Relativistic space-charge-limited current for massive Dirac fermions
RRelativistic space-charge-limited current for massive Dirac fermions
Y. S. Ang, ∗ M. Zubair, and L. K. Ang † SUTD-MIT International Design Center, Singapore University of Technology and Design, Singapore 487372
A theory of relativistic space-charge-limited current (SCLC) is formulated to determine the SCLCscaling, J ∝ V α /L β , for a finite bandgap Dirac material of length L biased under a voltage V . Ina one-dimensional (1D) bulk geometry, our model allows ( α , β ) to vary from (2,3) for the non-relativistic model in traditional solids to (3/2,2) for the ultra-relativistic model of massless Diracfermions. For a two-dimensional (2D) thin-film geometry, we obtain α = β that varies between 2 and3/2, respectively, at the non-relativistic and ultra-relativistic limits. We further provide a rigorousproof based on a Green’s function approach that for uniform SCLC model described by carrierdensity-dependent mobility, the scaling relations of the 1D bulk model can be directly mapped intothe case of 2D thin film for any contact geometries. Our simplified approach provides a convenienttool to obtain the 2D thin-film SCLC scaling relations without the need of explicitly solving thecomplicated 2D problems. Finally, this work clarifies the inconsistency in using the traditional SCLCmodels to explain the experimental measurement of 2D Dirac semiconductor. We conclude that thevoltage-scaling 3 / < α < . PACS numbers: 77.22.Jp, 72.10.-d, 73.63.-b, 73.50.-h
I. INTRODUCTION
Space-charge-limited current (SCLC) gives the maxi-mum current that can be transported across a solid oflength L with a biased voltage V , limited by the electro-static repulsion generated by the in-transit unscreenedcharge carriers that are in excess of the thermodynam-ically allowed population . In a trap-free bulk crystal,SCLC exhibits a signature current-voltage ( J - V ) char-acteristics of J MG ∝ V /L known as the Mott-Gurney(MG) law , which is the solid-state counterpart of theSCLC in vacuum as given by the Child-Langumir (CL)law: J CL ∝ V / /L in classical regime and J CL ∝ V / /L in quantum regime . Including defect states ortraps in solids, SCLC becomes trap-limited as describedby the Mark-Helfrich (MH) law : J MH ∝ V l +1 /L l +1 ,where l = T c /T , T is temperature and T c is a parame-ter characterizing the exponential spread in energy of thetraps. Due to the geometrical effect , the 1D SCLC valueis enhanced as a result of finite emission area and weak-ened Coulomb screening in high aspect-ratio nanowire .Furthermore, SCLC is an important tool to probe thetrap characteristics in solids, and also for photocurrentmeasurement since the extraction efficiency of photogen-erated carriers is fundamentally limited by SCLC .For organic semiconductors, field-dependent anddensity-dependent mobility SCLC models are com-monly employed to characterize the SCLC carried bythe holes. Similarly, SCLC of electrons was found tobe universally described by a trap model with Gaus-sian energy distribution in a large class of organicsemiconductors . Recently it is demonstrated thatthe magnitude of electron SCLC can be significantlyenhanced via the dilution of traps in conjugated poly-mer blends of only 10% of active semiconductor , whichopens up an exciting possibility of high-efficiency andlow-cost organic light emitting diode. SCLC in the trap-limited regime was re-formulated with inclusion ofthe interplay between dopants and traps, Poole-Frenkeleffect and quantum mechanical tunneling, which hassolved the long-standing problem of the enormouslysharp current rise at the trap-limited regime and demon-strated that an exponentially distributed trap is not nec-essarily required to explain the power-law sharp rises ofSCLC in the trap-limited regime. Remarkably, the modelsuccessfully reproduced the anomalous noise-spectrumpeak observed in .In spite of SCLC being a classic model first derived in1940s, it remains an active topic for organic materials andnanowires as mentioned above. With the advances in fab-ricating novel 2D Dirac materials , it is of the inter-ests to revisit the SCLC model for these 2D Dirac mate-rials. To our best knowledge, there is no theory or modelto deal with the SCLC transport in Dirac materials. Re-cent experiments reported a typical J - V characteristic inthe form of MH law for highly disordered materials likereduced graphene oxide . On the other hand, SCLCin crystalline monolayer MoS and hBN was foundto exhibit an unusual power law dependence of J ∝ V α with 1 . (cid:46) α (cid:46) .
5, which was claimed to be originatedfrom the different levels of traps in different samples byusing the traditional MH law. This explanation is doubt-ful as the traditional MH law is only valid for α > T c > T , which implies that the voltage scaling from theMH law must be α ≥ . (cid:46) α (cid:46) . T c < T , the traps are narrowly distributed in energyspace and the SCLC essentially reduces to single-levelshallow trapping with α = 2 . Thus, the observation of α < , Gaussian disorder , field-dependent and density-dependent mobility.For a Dirac material with finite bandgap, the electronsmimics relativistic massive Dirac fermions whereas a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r the classical SCLC models are based on the conductionmodel of non-relativistic quasi-free electrons . In thiswork, we proposed a model of relativistic SCLC of mas-sive Dirac fermions, which can explain the peculiar α < T c < T used in the MH law in order to fit the experimen-tal data. According to our model, the J - V characteris-tics of SCLC in a 1D bulk geometry will vary betweenthe non-relativistic limit of J ∝ V /L to the ultra-relativistic limit of J ∝ V / /L (this is different fromthe CL law - see below for explanation). We present amaster equation which is in good agreement with the ex-perimental data and can be used to characterize the tran-sition between the Ohmic conduction and SCLC regime.By extending the bulk 1D model to a 2D thin film model,the scaling relation becomes J ∝ V α /L β with α = β varying between 3/2 and 2, respectively, at the ultra-relativistic and non-relativistic limits. In doing so, weprove rigorously, using a Green’s function approach ,that the 1D bulk SCLC current-voltage scaling relationcan be directly mapped to 2D thin-film SCLC. It is shownthat for a general transport equation of J = enµ ( n ) E where µ ( n ) is a mobility that depends on carrier den-sity, n , and E is the electric field, the 1D bulk SCLCand 2D thin-film SCLC are linked by a universal SCLCscaling relation (see Section III). Our analysis provides aconvenient tool to deduce the 2D thin-film SCLC scalingrelation via simple 1D SCLC model without the need ofexplicitly solving the complicated 2D SCLC model. II. THEORY OF RELATIVISTICSPACE-CHARGE-LIMITED CURRENT
In this section, a relativistic SCLC model is devel-oped using semiclassical Boltzmann transport equation(BTE). For simplicity, we first consider the SCLC by as-suming a simple 1D Poisson equation, which allows semi-analytical scaling relations to be determined. In SectionIII, we shall show that the simple 1D SCLC scaling re-lation derived in this section can be directly mapped tothe case of 2D SCLC with thin-film geometry.
