕ′. The tuning range will become wider for higher temperature (e.g. T_{h} = 350 K) as compared to T_{h} = 300 K. In the limit of T_{R} = 0, we have γ = 0, , and Eq. (6) becomes , which depends only on α (or ϕ′).
With few reported values of the crossplane thermal conductivity for vdW heterostructure, we use two measurements (at 300 K)^{27,39} to illustrate the realistic cooling efficiency of our design. For Gr/WSe_{2}/Gr with d = 62 nm and κ = 0.048 W/m/K, we have T_{R} = 425 K and our design predicts a cooling efficiency of 26.11% of the Carnot efficiency with an optimal barrier ϕ′ ≈ 0 (Ohmic contact). For Gr/MoSe_{2}/Gr with d = 70 nm and κ = 0.0847 W/m/K, we have T_{R} = 496 K and the cooling efficiency is 21.94% of the Carnot efficiency with an ohmic contact too. If the T_{R} can be engineered to be less than 300 K, the maximum efficiency will occur at some optimal values of ϕ′ = 0.05 to 0.3 volt, which are also within the current tunable range for Gr/WSe_{2} and Gr/MoSe_{2} contacts^{24,40}.
For practical TEbased coolers used in various applications (e.g. airconditioned car seats, and semiconductor laser cooling), the efficiency is less than 15% of the Carnot efficiency^{41}. With the highest reported value of ZT = 2.4 for Bi_{2}Te_{3}/Sb_{2}Te_{3} superlattice structure^{6}, the efficiency will become 31.1% of the Carnot efficiency at the same temperature studied here. For practical applications, the refrigerator has to pump a heat flux of a few hundreds W/cm^{2}. For our design, the pumped heat current is estimated by where A_{G} = 0.01158 A/cm^{2}/K^{3}. At T_{c} = 260 K, T_{h} = 300 K, and ϕ′ = 0.05 volt, the estimated cooling power is up to 500 W/cm^{2}, which is larger than those of thinfilm Bi_{2}Te_{3}superlattice thermoelectric cooling devices^{42}. At higher range of T = 400 to 500 K, our design will give about 1.7 to 3 kW/cm^{2}.
Power generator
When the current flow is from hot side to cold side (e.g. T_{h} = 400 K and T_{c} = 300 K). the device will behave as a power generator. This current flow via an external circuit is then extracted as the power output by harvesting the thermal energy from the heated graphene electrode (hot side). The maximal value of the power generation efficiency is calculated by
The calculated results are plotted in Fig. 4 as a function of ϕ′ for T_{R} = 500 K down to 10 K at a fixed T_{c} = 300 K for two heat sources: T_{h} = 400 K (solid lines) and 500 K (dashed lines). At T_{h} = 400 K, the efficiency is from about 8% to 20% for T_{R} = 500 K down to 10 K. Assuming κ = 0.08 W/m/K and d = 89 nm, this corresponds to T_{R} = 450 K and η_{g} is about 8% (with ϕ′ ≈ 0), which is comparable to or better than some of highlyefficient thermal harvesting devices, such as (a) a twolayer WSe_{2} TEbased device (ZT = 1.6) having a maximum efficiency of 6.5%^{9}, (b) an electrochemical system for harvesting lowgrade waste heat energy (<100 °C) with efficiency less than 8%^{43}, and (c) a theoretical efficiency of 8% for ZT = 2.4 TE material at 400 K^{1}. Note that the efficiency for a power generator is very sensitive to heat source temperature. When T_{h} is increased from 400 K (solid lines) to 500 K (dash lines), the efficiency increases by a factor of about 2 as shown in Fig. 4.
Based on the available experimental data^{27,28,39}, we calculate (see Table 1) the efficiencies of harvesting heat at 400 K (cold side is kept at 300 K) for some practical design parameters: (a) Gr/MoS_{2}/Gr with d = 50 nm, (b) Gr/WS_{2}/Gr with d = 50 nm, (c) Gr/MoSe_{2}/Gr with d = 70 nm, and (d) Gr/WSe_{2}/Gr with d = 62 nm. The table shows that the operating range of T_{R} is from 850 K down to 428 K, with efficiency from 3.15% to 8.56%. The efficiency for both Gr/MoSe_{2}/Gr and Gr/WSe_{2}/Gr is about 7% to 8%, which is better than 3% to 4% generated by Gr/MoS_{2}/Gr and Gr/WS_{2}/Gr. Note that we only use T_{h} = 400 K as an example in the table. For other heat source temperatures T_{h} (with T_{c} = 300 K), the efficiency of the energy harvesting using these common vdW heterostructures can be estimated based on the fitted equation determining the crossplane thermal conductivity as a function of temperature by κ = a + bT + cT^{2} + dT^{3}. where a, b, c and d are the fitting parameters, and they can be found in the Supplementary materials (see S2).
