Generalized high-energy thermionic electron injection at graphene interface
PPublisher link: Phys. Rev. Appl. 12, 014057(2019)Generalized High-Energy Thermionic Electron Injection at Graphene Interface
Yee Sin Ang, ∗ Yueyi Chen, Chuan Tan, and L. K. Ang † Science and Mathematics, Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372
Graphene thermionic electron emission across high-interface-barrier involves energetic electronsresiding far away from the Dirac point where the Dirac cone approximation of the band structurebreaks down. Here we construct a full-band model beyond the simple Dirac cone approximationfor the thermionic injection of high-energy electrons in graphene. We show that the thermionicemission model based on the Dirac cone approximation is valid only in the graphene/semiconductorSchottky interface operating near room temperature, but breaks down in the cases involving high-energy electrons, such as graphene/vacuum interface or heterojunction in the presence of photonabsorption, where the full-band model is required to account for the band structure nonlinearity athigh electron energy. We identify a critical barrier height, Φ (c) B ≈ . I. INTRODUCTION
Recent theoretical and experimental developmentshave revealed graphene’s extraordinary potential in var-ious thermionic-based energy applications [1, 2]. Inthermionic emission process, electrons are thermally ex-cited to overcome the interface potential barrier andemitted across the interface. Collection of these emit-ted electrons at the anode forms an electricity throughan external load, thus achieving thermionic-based heat-to-electricity conversion. Recent experiment has demon-strated graphene monolayer as a highly efficient anodefor direct heat-to-electricity energy conversion with highconversion efficiency reaching 9.8%, which can be furtheroptimized by electrostatic gating and cathode-anode gapreduction [1]. Thermionic emission of electrons over aninsulating barrier can also be harnessed to achieve elec-tronic cooling effect [3–6]. The performance is, however,fundamentally limited by thermal backflow directed fromhot to cold electrodes. Recent advancements of 2D ma-terial van der Waals heterostructure [7–9] (VDWH) offernew opportunities in solid-state thermionic cooler. Thelayered structure of VDWH strongly impedes phononpropagation and effectively diminishes the thermal back-flow effect that is detrimental to the efficiency ofthermionic cooler [10–12]. Photon-enhanced thermionicinjection across graphene/WSe /graphene VDWH repre-sents another novel route towards the efficient broadbandphotodetection and harvesting of light energy in a com-pact solid-state platform [2]. ∗ yeesin [email protected] † ricky [email protected] Previous theoretical models [14–22] describes the out-of-plane thermionic electron emission from the surface ofgraphene via two key assumptions: (i) electron wavevec-tor component lying in the plane of graphene, denoted as k (cid:107) , is conserved during the out-of-plane emission process[20]; and (ii) electrons undergoing thermionic emissionare described by a conic energy band structure, com-monly known as the Dirac cone approximation [19, 20,23–25]. These assumptions break down in the case of 𝜀 ∥ 𝑘 ∥ b ca Φ 𝐵 𝜀 𝐹 Vacuum K K’ Γ 𝜀 ∥ 𝑘 ∥ Graphene 𝜀 ∥ 𝑘 ∥ DO S ( e V − m − ) × . . ε k (eV) d FIG. 1. Model of thermionic emission in graphene. (a) Banddiagram showing the thermionic electron emission based onthe full-band model; (b) Dirac cone at K and K (cid:48) valley ofthe first Brillouin zone; (c) full energy band of graphene; and(d) electronic density of states (DOS) showing linear Diracapproximation model (dashed) and full-band model (solid). a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l thermionic emission of high-energy electrons due to thefollowing reasons. Firstly, the conservation of ε (cid:107) is vio-lated due to the presence of electron-electron, electron-impurity, and electron-phonon scattering effect [26–30].Clear experimental evidence of k (cid:107) -nonconserving verti-cal electron transport due to electron-phonon scatter-ing has been observed in graphene heterostructures [29].At typical device operated at room temperature andabove, electrons in graphene are expected to undergostrong scatterings with phonons and impurities. Thus,at the high-temperature thermionic emission regime, k (cid:107) -conservation is expected to be strongly violated due tophonon [29] and impurity scattering [30] effects, whichimmediately implies the breakdown of assumption (i).Secondly, as the interface potential exceeds Φ B ≈ ε (cid:107) > B by up to ∼ k (cid:107) -nonconservation [25]; and (ii) band structurenonlinearity beyond the simple Dirac cone approxima-tion by using the full-band tight-binding energy disper-sion [34]. We found that the simple Dirac-based model isonly valid for solid-state graphene/semiconductor Schot-tky contact with low interface-barrier (Φ B < T ≈
300 K). At high-barrier (Φ B (cid:29) T > critical potential barrier height ,Φ ( c ) B ≈ . II. THEORY
The thermionic electrical and heat current densitiesfrom the surface of a 2D electronic system are [see Fig.1(a) for the energy band diagram], J = g s,v e (2 π ) (cid:88) i v ( i ) ⊥ ( k ( i ) ⊥ ) L ⊥ (cid:90) d k (cid:107) T ( i ) ( k (cid:107) , k ⊥ ) f ( ε k (cid:107) ) , (1a) Q = g s,v (2 π ) (cid:88) i v ( i ) ⊥ ( k ( i ) ⊥ ) L ⊥ × (cid:90) d k (cid:107) ( ε (cid:107) − ε F ) T ( i ) ( k (cid:107) , k ⊥ ) f ( ε k (cid:107) ) , (1b)where g s,v = 4 is the spin-valley degeneracy, L ⊥ is the 2Dmaterial thickness ( L ⊥ = 0 .