A. Boltzmann transport equation for conventionalsemiconductor
The starting point of the trap-free SCLC theory, i.e.the Mott-Gurney’s law, is the semiclassical BTE whichprovides a basic equation of current density governing thetransport of charge carriers. The diffusion component isusually not considered except in some cases of polymersdue to their highly disordered nature. For a quasi-staticsystem, the BTE in the linearized transport regime underrelaxation-time approximation is − e E (cid:126) · ∂f∂ k + v · ∂f∂ r = − f − f τ (1) where E is the electric field, k is the crystal momen-tum, r is the position vector, v is the carrier velocity, fis the out-of-equilibrium distribution function, f is theequilibrium Fermi-Dirac distribution function, and τ isa typical collision time scale. If the system is spatiallyhomogeneous the diffusion component of the transportcurrent, i.e. ∂f /∂ r , can be omitted. By assuming a 3Disotropic parabolic energy dispersion, one arrives at thewell-known drift-current density, J = e/ (3 π ) (cid:82) v k f d k ,for semiconductors: J D = τ e m n ( x ) E ( x ) (2)By connecting the drift-equation with the 1D Pois-son equation via charge density n ( x ), the Mott-Gurneycurrent-voltage scaling of J ∝ V can be recovered.For 2D gapped Dirac materials, Eq. (2) is no longervalid due to two reasons: (i) the dimensionality is re-duced to 2D; and (ii) the energy dispersion follows a rel-ativistic dispersion similar to that of the massive Diracfermions. In the following, we shall formulate the drift-equation for Dirac semiconductor based on BTE ap-proach and demonstrate that the SCLC mediated byrelativistic quasiparticles follows a completely differentcurrent-voltage scaling relation. B. Boltzmann transport equation for 2D Diracsemiconductor
For massive Dirac fermions, the energy dispersion is ε k = (cid:112) (cid:126) v F k + ∆ where v F is the Fermi velocity, k is the crystal momentum and 2∆ is the bandgap.The group velocity is v k = (cid:126) − dε k /dk = (cid:126) v F k/ε k .The density of states D ( ε ) = (cid:80) k δ ( ε − ε k ), is rewrit-ten as D ( ε ) = ( g sv ε/ π (cid:126) v F )Θ( ε − ∆) where g sv de-notes the spin-valley degeneracy and Θ( x ) is a Heav-iside function. The electron density at low temper-ature can then be obtained from the two-dimensional(2D) density of states n = (cid:82) D ( ε k ) dε k which gives n = ( g sv / π (cid:126) v F ) (cid:0) µ − ∆ (cid:1) and µ is the Fermi level.The general expression of the 2D linear current den-sity is J = ( τ e E/ π ) (cid:82) v k kdk ( − ∂f /∂ε k ) where f isthe Fermi-Dirac distribution function, τ is the scatteringtime and E is the externally applied electric field. In thelow temperature limit, the current density can be ana-lytically solved to give J = ( g sv τ e E/ π (cid:126) µ )( µ − ∆ ).Eliminating µ via n , we obtain J = (cid:114) eg sv π τ ev F (cid:126) en ( x ) (cid:112) en ( x ) + ρ c E ( x ) , (3)where ρ c ≡ eg sv ∆ / π (cid:126) v F is a bandgap-dependent characteristic charge density , n and E are re-expressedas functions of the transport direction, x . The term( en ( x ) + ρ c ) − / in Eq. (3) represents a major differencebetween the relativistic massive Dirac fermions and thatof the non-relativistic quasi-free electrons. As ρ c ∝ ∆ ,it can be seen from the ε k - k relation that the electronsare non-relativistic at very large ρ c or ∆ (cid:29) (cid:126) v F k . Forvanishingly small ρ c or ∆ (cid:28) (cid:126) v F k , the electrons ap-proach ultra-relativsitic limit and become massless Diracfermions.By expressing Eq. (1) in the Drude form of J = enµ D ( n ) E , a density-dependent Dirac mobil-ity is defined as µ D ≡ γ ( en ( x ) + ρ c ) − / where γ ≡ τ v F e / g / s,v /π / (cid:126) . Consequently, the relativis-tic SCLC of massive Dirac fermions belongs to theclass of density-dependent mobility SCLC. However, µ D ≡ γ ( en ( x ) + ρ c ) − / is unique to the massive Diracfermions. C. Relativistic SCLC in bulk geometry
We assume that the SCLC is carried solely by electronsinjected through an Ohmic contact. For simplicity, weemploy the 1D Poisson equation dE ( x ) /dx = en ( x ) /(cid:15)d where E ( x ) = dV ( x ) /dx , V ( x ) is the electrostatic poten-tial, and (cid:15) is the effective dielectric constant. Here, wefirst assume that the 3D carrier density, n D ( x ), is re-lated to n ( x ) via n D ( x ) = n ( x ) /d with d as an effectivethickness (see modification later). By coupling Eq. (3)with the Poisson equation via n ( x ), we obtain the gov-erning equation of the relativistic SCLC for Dirac solid: E ( x ) dE ( x ) dx = Jγ(cid:15)d (cid:114) (cid:15)d dE ( x ) dx + ρ c . (4)We first investigate the solutions of the 1D relativisticSCLC, i.e. Eq. (4), in two asymptotic limits: (i) non-relativistic SCLC regime ( ρ c (cid:29) en ( x )); and (ii) ultra-relativistic SCLC regime ( ρ c (cid:28) en ( x )), which allows Eq.(3) to be approximated, respectively, as J nr = 98 (cid:15)d τ ev F ∆ V L , ρ c (cid:29) en ( x ) , (5a) J r = 83 (cid:114) eg sv δ(cid:15) π τ ev F (cid:126) V / L , ρ c (cid:28) en ( x ) . (5b)The MG scaling is readily recovered from the non-relativistic charge dynamics at large ρ c . As the electronsreside just slightly above the bandgap where the inequal-ity (cid:126) v F k (cid:28) ∆ holds true, we have ε k ≈ m ∗ v F + (cid:126) k / m ∗ where m ∗ ≡ ∆ /v F . It can be shown that the corre-sponding current density is in the non-relativistic Drudeform, which recovers the MG scaling of J ∝ V /L .In the opposite limit of ρ c → (cid:126) v F k (cid:29) ∆ impliesultra-relativistic dynamics with a scaling of J ∝ V / /L as shown in Eq. 3(b). Coincidentally, this has the samescaling to the CL law although the underlying physicsis fundamentally different. For Dirac materials studiedthere, the ultra-relativistic SCLC is obtained from J r ∝ (cid:114) d V ( x ) dx dV ( x ) dx , (6) . . . . (a) φ J numerical J fit = a √ φ + b (b) φ J ( φ (cid:28) ) numerical J fit = c − c φ