Discussions
In our model, the tunneling of lowenergy electrons through the Schottky barrier at the interface will become important if the width of the barrier is small^{35}. This consideration imposes a lower limit to the layer thickness d in our design to ensure that the injection of the electrons from the graphene electrode across the barrier is governed by the overbarrier process (thermionic emission) as assumed in the model. This minimal d may be estimated by using , with m^{*} being the effective electron mass of the barrier layers^{44}, which gives 0.845m_{e}, 0.776m_{e}, 0.665m_{e} 0.643m_{e}, respectively, for MoS_{2}, MoSe_{2}, WS_{2} and WSe_{2}. For Gr/WS_{2}/Gr, it was reported that thermionic emission will be dominant for 5 layers or more^{35}. For Gr/WSe_{2}/Gr and Gr/MoSe_{2}/Gr, the minimal thickness is estimated to be, respectively 5 and 4 nm based on Simmons model^{45}.
For fewlayer TMDs (MoS_{2}, MoSe_{2} WS_{2} and WSe_{2}), their crossplane lattice thermal conductivity is very small compared to traditional TE materials, which is in the range of 0.01 to 0.1 W/m/K due to the localized lattice vibrations or the disorder within the TMDs^{27,28,29,39}. It was claimed that the reduction is also valid for other reassembled TMDs^{27}, and thus considering d = 50 nm and κ = 0.01 W/m/K, we have T_{R} around 272.6 K, which implies that it is positive to realize the calculated efficiencies of about 10% in harvesting waste heat at 400 K or 20% at 500 K as shown in Fig. 4 for T_{R} = 300 K.
As T_{R} scales as (κ/d)^{1/3}, to further reduce T_{R} down to 100 K or even 10 K, the lattice crossplane thermal conductivity κ must be reduced for a fixed thickness d. Further reducing d will not help as it will also promote the unwanted tunneling effect mentioned above. Thus, we conclude that our design will require vdW structure with low crossplane thermal conductivity and reasonable thickness to function as a highefficiency solid state thermionic device. A recent work for WSe_{2}^{46} may have presented such a solution in increasing spacing of the layers, and stacking disorder, and it is possible have a further reduction in the crossplane lattice thermal conductivity without reducing d. Another method to reduce the crossplane thermal conductivity is by using superlattice composed of WSe_{2} and MoSe_{2}. We speculate that a precise control of superlattice period thickness will be able to lead to a much lower crossplane thermal conductivity together with fineengineering thermal boundary resistance between different layers. If such extremely low crossplane thermal conductivity can be realized experimentally, it is possible to have very high efficiency energy harvesting as indicated in Fig. 4.
As mentioned before, the interface resistance between graphene and the TMDs has been neglected in our model. To justify this assumption, a Molecular dynamics simulation has been done. The simulation details can be found in the section of Method and Supplementary Materials (see S1). In Fig. 5a, multiple layers of MoSe_{2} (or WSe_{2}) are sandwiched by fewlayers graphene with three layers on top and three at the bottom. The temperature distribution along the crossplane direction of the hybrid multilayer structure, with an imposed heat flux J_{Q} = 0.5 GW/m^{2} on the top graphene layer, is demonstrated in Fig. 5b. Note that small temperature drops near the heat source and heat sink are mainly induced by the artificial temperature control. From Fig. 5b, we see large temperature reduction ΔT_{1} = 30 K and ΔT_{2} = 34 K occurring at the two interfaces between graphene and MoSe_{2}, respectively at the interface between layers 3 and 4, and also between layers 11 and 12. In comparison, the total temperature drop is about 2.8 K per layer for the eight layers of MoSe_{2} (one order of magnitude lower). The thermal conductance (or Kapitza conductance) for these two interfaces is about G = J/ΔT = 16.56 and 14.59 MW/m^{2}/K, respectively. Note that the difference of G at the upper and the lower interface can be attributed to the temperature dependence of G^{47,48}. These two values are larger than the crossplane thermal conductance of a fewlayer MoSe_{2} by one to two orders of magnitude using the experiment data^{39}. Similar calculation has been repeated for WSe_{2}, which shows similar results (not shown). In Fig. 5c, we show that the interface thermal conductance for WSe_{2} and MoSe_{2} is almost independent of the numbers of layers, and the average value is about G = 16.7 MW/m^{2}/K for Gr/MoSe_{2}/Gr and 17.12 MW/m^{2}/K for Gr/WSe_{2}/Gr. Surprisingly, we find that our calculated interface conductance across graphene and MoSe_{2} interface is very close to 25 MW/m^{2}/K measured for graphene contact interface^{48}. From these findings, it is important to note that it is valid to consider the layer’s resistance due to 2D TMDs in this paper as the first approximation. Subsequent improvement can be pursued through comparison with experimental verification (from other groups) on our calculated efficiency shown in Table 1 below. It is worth to mention that the calculated temperature gradient (around 50 K) across eight layers of MoSe_{2} (or WSe_{2}) is consistent with the previous experiment^{33}. So it is reasonable to