335 nm for graphene [36]), ε F is the Fermi level, f ( ε k (cid:107) ) is the Fermi-Dirac distri-bution function, k (cid:107) = ( k x , k y ) is the electron wave vec-tor component lying in the 2D plane, ⊥ denotes the di-rection orthogonal to the x - y plane of the 2D system, k ( i ) ⊥ is the quantized out-of-plane wave vector compo-nent of i -th subband, v ( i ) ⊥ ( k ( i ) ⊥ ) = (cid:113) mε ( i ) ⊥ is the cross-plane electron group velocity, m is the free electron mass, ε ( i ) ⊥ is the discrete bound state energy level, and thesummation, (cid:80) i · · · , runs over all of the i -th quantizedsubbands. The non-conservation of k (cid:107) during the out-of-plane thermionic emission process leads to the cou-pling between k (cid:107) and k ⊥ . Accordingly, the i -th subbandtransmission probability becomes T ( i ) ( k ⊥ , k (cid:107) ), i.e. thecross-plane electron tunneling is dependent on both k ⊥ and k (cid:107) . The k (cid:107) -nonconserving model has been exten-sively studied in previous theoretical works [6, 25, 37–39] and has been successfully employed in the analysisof thermionic transport experiments in graphene-baseddevices [2, 13, 19].For thermionic emission, the transmission probabilitycan be written as T ( i ) ( k ⊥ , k (cid:107) ) = λ Θ (cid:16) ε (cid:107) + ε ( i ) ⊥ − Φ B (cid:17) ,i.e. ε ( i ) ⊥ and ε (cid:107) are combined to overcome the inter-face barrier Φ B . Here λ is a parameter representing thestrength of k (cid:107) -non-conserving scattering processes. Theterm Θ( x ) denotes the Heaviside step-function. Equa-tions (1a) and (1b) can be simplified into a single-subband thermionic emission electrical and heat currentdensities for graphene, respectively, as J = g s,v ev ⊥ (2 π ) (cid:90) d k (cid:107) T ( ε (cid:107) ) ξ T ( ε (cid:107) ) , (2a) Q = g s,v v ⊥ (2 π ) (cid:90) d k (cid:107) ( ε (cid:107) − ε F ) T ( ε (cid:107) ) ξ T ( ε (cid:107) ) , (2b)where ξ T ( x ) ≡ exp (cid:16) − x − ε F k B T (cid:17) . In writing T ( ε (cid:107) ) = λ Θ (cid:0) ε (cid:107) ( k (cid:107) ) − Φ B (cid:1) , we have used the fact that the quan-tized subband energy level, ε (1) ⊥ , can be absorbed into ε (cid:107) by setting ε (1) ⊥ as the zero-reference of ε (cid:107) [25], anddenoted v ⊥ ≡ v (1) ⊥ ( k (1) ⊥ ). The Fermi-Dirac distri-bution function approaches the semiclassical Maxwell-Boltzmann distribution function, ξ T ( ε (cid:107) ), since the emit-ted electrons are in the non-degenerate regime, i.e. ε (cid:107) ≈ Φ B , and Φ B (cid:29) ε F for typical values of ε F < B = 4 . k (cid:107) -integration. The k (cid:107) -integral is transformedrewritten as d k (cid:107) / (2 π ) = D ( ε (cid:107) ) dε (cid:107) , where D ( ε (cid:107) ) isthe electronic density of states (DOS). In general, the k (cid:107) -nonconserving thermionic emission model in Eq. (2)can be solved for any 2D materials using the appropriateDOS. Here we shall focus on solving Eq. (2) for graphene.The full-band tight-binding model of graphene yields, D FB ( ε (cid:107) ) = D (cid:113) F ( ε (cid:107) /t (cid:48) ) K (cid:18) ε (cid:107) /t (cid:48) F ( ε (cid:107) /t (cid:48) ) (cid:19) < ε (cid:107) < t (cid:48) D √ ε (cid:107) /t (cid:48) K (cid:18) F ( ε (cid:107) /t (cid:48) ) ε (cid:107) /t (cid:48) (cid:19) t (cid:48) < ε (cid:107) < t (cid:48) , (3)where D ≡ A c g s,v π ε (cid:107) t (cid:48) , t (cid:48) = 2 . a = 0 .