20 40 60 80 1000 . . . . (c) φ J ( φ (cid:29) ) numerical J fit = c √ φ FIG. 1. J as a function of φ . (a) Numerical results of J over the full range of φ . The dashed line shows the empiricalfitting equation; numerical results with (b) φ (cid:28)
1; and at (c) φ (cid:29)
1. The fitting constants ( a , b , c , c , c ) are (1.067, 1.45,0.889,0.368,1.092). while SCLC in vacuum (or CL law) is obtained from J vac ∝ (cid:112) V ( x ) d V ( x ) dx . (7)A simple dimensional analysis immediately shows that J r and J vac are both proportional to V / albeit theirvery different origins. Nonetheless, Eq. (6) originatesfrom the J r ∝ (cid:112) n ( x ) dependence owing to the ultra-relativistic electron dynamics in Dirac solids while Eq.(7) originates from the J vac ∝ (cid:112) V ( x ) dependence owingto the energy balance of a non-relativistic free electronaccelerating in vacuum. This demonstrates the funda-mentally different mechanism behind the J ∝ V / scal-ing in the two cases.The two limits: α = 3 / α = 2 are, respec-tively, the extreme limits of the ultra-relativsitic and non-relativistic SCLC, an intermediate regime of 3 / < α < . < α < . and 1 . < α < . where the charge carriers are es-sentially massive Dirac fermions. The model proposedhere suggests that the α < intrinsic to therelativistic carriers without the unjustified or invalid as-sumption of introducing traps with T c < T , which is alsoinconsistent with the original formulation of the MH lawand other trap-limited SCLC models as discussed above.For convenience, we transform Eq. (4) into a dimen-sionless form of d V ( χ ) dχ d V ( χ ) dχ = J (cid:115) d V ( χ ) dχ + φ, (8)where V ( x ) ≡ V ( x ) /V and χ ≡ x/L . The normalizedcurrent J and the dimensionless parameter φ are J ≡ (cid:126) τ e / v F √ (cid:15)d JL V / , (9a) φ ≡ ρ c (cid:15)d L V . (9b)For a given value of φ and boundary conditions: V (0) = 0and V (1) = 1, Eq. (8) is solved numerically at various J .The corresponding space charge limited current (SCLC)is determined when the value of J will cause the onsetof V ( χ ) <
0, and the calculated SCLC J is plotted asa function of φ in Fig. 1(a), which exhibits contrastingbehaviour at φ (cid:28) φ (cid:29)
1. For φ (cid:28) φ (cid:29) J (cid:48) fit = c − c φ and J (cid:48)(cid:48) fit = c / √ φ , respectively,where ( c , c , c ) = (0 . , . , . φ -dependence can be understood from the dependenceof φ ∝ ρ c , which corresponds, respectively, to the ultra-relativistic SCLC at φ (cid:28) φ ), and the non-relativistic SCLC at φ (cid:29) φ ). By substitutingEq. (9) into the above mentioned fitting equations, J r ∝ V / /L and J nr ∝ V /L are recovered, thus confirmingthe analytical solutions given in Eq. (5).Figure 2 shows the smooth transition of V and L in be-tween the ultra-relativisitc and non-relativistic regimes.The dimensionless current density ˜ J L - ˜ V characteristic(at a fixed L ) exhibits a voltage scaling of α = 2 atlow- ˜ V and α = 3/2 at high- ˜ V . Here the dimension-less parameters are ˜ J L ≡ J /J , ˜ V ≡ V /V , J ≡ τ e / v F √ (cid:15)δ/ (cid:126) and V = ρ c L /(cid:15)δ . At a fixed L , we have˜ V ∝ /V ∝ /ρ c . and thus low- ˜ V and high- ˜ V corre-spond to the non-relativistic ( ˜ J L ∝ ˜ V ) and the ultra-relativistic ( ˜ J L ∝ ˜ V / ) regimes, respectively as shownin Fig. 2 (blue axis, (cid:3) -symbols). In the intermediateregime, ˜ J L - ˜ V deviates from the simple power law, andapplying a fitting would lead to a sub-quadratic scalingin the range of 3 / < α < J V - ˜ L characteristic (at afixed V ) shows a length scaling of J ∝ ˜ L − β of β = 2 atsmall ˜ L and β = 3 at large ˜ L , as shown in Fig. 2 (greenaxis, (cid:52) -symbols). Here, the dimensionless parametersare ˜ J V = J/ ¯ J , ˜ L = L/L , ¯ J = τ e / v F ρ c V / / (cid:126) √ (cid:15)d and L = (cid:112) (cid:15)dV /ρ c . As ˜ L ∝ √ ρ c , small ˜ L andlarge ˜ L corresponds to the non-relativistic and the ultra-relativistic regimes, respectively. The β < α < φ can be accurately fitted by J fit ( φ ) = a √ φ + b where( a, b ) = (1 . , . J and φ , this empirical relation is univer-sally valid and thus we derive a master equation that uni-versally describes the SCLC transport over a wide range 10 − − − −
214 3 ˜ V ˜ J L − − − − ˜ L ˜ J V FIG. 2. Normalized voltage (blue axes, (cid:3) -symbols) and length(green axes, (cid:52) -symbols) characteristics of the current density.The broken (red) and the dotted (black) guide-line representsultra-relativistic and non-relativistic limit, respectively. Thelabels of the guide-lines, i.e. 1, 2, 3 and 4, correspond tothe scalings ˜ J L ∝ V , ˜ J L ∝ V / , ˜ J V ∝ L and ˜ J V ∝ L ,respectively. of parameters: V I = Λ( ρ c , (cid:15)d, τ, L, W ) V + Ω( (cid:15)d, τ, L, W ) , (10)where I = J × W is the total current, W is the devicewidth, andΛ( ρ c , (cid:15)d, τ, L, W ) ≡ ρ c ( (cid:15)d ) (cid:126) L a τ v F e W , (11a)Ω( (cid:15)d, τ, L, W ) ≡ b (cid:126) L a τ v F e W (cid:15)d . (11b)It is important to emphasize that Eq. (10) is extremelyusefully if it is used to fit with the experimental I - V mea-surement (in the form of V /I as a function of 1 /V ) todetermine the values of Λ and Ω, which can be subse-quently used to determine the collision time scale τ byusing Eqs. (9) if the other parameters are known.For the ultra-relativistic limit at ρ c →
0, Eq. (10) be-comes V /I ≈ Ω which confirms the predicted ultra-relativistic scaling of ( α, β ) = (3 / , ρ c (cid:29)