believe that more significant temperature gradient across more layers of MoSe_{2} (or WSe_{2}) can be established. Compared with larger temperature drop across Gr/TMDs/Gr structure, the little temperature drop across Graphene layers can be ignored. In other words, the crossplane thermal conductivity of graphene layers is much higher than that of TMDs. Therefore solid thermionic converter based on Gr/TMDs/Gr structure has better performance than pure graphene layers (Gr/Gr/Gr). In the Table 1, we study some wellknown TMDs (MoS_{2}, MoSe_{2}, WS_{2} and WSe_{2}) using the reported experimental parameters, it is found that MoSe_{2} and WSe_{2} are better candidates. They are able to harvest waste heat at 400 K with about 7% to 8% in efficiency. To further increase the efficiency of a power generator, the current model can be also extended to include the contribution from solar energy and other thermal effects^{49,50,51}. As a refrigerator, the cooling efficiency is about 0.22 to 0.26 of the Carnot efficiency for a temperature difference of 40 K between 260 and 300 K. Note that the effects of finite electrical conductivity and thermal conductivity in the lateral direction on the proposed device’s performance are not yet considered. The treatment of full model requires solving the coupling of Schrodinger and Dirac system, which is beyond the scope of this paper.
Similar to conventional TE devices, thermionic devices have lot of advantages, such as no moving parts, no noise, high reliability, long service time and so on. These features enables many possible applications for thermionic power generation and thermionic cooling. The most promising application for TIC is wasted heat recovery from vehicles to improve fuel economy. Other potential applications include harvesting industrial waste heat (e.g. steel rolling mill, cement, glass manufacture plant, etc.) and domestic heat (e.g. water heater). for electricity or charging batteries. While for thermionic cooling, most possible application is onchip cooling of nanoelectronics devices. But the application can be extended to integrate with targeted devices to maintain the lowtemperature environment for semiconductor laser, medical and scientific equipments. Finally, the vertical transport of charge transport at the graphenesemiconductor interface remains interesting. Modified Schottky models have been formulated to study the inhomogeneous of Schottky barrier at the interface^{52} and also the smooth transition between the T^{3} and T^{2} temperature scaling^{53}.
Methods
The interlayer distances between the graphene and MoSe_{2} (or WSe_{2}) are set to 3.35 Å which is interlayer distance of bulk graphite. The interlayer distance between the MoSe_{2} (or WSe_{2}) layers is 3.11 Å (or 3.14 Å) according to previous firstprinciple calculation^{54}. The inplane dimensions of the layer considered here are 69 Å × 70 Å. Periodic boundary conditions are applied along the inplane directions and free boundary conditions are applied along the crossplane direction of heterostructure. In the simulation, the initial configuration is equilibrated by using the constant volume and temperature (NVT) ensemble at a temperature T for 50 ps with a time step Δt = 0.5 fs. Upon realization of the equilibrium state, the system is switched to the constant volume and energy (NVE) ensemble to maintain the energy conservation condition. A constant heat flux is then imposed into the system at each time step by adding a small amount of heat Δε into the upmost graphene layer (layer 1). In doing so, we reduce the same amount of energy from the graphene layer at the bottom (layer 14). The simulation is conducted until a stable temperature gradient is established along the heat flux direction. For a hybrid system consisting of different interfaces, a temperature drop ΔT_{ln} at the interface is usually developed, which gives a measurement of the interface thermal conductance or the Kapitza conductance G = J/ΔT_{ln}, where J = Δε/AΔt with A denoting the crosssection area.
Additional Information
How to cite this article: Liang, S.J. et al. Thermionic Energy Conversion Based on Graphene van der Waals Heterostructures. Sci. Rep. 7, 46211; doi: 10.1038/srep46211 (2017).
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Acknowledgements
This work was supported by SUTDMIT IDC Grant (IDG21200106 and IDD21200103) and Singapore MOE T2 Grant (T2MOE1401). L. K. Ang acknowledges the support of AFOAR AOARD grant (FA23861414020).
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Affiliations
SUTDMIT International Design Center (IDC), Singapore University of Technology and Design (SUTD), 8 Somapah road, 487372, Singapore
 ShiJun Liang
 & L. K. Ang
School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798, Singapore
 Bo Liu
 & Kun Zhou
Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
 Wei Hu
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Contributions
S.J.L. conceived the project, B.L. and K.Z. performed the molecular dynamics simulation, W.H. did the firstprinciple calculations. S.J.L. and L.K.A. wrote the manuscript with contributions from the other authors.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to ShiJun Liang or L. K. Ang.
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