142 nm, A c = 3 a √ / F ( x ) ≡ (1 + x ) − ( x − /
4, and K ( m ) ≡ (cid:82) dx (cid:2) (1 − x )(1 − mx ) (cid:3) − / is the completeElliptic integral of the first kind [40].The conduction and valence band touches at the K and K (cid:48) points in the first Brillouin zone, commonly known asthe Dirac cone [Figs. 1(b) and (c)]. At the vicinity ofDirac cone, ε (cid:107) can be expanded up to the first order in | k (cid:107) | to yields a pseudo-relativistic relation, ε (cid:107) = (cid:126) v F | k (cid:107) | ,where v F = 10 m/s. The corresponding DOS is D D ( ε (cid:107) ) = g s,v ε (cid:107) π (cid:126) v F , (4)which exhibits a monotonous linear relation with ε (cid:107) . This linear energy dispersion and the corresponding densityof states have led to many unusual physical phenomenain graphene, such as Klein tunnelling effect [41], room-temperature quantum Hall effect [42], exceptionally largeelectron mobility mobility, gate-tunable optical and plas-monic responses [44], strong optical nonlinearity [45] andthe emergence of new electromagnetic modes [46]. Itshould be noted that the D D ( ε (cid:107) ) is in good agreementwith D FB ( ε (cid:107) ) only for ε (cid:107) < ε (cid:107) > D D ( ε (cid:107) ) severely overestimates the actualDOS calculated from the full-band model [see Fig. 1(d)].The question of how the high-energy discrepancy between D D ( ε (cid:107) ) and D FB ( ε (cid:107) ) affect the thermionic emission ofenergetic electrons with ε (cid:107) > ε (cid:107) > J D = λv ⊥ L ⊥ g s,v e ( k B T ) π (cid:126) v F (cid:18) B k B T (cid:19) ξ T (Φ B ) , (5a) Q D = λv ⊥ L ⊥ g s,v ( k B T ) π (cid:126) v F Λ ξ T (Φ B ) , (5b)where Λ ≡ (Φ B /k B T ) + (2 − ε F /k B T ) (1 + Φ B /k B T ).On the other hand, using the DOS in Eq. (3), the full-band equivalence of Eq. (5) is J FB = λ v ⊥ L ⊥ g s,v eπ t (cid:48) I (Φ B ) , (6a) Q FB = λ v ⊥ L ⊥ g s,v π t (cid:48) I (Φ B ) , (6b)where I µ (Φ B ) can be numerically solved from I µ (Φ B ) ≡ Θ ( t (cid:48) − Φ B ) (cid:90) t (cid:48) Φ B ε (cid:107) ( ε (cid:107) − ε F ) µ dε (cid:107) (cid:113) F (cid:0) ε (cid:107) /t (cid:48) (cid:1) K (cid:32) ε (cid:107) /t (cid:48) F (cid:0) ε (cid:107) /t (cid:48) (cid:1) (cid:33) ξ T ( ε (cid:107) )+ (cid:90) t (cid:48) t ε (cid:107) ( ε (cid:107) − ε F ) µ dε (cid:107) (cid:112) ε (cid:107) /t (cid:48) K (cid:32) F (cid:0) ε (cid:107) /t (cid:48) (cid:1) ε (cid:107) /t (cid:48) (cid:33) ξ T ( ε (cid:107) ) . (7)The second term of Eq. (7) is set to 3 t (cid:48) as D Gr ( ε (cid:107) ) reachesthe maximum of the conduction band at ε (cid:107) ≤ t (cid:48) . Itshould be noted that J D in Eq. (5a) has been rigor-ously studied in graphene-based Schottky contacts and agood agreement between theory and experimental datais demonstrated [13, 19]. Such good agreement immedi-ately suggests the need to extend Eq. (5) via the full-band model in Eq. (3), so to obtain a generalized theo-retical framework that encompasses both the low-energy thermionic emission in graphene-based Schottky contactand the high-energy counterpart in graphene/vacuum in-terface. III. RESULTS AND DISCUSSIONS
In Fig. 2, the analytical results of the Dirac cone ap-proximation, J D and Q D , and the numerical results of
260 280 300 320 34000 . . . . . Φ B = 0 . J ( A / c m )
260 280 300 320 3400.950.960.97 . . . . . Φ B = 4 .