0, Eq. (10) reduces to V /I ≈ Λ /V which recovers the classical MG scalingof ( α, β ) = (2 ,
3) as expected. Therefore, the interme-diate relativistic SCLC will produce a positive intercepton the vertical-axis of V /J -1 /V characteristic whereasthe SCLC with non-relativistic scaling will have a zerointercept as shown in Fig. 3(b).Interestingly, the V /I -1 /V characteristic [suggestedin Eq. (10)] provides a convenient tool to represent theSCLC data that can be generally applied to any solids.To illustrate this point, we consider a trap-free solidin which the conduction transits from Ohmic to SCLCat increasing V as showin in Fig. 3a. In the Ohmicregime (large 1 /V or small V ) where J ∝ V , we have V /I ∝ (1 /V ) − , i.e. V /I decreases with increas-ing 1 /V (green dash-dotted line in Fig. 3a). In con-trast, in the SCLC regime (small 1 /V or large V ) where I ∝ V , we have V /I ∝ /V , i.e. V /I increaseslinearly with 1 /V (blue dashed line in Fig 3a). Thesecontrasting behaviors lead to a transitional peak inthe intermediate regime that clearly separates the SCLC-dominated and Ohmic-dominated conduction as shownin Fig. 3(a). This finding is confirmed by using var-ious experimental data (color symbols) for MoS fromRef. at different temperature as shown in Fig. 3a.The above mentioned transitional peak between SCLC-dominated regime (dashed curve) and Ohmic-dominatedregime (dash-dotted curve) can be clearly observed at alltemperatures.A zoom-in view at the small-1 /V SCLC-dominatedregime is shown in Fig. 3(b), which indicates that theexperimental results (symbols) can be explained by lin-ear fitting so to obtain the voltage scaling, which rangesfrom α = 2.11 down to 1.67 according to Eq. (10). The α scaling decreases to α < α . For the three α < y -axis at1 /V → α approaches 2, the in-tercepts diminishes and becomes approximately zero at α = 2. These observations are in good agreement withthe predicted intercepts of Eq. (10), as discussed above.Note that Eq. (10) breaks down in the case of α > III. SIMPLIFIED MODEL OF UNIFORM SCLCINJECTION IN 2D DIRAC SEMICONDUCTOR
The relativistic SCLC model derived above is based onsolving the 1D Poisson equation. As Dirac semiconductoris a 2D thin film, thus a 2D thin-form model is required.In this Section, we provide a simplified formalism of 2Dthin-film relativistic SCLC model without the need toexplicitly solve for the 2D model . A. Universal model of D -dimensional uniformSCLC injection in solid with density-dependentmobility The SCLC has been previously formulated for thin-film and nanowire using an integral form of 2D elec-trostatic Poisson equation . Here, we shall formulatea thin-film SCLC relativistic model under the assump-tion of carrier density-dependent mobility to illustratethat the general SCLC scaling properties for Dirac semi- 10 (a) OhmicSCLC . . . α = 1 . α = . α = . (b) 1 /V ( V − ) V / I ( V / µ A ) FIG. 3. Plot of V /I against 1 /V in the SCLC regime us-ing MoS experimental data from Ref. with (T, α ) of (285K,1.67) (blue square), (265K, 1.73) (green circle), (245K, 1.82)(yellow triangle), (205K, 2.00) (red diamond) and (185K,2.11) (black star). (a) The entire V /I range over 0.4 < /V < J ∝ V ) and SCLC( J ∝ V ) fitted to the T = 205 K data. The horizontalgray line indicate the transitional regime of J ∝ V / (1 < /V <
6) separating the green-shaded SCLC-dominatedand the yellow-shaded Ohmic-dominated regime. (b) TheSCLC-dominated regime at small 1 /V < conductor, which has a density-dependent mobility of µ D = γ ( en ( x ) + ρ c ) − / .We consider the D -dimension transport in a solidwith density-dependent mobility in a general form of µ = µ f [ n ( x )] /f where f [ n ( x )] is a density-dependentterm and f is a constant factor. The dimensionality of D = 1 and D = 2 corresponds to bulk and 2D thin film,respectively. In the following analysis, we consider thecase of uniform SCLC injection where the two electrodesare separated by a fixed spacing of L as shown in Fig.4(a). For uniform SCLC injection along the x -direction,the electric field profiles, i.e. E D for bulk ( D = 1) andthin film ( D = 2) can be written, respectively, as E ( x ) = e(cid:15) (cid:90) dx (cid:48) ∂G ( x, x (cid:48) ) ∂x n ( x (cid:48) ) , (12a) E ( x, y ) = e(cid:15) (cid:90) dx (cid:48) (cid:90) dy (cid:48) ∂G ( x, y, x (cid:48) , y (cid:48) ) ∂x δ ( y (cid:48) ) n ( x (cid:48) ) , (12b)where n ( x ) and n ( x ) denote the surface and volumecarrier density respectively. G ( x, x (cid:48) ) and G ( x, y, x (cid:48) , y (cid:48) )is, respectively, the 1D and 2D Green’s function that aredependent on the geometry of contacts. Figures 4(b) and(c) shows a 2D thin film with two possible contact ge-ometries, i.e. edge and strip contacts, respectively . Byeliminating the y (cid:48) -integration via δ ( y (cid:48) ) and suppressingthe argument of y = 0 in Eq. (12b) for simplicity, Eq.(12) can be written compactly as E D ( ξ ) = e(cid:15) (cid:90) dξ (cid:48) ∂G D ( ξ, ξ (cid:48) ) ∂ξ n ν ( ξ (cid:48) ) , (13)where we have introduced dimensionless variable as ξ ≡ x/L , and the subscript of ν = 2 , J D = en ν ( ξ ) µ f [ n ν ( ξ )] f E D ( ξ ) , (14)where the subscript D = 1 , µ [ n ν ( ξ )] ≡ µ f [ n ν ( ξ )] /f where f [ n ν ( ξ )]is a n ν ( ξ )-dependent term and f is a normalizationconstant. From Eq. (14), the bias voltage relation: V = (cid:82) L E D ( x (cid:48) ) dx (cid:48) is written as J D f (cid:90) dξ (cid:48) n ν ( ξ (cid:48) ) f [ n ν ( ξ (cid:48) )] = eµ VL . (15)The solution of Eq. (15) gives the equation of SCLC.Its full solution require the knowledge of n ν ( ξ ) over theintervals from ξ = 0 to ξ = 1, which can be obtainedby solving the nonlinear integral equation in Eq. (13).Nonetheless, the scaling relations, i.e. J D - V