260 280 300 320 34000 . . . J D J FB Q ( W / c m ) Φ B = 4 . T (K) Q FB Q FB Q D Q D J FB J D Φ B = 0 . a bc d × ×
260 280 300 320 340 1000 1200 1400 1600 1800 J D / J F B J D / J F B Q D / Q F B Q D / Q F B . . . . . . J D / J F B T = 300 K T = 500 K T = 1000 K1 2 3 40 . . . . . . Φ B (eV) Q D / Q F B ef FIG. 2. Comparison of Dirac approximation and full-band thermionic models. (a) and (b) shows the electrical and heat currentdensities at Φ B = 0 . B = 4 . J D / J FD and Q D / Q FD . The Φ B -dependence of (e) J D / J FD ; and (f) Q D / Q FD at different temperatures, T = 300 , , the full-band model, J FB and Q FB , are plotted for twovalues of Φ B which are typical for graphene-based Schot-tky contact [19, 35] and for graphene/vacuum thermionicemitter [20]: (i) the low-barrier graphene/semiconductorSchottky diode regime (Φ B = 0 . B = 4 . T > v ⊥ = 3 . × m/s and L ⊥ = 0 .
335 nm for graphene [25], and scatter-ing strength [27] of λ = 10 − . In the low-barrier regime,both Dirac and full-band models produce nearly identicalelectrical [Fig. 2(a)] and heat [Fig. 2(c)] current densitiesin the typical room temperature operating regime for aSchottky diode. The Dirac model slightly underestimatesthe electrical and heat current densities by approximately5% [see insets of Figs. 2(a) and (c)]. Conversely, in thehigh-barrier graphene/vacuum regime, the Dirac modelseverely overestimates the electrical and heat currentdensities by ∼
60% in the high-temperature range from1000 K to 1800 K [see Figs. 2(b) and (d)]. This rather siz-able discrepancy immediately reveals the incompatibilityof Dirac cone approximation in the thermionic emissionof high-energy electrons occurring at graphene/vacuuminterface [10–12] or in the presence of photon absorp-tion [2, 13]. This fallacy of Dirac cone approximationarises from the fact that graphene band structure be-comes highly nonlinear at high electron energy ε (cid:107) > < > B = 0 . B = 4 . J D / J FB [Fig. 2(e)] and Q D / Q FB [Fig. 2(f)], are weakly de-pendent on temperature, and both ratios are approx-imately equal, i.e. J D / J FB ≈ Q D / Q FB . For Φ B ly-ing approximately between 0 . . J D / J FB ≈ Q D / Q FB <
1, which signifies the un-derestimation of the thermionic emission current den-sities by the Dirac cone approximation in comparisonwith the full-band model. Such underestimation peaksat Φ B ≈ . J D / J FB ≈ Q D / Q FB ≈ .
35. Inter-estingly, there exists a critical barrier height, Φ (c) B ≈ . . − − − . . − − − Φ B = 0 . B = 4 . a b l n ( J F D / T ) /T (10 − K − )1 /T (10 − K − ) FIG. 3. Modified Arrhenius plot of ln ( J FB /T ) ∝ − /T for(a) Φ B = 0 . B = 4 . J D / J FB ≈ Q D / Q FB < J D / J FB ≈ Q D / Q FB > (c) B correspondsto an accidental DOS averaging effect at which the un-derestimation of the electron population available forthermionic emission due to the Dirac cone approximationat energy slightly above Φ B is exactly compensated bythe overestimation of that at higher energy. Beyond thiscritical barrier height, the overestimation of the electronpopulation available for thermionic emission due to Diraccone approximation becomes increasingly severe, whichdirectly leads to the monotonously increasing trend inboth J D / J FB and Q D / Q FB as Φ B > Φ (c) B .We now investigate the current-temperature scaling ofthe full-band thermionic emission model. For 2D materi-als, the current-temperature scaling of thermionic emis-sion in the out-of-plane direction follows the universalscaling law, ln ( J /T ) ∝ − /T [25], rather than theclassic Richardson-Dushman scaling law of ln (cid:0) J /T (cid:1) ∝− /T for 3D materials [48, 49]. In Fig. 4, the numericalvalue of ln ( J FD /T ) is plotted against 1 /T . The numer-ical results fit excellently into a straight line for bothlow-barrier [Fig. 3(a)] and high-temperature [Fig. 3(b)]regimes, thus confirming the expected universal scalingbehavior in the full-band model of graphene.To further investigate the impact of the full-bandmodel on the modeling of graphene-based thermionicenergy device, we calculate the thermionic cooling ef-ficiency of a graphene thermionic cooler (see inset ofFig. 4(c) for a schematic drawing of the energy banddiagram). Here, the thermionic cooler is composed oftwo graphene electrodes in parallel-plate configurationswhere a bias voltage, V , is used to modulate the netemitted electrical and heat current to achieve cooling[20]. The hot (cold) graphene electrode temperature isdenoted as T H ( T C ). In Figs. 4(a) and (b), we see thatthe Dirac model overestimates both electrical and heatcurrent densities immediately after the onset of cool-ing effect where ∆ Q >
0. The coefficient of perfor-mance (COP) of the thermionic cooler is calculated as η ( V, T c , T h ) = ∆ Q/V ∆ J , ∆ Q = Q ( T c ) − Q ( T H ) and∆ J ≡ J ( T c ) − J ( T H ) are the net heat and electrical a ∆ Q ( W / c m ) . . ∆ J ( A / c m ) . . . . . . . c Φ 𝐵 𝜀 𝐹 Vacuum 𝑒𝑉 𝑇 𝐶 𝑇 𝐻 V (V) η / η c ∆ J D ∆ J D ∆ J FB ∆ J FB ∆ Q D ∆ Q FB η D /η c η FB /η c × − × − ∆ J ∆ Q , . . . . - . . . . - . b FIG. 4. Net (a) heat and (b) electrical current densities, and(c) efficiency of thermionic cooler calculated with T H = 1600K, T C = 1400 K, and ε F = 0 . T C )and hot ( T H ) graphene electrodes and biased by, V . current densities, respectively. In Fig. 4(c), the COP,normalized by Carnot efficiency η c ≡ T C / ( T H − T C ), isplotted as a function of V with T c = 1400 K, T H = 1600K and ε F = 0 . V ≈ .