and J D - L ,can be readily deduced via a simple dimensional analy-sis without explicitly solving Eqs. (13) and (15).To illustrate this, we first combine Eqs. (13) and (14)to obtain1 = e µ (cid:15)f J D n ν ( ξ ) f [ n ν ( ξ )] (cid:90) dξ (cid:48) ∂G D ( ξ, ξ (cid:48) ) ∂ξ n ν ( ξ (cid:48) ) . (16)From the Poisson equation, i.e. ∇ G D ( r , r (cid:48) ) = δ D ( r )where δ D ( r ) is a D -dimensional Dirac delta function,the physical dimension of G D can be obtained as[ G D ( ξ, ξ (cid:48) )] = L − D , where L denotes the fundamentaldimension of length and [ X ] denotes the unit of physi-cal quantity X . Correspondingly, the partial derivative, ∂G D /∂ξ , in Eq. (16) can be non-dimensionalized as ∂G D ( ξ, ξ (cid:48) ) ∂ξ = L − D ∂ G D ( ξ, ξ (cid:48) ) ∂ξ , (17)where G D ( ξ, ξ (cid:48) ) is a dimensionless Green’s function. Wenow rewrite Eqs. (15) and (16) as J D f (cid:90) dξ (cid:48) n ν ( ξ (cid:48) ) f ν = eµ VL , (18a)1 = (cid:18) e µ L − D (cid:15)f J D (cid:19) n ν ( ξ ) f ν (cid:90) dξ (cid:48) ∂ G D ( ξ, ξ (cid:48) ) ∂ξ n ( ξ (cid:48) ) , (18b) 𝐿𝑥𝑧 𝑦 (𝑎) (𝑏) (𝑐)𝑦 𝑧 𝑥
FIG. 4. Schematic drawings of the device and contact ge-ometries. Top view of (a) constant- L contact geometry foruniform SCLC injection. Current is uniformly injected alongthe x -direction. For D = 1 bulk geometry, the constant- L ge-ometry is invariant along both z -and y -directions whereas for D = 2 thin-film geometry, the structure is only invariant alongthe z -direction. (c) and (d) shows the side view of constant- L
2D thin film with edge and strip contacts, respectively. where f ν ≡ f [ n ν ( ξ )] for simplicity. A direct inspectionof Eq. (18a) shows that after the integral (cid:82) dξ (cid:48) ( · · · ) isfully converted into a dimensionless form, the J D - V and J D - L scaling relations can be unambiguously determined.In this case, (cid:82) dξ (cid:48) ( · · · ) becomes a dimensionless numericfactor that does not play any roles in the J D - V and J D - L scaling relations.The non-dimensionalization of Eq. (18a) can be ac-complished by appropriately regrouping the constantterm in Eq. (18b), i.e. (cid:0) e µ L D − /(cid:15)f J D (cid:1) into eachof the n ν ( x ) and f ν terms in the right-hand side of Eq.(18b) such that dimensionless terms N ν ( ξ ) and F ν canbe defined, respectively, for n ν ( ξ ) and f ν . In general, theregrouping of (cid:0) e µ L D − /(cid:15)f J D (cid:1) can be expressed in anarbitrary form of e µ L − D (cid:15)f J D ≡ A J D ( f ν ) ˜ A J D ( f ν ) B L,D ( f ν ) ˜ B L,D ( f ν ) (19)where A J D ( f ν ) and ˜ A J D ( f ν ) are terms containing J D and B L,D ( f ν ) and ˜ B L,D ( f ν ) are terms containing L − D . Theroles of A ’s and B ’s are to pair up with n ( ξ ) and f ν in Eq.(18b) such that the resulting terms are dimensionless.In the following, we suppress the argument of A ’s and B ’s for simplicity. As the explicit form of f ν determinesthe regrouping of (cid:0) e µ L D − /(cid:15)f J D (cid:1) , A ’s and B ’s areboth f ν -dependent. Furthermore, A ’s are D -independentand B ’s are D -dependent as L − D is deliberately dis-tributed only into B ’s. We can now recast Eq. (18b)as 1 = N ν ( ξ ) F ν (cid:90) dξ (cid:48) ∂ G D ∂ξ N ( ξ (cid:48) ) (20)where all terms are dimensionless via the following group-ing N ν ( ξ ) ≡ ( A J D B L,D ) n ν ( ξ ) , F ν ≡ (cid:16) ˜ A J D ˜ B L,D (cid:17) f ν . (21)With N ν ( ξ ) and F ν now being dimensionless, Eq. (18a)can be rewritten as: J D A J D ˜ A J D B L,D ˜ B L,D (cid:90) dξ (cid:48) N ν ( ξ (cid:48) ) F ν = eµ VL , (22)or more compactly as J D A J D ˜ A J D = ψ G D eµ B L,D ˜ B L,D
VL , (23)where ψ G D is a dimensionless numeric factor dependenton the D and G D , i.e. ψ G D ≡ (cid:18)(cid:90) dξ (cid:48) N ν ( ξ (cid:48) ) F ν (cid:19) − , (24)which can be explicitly solved from Eq. (18b) and itaffects only the overall magnitude of SCLC without af-fecting its voltage and length scaling relations. Thus the J D - V and J D - L scaling relations are determined by J D A J D ˜ A J D ∝ B L,D ˜ B L,D
VL . (25)
B. Derivation of 2D thin-film SCLC scalingrelations
Equation (25) represents a universal SCLC scaling re-lations for uniform SCLC injection into either a D = 1(bulk) or D = 2 (thin film) based solid of length L witharbitrary density-dependent mobility µ [ n ν ( ξ )]. Severalremarkable properties can be extracted from Eq. (25): (i)The J D - V scaling, i.e. J D A J D ˜ A J D ∝ V , is determinedsolely by the µ [ n ν ( ξ )] and is completely independent onthe device geometry ( G D ) and dimensionality ( D ) ; (ii)the J D - L scaling, on the other hand, is affected by both f ν and D ; (iii) For a fixed D (= 1 for bulk or = 2 forthin film), the geometry of contacts affects only ψ G D andhence both SCL J D - V and J D - L scaling are universalindependent on contact.From Eq. (19), we see the constant term carriers alength scale dependence of L − D , and thus Eq. (19) willbe independent of L for D = 2 (for a thin film setting)resulting in B L, = ˜ B L, = 1. In this 2D thin film limit,Eq. (23) gives the SCLC relation for a thin film: J A J ˜ A J = ψ G eµ (cid:18) VL (cid:19) . (26) Note that Eq. (26) includes that both J - V and J - L follow the same scaling relation.Togther with property (i) and Eq. (26), it shows apowerful tool that can be used to directly map the scalingrelation of a simple bulk SCLC model into 2D thin-filmSCLC model. By virtual of Property (i), we concludethat the voltage scaling for thin-film ( J - V ) is identical tothe bulk J - V scaling, thus the voltage scaling obtainedin Sec. II are valid for thin film as well. For a thinfilm, Eq. (26) also dictates that the length ( J - L ) scalingrelation is identical to the J - V .To summarize this session, we provide a rigorousderivation of scaling laws (both voltage and length) foruniform SCL injection into a 2D thin film seeting. Theseproperties allow the J - V and J - L scaling relations of a2D thin-film to be fully determined from a simple 1D bulkSCLC model that have been shown in Sec. II. The full J - V and J - L scaling relations are thus obtained with-out the need of explicitly solving the complicated coupledequations in Eq (18). In Appendix A, two examples of2D thin-film SCLC are analyzed using our simplified for-malism developed here. C. Derivation of SCLC scaling relations and fullnumerical solutions of Eq. (18) for 2D Diracsemiconductor with traps
The relativistic SCLC scaling relation of Dirac semi-conductor in 2D thin-film geometry can be readily deter-mined by using the simple derivation developed above.Since the 1D SCLC scaling relations takes the form of J ∝ V / and J ∝ V respectively for the ultra-relativistic and non-relativistic regimes, for 2D thin film,our simple analysis yields J ∝ (cid:18) VL (cid:19) α , (27)where α = 2 and α = 3/2 are for the non-relativisticand ultra-relativistic limit, respectively. In the interme-diate regime, the scaling follows an approximate power-law form with α varies continuously from 2 → / n ls ( ξ ) (cid:0) − ξ (cid:1) / (cid:112) C l n ls ( ξ ) + n c = (cid:90) − (cid:32) γe C l J (cid:15) n s ( ξ (cid:48) ) (cid:0) − ξ (cid:48) (cid:1) ξ − ξ (cid:48) + (cid:112) C l n ls ( ξ ) + n c πn ls ( ξ (cid:48) ) (cid:33) dξ (cid:48) , (28a) V = J LγeC l (cid:90) − (cid:112) C l n ls ( ξ ) + n c n ls ( ξ ) dξ, (28b) Φ l =1 ∝ V − / l =1 Φ l =1 ∝ V − l =1 V l =1 Φ l = FIG. 5. Numerical solution of the trap-free ( l = 1) 2D thinfilm of Dirac semiconductor. The dashed and dotted linesdenotes Φ l =1 ∝ V − l =1 and Φ l =1 ∝ V − / l =1 , respectively. AsΦ l =1 ∝ /J and V l =1 ∝ V /L , the small- V l =1 and large- V l =1 regime corresponds to J ∝ ( V /L ) (non-relativistic) and J ∝ ( V /L ) / (ultra-relativistic), respectively. Note that thedata points exhibits oscillations at small V l =1 due to numeri-cal error. where l ≡ T c /T > n s ( ξ ) is the 2D carrier density, n c ≡ ρ c /e , C l ≡ N /N lt with N as the effective den-sity of states at the conduction band-edge and N t as thetrap density, ξ and ξ (cid:48) are dimensionless variables. Inthe ultra-relativistic and non-relativistic SCLC regimes,we obtain J ∝ ( V /L ) l/ and J ∝ ( V /L ) l +1 , re-spectively. By setting l = 1 (which corresponding to atrap-free case), we obtain J ∝ ( V /L ) (non-relativistic)and J ∝ ( V /L ) / (ultra-relativistic), thus confirmingthe simple derivation in Eq. (27). This agreementdemonstrates that the unconventional J D - V scaling of3 / < α < , the variationof α ≈ . α ≈ T < T c (valid) to T > T c (invalid). Our relativistic SCLC model with exponen-tial traps presented here is however ever to take accountfor such temperature dependence without imposing theinvalid T c < T condition.To further confirm that the analytical relativisticSLCC scaling relation obtained above, the integral equa-tion in Eq. (28a), which belongs to the class of nonlinearCauchy singular integral equation is numerically solvedfor the trap-free case of l = 1. The numerical solved n s ( ξ ) [from Eq. (28a)] is then integrated in Eq. (28b)to obtain the J as a function of V . For simplicity, Eq.(28) is solved in terms of a dimensionless variables, i.e. V l =1 ∝ V /L and Φ l =1 ∝ /J [see Appendix C and Eqs.(C6) and (C7) for the definition of V l and Φ l ]. The nu-merical results (red circles) of Φ l =1 ( ∝ /J ) as a functionof V l is shown in Fig. 5 show good agreement with thederived scaling laws: Φ l =1 ∝ V − l =1 (dotted lines) at small voltage of V l =1 < l =1 ∝ V − / l =1 at large voltageof of V l =1 >
2. Thus, the comparison confirms the twocorresponding analytical scaling laws [see Eq. (27)] forspace charge limited conduction in a 2D thin-film Diracsolid with finite bangap: J ∝ ( V /L ) and J ∝ ( V /L ) / respectively for the non-relativistic and ultra-relativisticlimits. More importantly, the unconventional relativisticSCLC scaling of 3 / < α < .Finally, we discuss the screening effect on the relativis-tic SCLC in 2D Dirac materials. In 2D Dirac materi-als, the charge transport is sensitively influenced by thesubstrate screening and excess charge screening inducedby gate-electrode in field-effect-transistor geometry. De-spite these screening effects, SCLC was unambiguouslyobserved in experiments as reported in Refs. [31] and[32]. These experimental observations suggest that thescreening effect cannot entirely remove SCLC in 2D ma-terials. In a previous theoretical work , it is demon-strated that the surrounding dielectric screening will af-fect the transport properties of 2D thin film by modifyingthe relaxation time τ and a significant mobility enhance-ment can be achieved via high- dielectric substrate withvanishingly thin membrane. This theoretical predictionwas experimentally confirmed in monolayer MoS2 withhigh- substrate . In relevance to our SCLC model, wepoint out that as the substrate screening effect altersonly the relaxation time τ which comes into the SCLCpicture as a proportionality constant, it can be reason-ably expected that only the magnitude of the SCLC willbe altered while the new scaling laws reported here willremain unchanged. A microscopic theory of substratescreening can be formulated via first-principle calcula-tion that takes into account the complex many-body in-teractions at the interface between 2D materials and thesubstrate . The complete microscopic quantum pictureod dielectric screening is beyond the scope of this work. IV. CONCLUSION