65 V. Around the maximal efficiencypoint η max , the full-band model yields η max = 0.75 at 0.7V, compared to η max = 0.7 at 0.8 V as predicted by theless accurate Dirac model. These discrepancies can havea significant impact on the practical design of graphene-based thermionic cooler as it can affect multiple values,such as net transported heat current density and the opti-mal bias voltage, that are crucially important for the op-timization of device figures of merit. Finally, we remarkthat the simplistic graphene thermionic cooler model re-ported in Fig. 4 is aimed to illustrate the discrepancybetween the Dirac cone approximation and the full-bandmodel. Realistic modeling of graphene-based thermionicenergy device should include important effects, such asimage potential lowering [20], blackbody radiation [17],space charge [47–50], electric-field-induced Fermi levelshifting [51], secondary electron emission [52, 53], andcarrier scattering effects [26, 27, 29, 30]. Such detailedmodeling is beyond the scope of this work and shall formthe subjects of future works. Importantly, the general-ized 2D thermionic emission model of graphene developedhere shall provide a theoretical foundation that may bedirectly useful for both the theoretical and experimentalstudies of the above-mentioned effects.Finally, for completeness, we compare themodel developed above with the widely-used clas-sic Richardson-Dushman (RD) model [54], i.e. J RD = A RD T exp ( − Φ B /k B T ) where A RD ≈ − K − is the RD constant. Due to the strongdominance of the exponential term at the typical oper-ating regime of Φ B (cid:29) k B T , the experimental data ofgraphene thermionic emission could still be fitted usingthe classic RD model [55, 56], especially through theRD scaling law ln (cid:0) J RD /T (cid:1) ∝ − /T , without yieldingsignificant errors in the extraction of Φ B [21]. However,the magnitude of current density can deviate by severalorders of magnitude when an inappropriate model isused [21]. Previous experiments have demonstratedthat the extracted pre-exponential term can differ bynearly two orders of magnitude between the classic RDmodel and the 2D graphene thermionic emission modelwith Dirac cone approximation [19, 57]. Such largedeviation can severely impact applications of the modelin cases where the magnitude of the emission currentdensity is important. We quantitatively compare theclassic RD model and the generalized full-band modeldeveloped in this work by defining the ratio, J RD / J FB .For the two cases studied above, i.e. Φ B = 0 . T = 300 K and Φ B = 4 . T = 1200 K for high-temperature graphene/vacuum field emitter, we obtain J RD / J FB ≈ . × and J RD / J FB ≈ .