In summary, we have proposed a theory of relativisticspace charge limited conduction (SCLC) in Dirac solidswith new scaling laws for both bulk and thin file model.For the one-dimensional (1D) bulk model, the scalinglaws are J ∝ V α /L β with 3 / < α < < β < J ∝ ( V /L ) α for uni-form SCLC injection with α remaining the same as thecase of 1D bulk model under the assumption of densitydependent mobility. Both scaling laws have been verifiedwith numerical calculations and have good agreementswith experimental results. The important finding fromthis paper is the new voltage scaling of α < in relativisticDirac semiconductor such as MoS . The widely-studiedphotoresponse of MoS can be readily used as an addi-tional platform to verify the proposed relativistic SCLCmodel here. ACKNOWLEDGMENTS
We thank Subhamoy Ghatak for providing us the ex-perimental data of MoS and insightful discussions. Wethank Chun Yun Kee, Kelvin J. A. Ooi and Shi-Jun Liangfor helpful discussions. This work is supported by Singa-pore Ministry of Education T2 grant (T2MOE1401) andUSA AFOAR AOARD Grant (FA2386-14-1-4020). Appendix A: Two examples of 2D thin-film SCLCusing the simplified formalism
In this Appendix, we illustrates the simple derivationof the 2D SCLC scaling relations developed in SectionsIII-A and III-B using two examples. In the first example,we consider a trivial case with f ν /f = 1, i.e. the mobilityis independent of carrier density. For bulk model of D =1, Eq. (18b) becomes1 = (cid:18) e µ L(cid:15)J (cid:19) (cid:90) dξ (cid:48) ∂ G ( ξ, ξ (cid:48) ) dξ n ( ξ (cid:48) ) , (A1)which can be fully non-dimensionalized by defining A J =( e µ /(cid:15)J ) / , ˜ A J = 1, B L, = L / and ˜ B L, =1. From Eq. (23), we obtained J (cid:0) e µ /(cid:15)J (cid:1) / = ψ G eµ V /L / , which can be rearranged to give the well-knwon bulk MG law of J = ψ G (cid:15)µ V L . (A2)The numerical factor can be solved as ψ G = 9 / . We can nowmap the J - V bulk SCLC scaling relation to the 2D thin-film case, which yields J ∝ V . Furthermore, as J - L scales equally with J - V , the 2D thin film SCLC scalingrelation can now be fully determined as J ∝ ( V /L ) .One can verify this scaling relation by explicitly solvingEq. (26) with A J = ( e µ /J ) / and ˜ A J = 1. Thisgives J (cid:0) e (cid:15)µ /(cid:15)J (cid:1) / = ψ G e(cid:15)µ V /L , which can be re-arranged to give the well-known 2D thin-film SCLC , i.e. J = ψ G (cid:15)µ (cid:18) VL (cid:19) , (A3) where ψ G is a G -dependent numeric factor.In the second example, we consider a carrier den-sity dependent mobility in a power-law form, i.e. f ν = n ν ( ξ ) l − and f = n l − where n and l are some con-stants. This particular form of µ is equivalent to Mark-Helfrich’s exponential-trap model with l >
1. Forthis particular form of f ν /f , Eq. (18b) can be fullynon-dimensionalized by re-grouping the constant factor( e (cid:15)µ L/n l − J ) via the following definitions: A J =( e (cid:15)µ /n l − J ) / ( l +1) , ˜ A J = ( e (cid:15)µ /n l − J ) ( l − / ( l +1) , B L, = L / ( l +1) and ˜ B L, = L ( l − / ( l +1) . The bulk SCLCcan then be obtained from Eq. (23) as J (cid:18) e µ n l − (cid:15)J (cid:19) ll +1 = ψ G eµ VL l +1 l +1 , (A4)which can be simplified as J = ψ l +1 G (cid:0) en l (cid:1) l − (cid:15) l µ V l +1 L l +1 , (A5)and is in agreement with the Mark-Helfrich’s exponentialtrap model . To generalize the bulk SCLC to the caseof 2D thin film, we again utilize the facts that: (i) J - V follows the same scaling as J - V ; and (ii) J - L scalesequally with J - V This gives J ∝ ( V /L ) l +1 , which is inagreement with the explicitly solution of Eq. (26), i.e. J = ψ l +1 G (cid:0) en l (cid:1) l − (cid:15) l µ (cid:18) VL (cid:19) l +1 . (A6) Appendix B: Derivation of 2D relativistic SCLCmodel
In this Appendix, we provide a full derivation of theMark-Helfrich’s SCLC model and Dirac semiconductor in2D thin-film geometry based on Grinburg’s formalism .
1. Mark-Helfrich’s trap model of SCLC in 2Dthin-film geometry
In the presence of traps that follows an exponential en-ergy distribution , the free and trapped carrier densitiesare related by n f ( x ) = (cid:0) N /N lt (cid:1) n ls ( x ) ≡ C l n ls ( x ) where n f ( x ) is the free carrier density, n s ( x ) is the trappedcarrier density, N is the effective density of states atthe conduction band edge, N t is the trap density and C l ≡ N /N lt . Here, l ≡ T c /T ≥ T c is a charac-teristic temperature representing the exponential spreadin energy of the traps. The charge density in a 2D thin-film is given as ρ ( x, y ) = eδ ( y ) [ − n s ( x ) + P s δ ( L − x )] , (B1)where δ ( y ) is a Dirac delta function and P s = (cid:82) L n s ( x ) dx is the charge density induced on the annode by the total0 n s ( x ) residing in the thin film. Note that y representthe direction that is out-of-plane of the thin film. Forthin-edge contacts, the corresponding Green’s function is G ( x − x (cid:48) , y − y (cid:48) ) = − π ln (cid:2) ( x − x (cid:48) ) + ( y − y (cid:48) ) (cid:3) / , (B2)and the scalar potential can then be solved as φ ( x, y ) = − π (cid:90) ∞−∞ dy (cid:48) (cid:90) L dx (cid:48) ln (cid:2) ( x − x (cid:48) ) + ( y − y (cid:48) ) (cid:3) / (cid:18) − πe(cid:15) (cid:19) δ ( y (cid:48) ) [ − n s ( x ) + P s δ ( L − x )] . (B3)Simplify φ ( x, y = 0) and knowing that E x ( x,
0) = − dφ/dx , we obtain E x ( x,
0) = 2 e(cid:15) ( L − x ) (cid:90) L L − x (cid:48) x − x (cid:48) n s ( x (cid:48) ) dx (cid:48) . (B4)By defining ξ = x/L and ξ (cid:48) = x (cid:48) /L , we obtained E x ( ξ,