6, respec-tively. This exceedingly large ratios of J RD / J FB (cid:29) IV. CONCLUSION
In conclusion, we have revealed the fallacy of Diraccone approximation in the modeling of high-barrier high-temperature thermionic emission in graphene. While theclassic Richardson-Dushman and the Dirac approxima-tion models [19, 20, 25] remain usable for the simple anal-ysis of experimental data [55–57], the full-band modeldeveloped here should be used in the case of high-energythermionic electron emission in graphene. The proposedfull-band model is especially critical for the computa-tional design and modeling of graphene-based thermionicenergy devices where the magnitude of the emission cur-rent densities is required to be determined accurately. Asthe 2D thermionic emission formalism developed abovecan be readily generalized to other 2D materials, our find-ings shall provide an important theoretical foundation forthe understanding of thermionic emission physics in 2Dmaterials.This work is supported by A*STAR AME IRG(A1783c0011) and AFOSR AOARD (FA2386-17-1-4020).Y. 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Ang, ‘Universalscaling laws in Schottky heterostructures based on two-dimensional materials’, Phys. Rev. Lett. 121, 056802(2018).[26] S. V. Meshkov, ‘Tunneling of electrons from a two-dimensional channel into the bulk ’, Zh. Eksp. Teor. Fiz.91,2252 (1986).[27] K. J. Russell, F. Capasso, V. Narayanamurti, H. Lu, J.M. O. Zide, and A. C. Gossard, ‘Scattering-assisted tun-neling: Energy dependence, magnetic field dependence,and use as an external probe of two-dimensional trans-port’, Phys. Rev. B 82, 115322 (2010).[28] V. Perebeinos, J. Tersoff, and Ph. Avouris, ‘Phonon-mediated interlayer conductance in twisted graphene bi-layers’, Phys. Rev. Lett. 109, 236604 (2012).[29] E. E. Vdovin, A. Mishchenko, M. T. Greenaway, M. J.Zhu, D. Ghazaryan, A. Misra, Y. Cao, S. V. Morozov, O.Makarovsky, et al, ‘Phonon-assisted resonant tunnelingof electrons in grapheneboron nitride transistors’, Phys.Rev. Lett. 116, 186603 (2016).[30] Y. Liu, Z. Gao, Y. Tan, and F. Chen, ‘Enhancement ofout-of-plane charge transport in a vertically stacked two-dimensional heterostructure using point defects’, ACSNano 12, 10529 (2018).[31] J. Voss, A. Vojvodic, S. H. Chou, R. T. Howe, andF. Abild-Pedersen, ‘Inherent enhancement of electronicemission from hexaboride heterostructure’, Phys. Rev.Applied 2, 024004 (2014).[32] Y.-J. Yu, Y. Zhao, S. Ryu, L. E. Brus, K. S. Kim, andP. Kim, ‘Tuning the graphene work function by electricfield effect’, Nano Lett. 9, 3430 (2009).[33] H. Yuan et al, ‘Engineering ultra-low work function ofgraphene’, Nano Lett. 15, 6475 (2015).[34] S. Reich, J. Maultzsch, C. Thomsen, and P. Ordejon,‘Tight-binding description of graphene’, Phys. Rev. B 66,035412 (2002). [35] S. Tongay, M. Lemaitre, X. Miao, B. Gila, B. R. Ap-pleton, and A. F. Hebard, ‘Rectification at graphene-semiconductor interfaces: Zero-gap semiconductor-baseddiodes’, Phys. Rev. X 2, 011002 (2012).[36] Z. H. Ni et al, ‘Graphene thickness determination usingreflection and contrast spectroscopy’, Nano Lett. 7, 2758(2007).[37] D. Vashaee, and A. Shakouri, ‘Electronic and thermo-electric transport in semiconductor and metallic super-lattices’, J. Appl. Phys. 95, 1233 (2004).[38] M. F. ODwyer, R. A. Lewis, C. Zhang, and T. E.Humphrey, ‘Efficiency in nanostructured thermionic andthermoelectric devices’, Phys. Rev. B 72, 205330 (2005).[39] R. Kim, C. Jeong, and M. S. Lundstrom, ‘On momentumconservation and thermionic emission cooling’, J. Appl.Phys. 107, 054502 (2010).[40] M. Abramowitz, and I. A. Stegun, ‘Handbook of math-ematical functions: with formulas, graphs, and mathe-matical tables’, Dover Publications (USA, 1965).[41] M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, ‘Chi-ral tunnelling and the Klein paradox in graphene, Nat.Phys. 2, 620 (2006).[42] K. S. Novoselov et al, ‘Room-temperature quantum Halleffect n graphene, Science 315, 1379 (2007).[43] X. Du, I. Skachko, A. Barker, and E. Y. Andrei, ‘ Ap-proaching ballistic transport in suspended graphene, Nat.Nanotechnol. 3, 491 (2008).[44] A. N. Grigorenko, M. Polini, and K. S. Novoselov,‘Graphene Plasmonics, Nat. Photon. 6, 749 (2012).[45] E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, andS. A. Mikhailov, ‘Coherent nonlinear optical response ofgraphene, Phys. Rev. Lett. 105, 097401 (2010).[46] S. A. Mikhailov, and K. Ziegler, ‘New electromagneticmode in graphene, Phys. Rev. Lett. 99, 016803 (2007).[47] Y. S. Ang, M. Zubair, and L. K. Ang, ‘Relativistic space-charge-limited current for massive Dirac fermions’, Phys.Rev. B 95, 165409 (2017).[48] P. Zhang, A. Valfells, L. K. Ang, J. W. Luginsland, andY. Y. Lau, ‘100 years of the physics of diodes’, Appl.Phys. Rev. 4, 011304 (2017).[49] T. Shinozaki, S. Hagiwara, N. Morioka, Y. Kimura, andK. Watanabe, ‘Real-time first-principles simulations ofthermionic emission from N-doped diamond surfaces’,Appl. Phys. Expr. 11, 064301 (2018).[50] L. K. Ang, T. J. T. Kwan, Y. Y. Lau, ‘New scaling ofChild-Langmuir law in the quantum regime’, Phys. Rev.Lett. 91, 208303 (2003).[51] I. Meric, M. Y. Han, A. F. Young, B. Ozyilmaz, P. Kim,and K. L. Shepard, ‘Current saturation in zero-bandgap,top-gated graphene field-effect transistors’, Nat. Nan-otech. 3, 654 (2008).[52] Y. Ueda, Y. Suzuki, and K. Watanabe, ‘Time-dependentfirst-principles study of angle-resolved secondary electronemission from atomic sheets’, Phys. Rev. B 97, 075406(2018).[53] Y. Ueda, Y. Suzuki, and K. Watanabe, ‘Secondary-electron emission from multi-layer graphene: time-dependent first-principles study’, Appl. Phys. Expr. 11105101 (2018).[54] O. W. Richardson, ‘Some applications of the electron the-ory of matter’, Phil. Mag. 23, 594 (1912); S. Dushman,‘Electron emission from metals as a function of temper-ature’, Phys. Rev. 21, 623 (1923).[55] F. Zhu et al, ‘Heating graphene to incandescene and the[16] S. Misra, M. Upadhyay Kahaly, and S. K. Mishra,‘Photo-assisted electron emission from illuminated mono-layer graphene’, J. Appl. Phys. 121, 205110 (2017).[17] X. Zhang, Y. Pan, and J. Chen, ‘Parametric OptimumDesign of a Graphene-Based Thermionic Energy Con-verter’, IEEE Trans. Electron. Dev. 64, 4594 (2017).[18] X. Zhang, Y. Zhang, Z. Ye, W. Li, T. Liao, and J. Chen,‘Graphene-based thermionic solar cells’, IEEE Electron.Dev. Lett. 39, 383 (2018).[19] D. Sinha, and J.-U. Lee, ‘Ideal graphene/silicon Schottkyjunction Diodes’, Nano Lett. 14, 4660 (2014).[20] S.-J. Liang, and L. K. Ang, ‘Electron Thermionic Emis-sion from Graphene and a Thermionic Energy Con-verter’, Phys. Rev. Appl. 3, 014002 (2015).[21] Y. S. Ang, and L. K. Ang, ‘Current-Temperature Scal-ing for a Schottky Interface with Nonparabolic EnergyDispersion’, Phys. Rev. Appl. 6, 034013 (2016).[22] Y. S. Ang, S.-J. Liang, and L. K. Ang, ‘Theoretical mod-eling of electron emission from graphene’, MRS Bullet.42, 505 (2017).[23] M. Trushin, ‘Theory of thermionic emission from a two-dimensional conductor and its application to a graphene-semiconductor Schottky junction’, Appl. Phys. Lett. 112,171109 (2018).[24] M. Trushin, ‘Theory of photoexcited and thermionicemission across a two-dimensional graphene-semiconductor Schottky junction’, Phys. Rev. B97, 195447 (2018).[25] Y. S. Ang, H. Y. Yang, and L. K. Ang, ‘Universalscaling laws in Schottky heterostructures based on two-dimensional materials’, Phys. Rev. Lett. 121, 056802(2018).[26] S. V. Meshkov, ‘Tunneling of electrons from a two-dimensional channel into the bulk ’, Zh. Eksp. Teor. Fiz.91,2252 (1986).[27] K. J. Russell, F. Capasso, V. Narayanamurti, H. Lu, J.M. O. Zide, and A. C. Gossard, ‘Scattering-assisted tun-neling: Energy dependence, magnetic field dependence,and use as an external probe of two-dimensional trans-port’, Phys. Rev. B 82, 115322 (2010).[28] V. Perebeinos, J. Tersoff, and Ph. Avouris, ‘Phonon-mediated interlayer conductance in twisted graphene bi-layers’, Phys. Rev. Lett. 109, 236604 (2012).[29] E. E. Vdovin, A. Mishchenko, M. T. Greenaway, M. J.Zhu, D. Ghazaryan, A. Misra, Y. Cao, S. V. Morozov, O.Makarovsky, et al, ‘Phonon-assisted resonant tunnelingof electrons in grapheneboron nitride transistors’, Phys.Rev. Lett. 116, 186603 (2016).[30] Y. Liu, Z. Gao, Y. Tan, and F. Chen, ‘Enhancement ofout-of-plane charge transport in a vertically stacked two-dimensional heterostructure using point defects’, ACSNano 12, 10529 (2018).[31] J. Voss, A. Vojvodic, S. H. Chou, R. T. Howe, andF. Abild-Pedersen, ‘Inherent enhancement of electronicemission from hexaboride heterostructure’, Phys. Rev.Applied 2, 024004 (2014).[32] Y.-J. Yu, Y. Zhao, S. Ryu, L. E. Brus, K. S. Kim, andP. Kim, ‘Tuning the graphene work function by electricfield effect’, Nano Lett. 9, 3430 (2009).[33] H. Yuan et al, ‘Engineering ultra-low work function ofgraphene’, Nano Lett. 15, 6475 (2015).[34] S. Reich, J. Maultzsch, C. Thomsen, and P. Ordejon,‘Tight-binding description of graphene’, Phys. Rev. B 66,035412 (2002). [35] S. Tongay, M. Lemaitre, X. Miao, B. Gila, B. R. Ap-pleton, and A. F. Hebard, ‘Rectification at graphene-semiconductor interfaces: Zero-gap semiconductor-baseddiodes’, Phys. Rev. X 2, 011002 (2012).[36] Z. H. Ni et al, ‘Graphene thickness determination usingreflection and contrast spectroscopy’, Nano Lett. 7, 2758(2007).[37] D. Vashaee, and A. Shakouri, ‘Electronic and thermo-electric transport in semiconductor and metallic super-lattices’, J. Appl. Phys. 95, 1233 (2004).[38] M. F. ODwyer, R. A. Lewis, C. Zhang, and T. E.Humphrey, ‘Efficiency in nanostructured thermionic andthermoelectric devices’, Phys. Rev. B 72, 205330 (2005).[39] R. Kim, C. Jeong, and M. S. Lundstrom, ‘On momentumconservation and thermionic emission cooling’, J. Appl.Phys. 107, 054502 (2010).[40] M. Abramowitz, and I. A. Stegun, ‘Handbook of math-ematical functions: with formulas, graphs, and mathe-matical tables’, Dover Publications (USA, 1965).[41] M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, ‘Chi-ral tunnelling and the Klein paradox in graphene, Nat.Phys. 2, 620 (2006).[42] K. S. Novoselov et al, ‘Room-temperature quantum Halleffect n graphene, Science 315, 1379 (2007).[43] X. Du, I. Skachko, A. Barker, and E. Y. Andrei, ‘ Ap-proaching ballistic transport in suspended graphene, Nat.Nanotechnol. 3, 491 (2008).[44] A. N. Grigorenko, M. Polini, and K. S. Novoselov,‘Graphene Plasmonics, Nat. Photon. 6, 749 (2012).[45] E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, andS. A. Mikhailov, ‘Coherent nonlinear optical response ofgraphene, Phys. Rev. Lett. 105, 097401 (2010).[46] S. A. Mikhailov, and K. Ziegler, ‘New electromagneticmode in graphene, Phys. Rev. Lett. 99, 016803 (2007).[47] Y. S. Ang, M. Zubair, and L. K. Ang, ‘Relativistic space-charge-limited current for massive Dirac fermions’, Phys.Rev. B 95, 165409 (2017).[48] P. Zhang, A. Valfells, L. K. Ang, J. W. Luginsland, andY. Y. Lau, ‘100 years of the physics of diodes’, Appl.Phys. Rev. 4, 011304 (2017).[49] T. Shinozaki, S. Hagiwara, N. Morioka, Y. Kimura, andK. Watanabe, ‘Real-time first-principles simulations ofthermionic emission from N-doped diamond surfaces’,Appl. Phys. Expr. 11, 064301 (2018).[50] L. K. Ang, T. J. T. Kwan, Y. Y. Lau, ‘New scaling ofChild-Langmuir law in the quantum regime’, Phys. Rev.Lett. 91, 208303 (2003).[51] I. Meric, M. Y. Han, A. F. Young, B. Ozyilmaz, P. Kim,and K. L. Shepard, ‘Current saturation in zero-bandgap,top-gated graphene field-effect transistors’, Nat. Nan-otech. 3, 654 (2008).[52] Y. Ueda, Y. Suzuki, and K. Watanabe, ‘Time-dependentfirst-principles study of angle-resolved secondary electronemission from atomic sheets’, Phys. Rev. B 97, 075406(2018).[53] Y. Ueda, Y. Suzuki, and K. Watanabe, ‘Secondary-electron emission from multi-layer graphene: time-dependent first-principles study’, Appl. Phys. Expr. 11105101 (2018).[54] O. W. Richardson, ‘Some applications of the electron the-ory of matter’, Phil. Mag. 23, 594 (1912); S. Dushman,‘Electron emission from metals as a function of temper-ature’, Phys. Rev. 21, 623 (1923).[55] F. Zhu et al, ‘Heating graphene to incandescene and the