0) = 2 e(cid:15) (1 − ξ ) (cid:90) − ξ (cid:48) ξ − ξ (cid:48) n s ( ξ (cid:48) ) dξ (cid:48) . (B5)We now consider a current density equation in Drude’s form, i.e. J = en f ( x ) µE x ( x, . (B6)By combining Eq. (B5) and (B6), we obtain1 = 2 e µC l J(cid:15) n s ( x ) l − ξ (cid:90) − ξ (cid:48) ξ − ξ (cid:48) n s ( ξ (cid:48) ) dξ (cid:48) . (B7)Equation (B7) can be rearranged as followed:1 = (cid:18) e µC l J(cid:15) (cid:19) ll +1 n ls ( ξ )1 − ξ (cid:90) − ξ (cid:48) ξ − ξ (cid:48) (cid:18) e µC l J(cid:15) (cid:19) l +1 n s ( ξ (cid:48) ) dξ (cid:48) . (B8)By defining ν s ( ξ ) ≡ (cid:18) e µC l J(cid:15) (cid:19) l +1 n s ( ξ ) , (B9)Eq. (B8) becomes1 = ν ls ( ξ )1 − ξ (cid:90) − ξ (cid:48) ξ − ξ (cid:48) ν s ( ξ (cid:48) ) dξ (cid:48) , (B10)which is an integral equation that can be solved toobtain ν ( ξ ). The bias voltage can be obtained from V = (cid:82) E x ( ξ, dξ and Eq. (B6) as V = JLeµC l (cid:90) dξn ls ( ξ ) . (B11) To obtain the exponential trap-limited SCLC in 2D thinfilm geometry with edge-contact, Eqs. (B9) and (B11)are combined to give V = JL(cid:15)µC l (cid:18) e µC l J(cid:15) (cid:19) l +1 (cid:90) dξν s ( ξ ) . (B12)With the definition of λ ≡ (cid:82) dξ/ν s ( ξ ), which is a con-stant that can be solved from the integral equation in Eq.(B10), we obtain J = (cid:16) (cid:15) λ (cid:17) l e µC l (cid:18) VL (cid:19) l +1 . (B13)Equation (B13) gives the exponential trap-limited SCLC of a 2D thin film with edge-contact geometry [see Fig.4(b)]. For strip-geometry [see Fig. 4(c)], the electric field is given as E x ( ξ ) = 2(1 − ξ ) / (cid:32) e(cid:15) (cid:90) − n s ( ξ (cid:48) ) (cid:0) − ξ (cid:48) (cid:1) / ξ − ξ (cid:48) dξ (cid:48) + VπL (cid:33) , (B14)1where ξ ≡ (2 x − L ) /L and ξ (cid:48) ≡ (2 x (cid:48) − L ) /L . Using similar procedure, we obtain Jeµ = 2 C l n ls ( ξ )(1 − ξ ) / (cid:32) e(cid:15) (cid:90) − n s ( ξ (cid:48) ) (cid:0) − ξ (cid:48) (cid:1) / ξ − ξ (cid:48) dξ (cid:48) + 1 πL JLeµC l (cid:90) dξn ls ( ξ ) (cid:33) , (B15)which can be simplified to 1 = ν ls ( ξ )(1 − ξ ) / (cid:32)(cid:90) − ν s ( ξ (cid:48) ) (cid:0) − ξ (cid:48) (cid:1) / ξ − ξ (cid:48) dξ (cid:48) + 1 π (cid:90) − dξν ls ( ξ ) (cid:33) . (B16)From Eq. (B12), the SCLC current density equation isobtained as J = (cid:16) (cid:15) λ (cid:48) (cid:17) l e µC l (cid:18) VL (cid:19) l +1 , (B17)where the numeric factor λ (cid:48) ≡ (cid:82) − dξν s ( ξ ) can be ob-tained by solving ν s ( ξ ) from Eq. (B16). In summary, the2D thin-film uniform injection of SCLC in the presenceof exponential traps follow the following scaling relationof J ∝ (cid:18) VL (cid:19) l +1 , (B18) for both edge-and strip-contact geometries. More impor-tantly, this scaling relation is in agreement with Eq. (A6)obtained using the simplified formalism.
2. Relativistic SCLC model for 2D massive Diracfermions
For 2D Dirac semiconductor, we obtain1 = 2 γe C l n ls ( ξ ) J(cid:15) (cid:112) C l n ls ( ξ ) + n c (cid:90) − ξ (cid:48) ξ − ξ (cid:48) n s ( ξ (cid:48) ) dξ (cid:48) , (B19)and (cid:113) C l n ls ( ˜ ξ ) + n c (cid:16) − ˜ ξ (cid:17) / n ls ( ˜ ξ ) = (cid:90) − γe V l J(cid:15) n s ( ˜ ξ (cid:48) ) (cid:16) − ˜ ξ (cid:48) (cid:17) ˜ ξ − ˜ ξ (cid:48) + (cid:113) C l n ls ( ˜ ξ ) + n c πn ls ( ˜ ξ (cid:48) ) d ˜ ξ (cid:48) , (B20)respectively for edge-contact and strip-contact. The ap-plied bias voltage for edge-and strip-contact geometriesbecome, respectively, V = JLγeC l (cid:90) (cid:112) C l n ls ( ξ ) + n c n ls ( ξ ) dξ, (B21)and V = JLγeC l (cid:90) − (cid:113) C l n ls ( ˜ ξ ) + n c n ls ( ˜ ξ ) d ˜ ξ. (B22)The coupled Eqs. (B19) to (B22) can be solved toobtain the relativistic SCLC in 2D thin-film geometry.Equations (B19) to (B22) has to be solved numerically.Nonetheless, in the non-relativistic and ultra-relativisticlimits, semi-analytical scaling relations can be derived.We first consider the non-relativistic limit of n c (cid:29) n ls ( ξ ) for all ξ with edge-contacts, Eqs. (B19) and (B21) canbe approximated, respectively, by1 = 2 γe C l n ls ( ξ ) J(cid:15)n / c (cid:90) − ξ (cid:48) ξ − ξ (cid:48) n s ( ξ (cid:48) ) dξ (cid:48) , (B23)and V = JLn / c γeC l (cid:90) dξn ls ( ξ ) , (B24)By defining ν s ( ξ ) ≡ (cid:18) γe C l J(cid:15)n / c (cid:19) l +1 n s ( ξ ) , (B25)we obtain J = 1 λ l +1 (cid:16) (cid:15) (cid:17) l γC l n l/ c e l − (cid:18) VL (cid:19) l +1 , (B26)2where λ ≡ (cid:82) dξ/ν ls ( ξ ) is a numerical factor which canbe solved from the nonlinear integral equation in Eq.(B19). By setting l = 1, the current-voltage scaling re-lation agrees with the 1D bulk model as shown in Eq.(5a) of the main text. The current-voltage scales equallywith the current-length which is also in agreement withthe simplified derivation of 2D thin-film SCLC scalingrelation presented in Eq. (27) of the main text.In the ultra-relativistic limit of n c →
0, Eqs. (B19)and (B21) become, respectively,1 = ν l/ s ( ξ ) (cid:90) − ξ (cid:48) ξ − ξ (cid:48) ν s ( ξ (cid:48) ) dξ (cid:48) , (B27)and V = JLγe (cid:32) γe C / l J(cid:15) (cid:33) l/ l/ (cid:90) dξν l/ s ( ξ ) , (B28) where ν s ( ξ ) ≡ (cid:32) γe C / l J(cid:15) (cid:33) l/ n s ( ξ ) , (B29)which can be rearranged to give J = 1 λ (cid:48) l/ (cid:16) (cid:15) (cid:17) l/ γe − l/ C l/ l (cid:18) VL (cid:19) l +1 . (B30)The numerical factor, λ (cid:48) ≡ (cid:82) dξ/ν l/ s ( ξ ), can again besolved from Eq. (B19). For l = 1, the current-voltagescaling relation agrees with the ultra-relativistic resultsin Eq. (5b) of the main text. In the intermediate regime,the scaling relation can be approximated by J ∝ ( V /L ) Λ , (B31)where Λ = l/ l + 1. Appendix C: Equations (B28) to (B32) in dimensionless form
Equations. (B28) to (B32) can be transformed into a dimensionless form for numerical solution in Fig. 5 of maintext. For edge-contacts , we obtain 1 = Φ l − ξ f ls ( ξ ) (cid:112) f ls ( ξ ) + 1 (cid:90) − ξ (cid:48) ξ − ξ (cid:48) f s ( ξ (cid:48) ) dξ (cid:48) , (C1)and V l = 1Φ l (cid:90) (cid:112) f ls ( ξ ) + 1 f ls ( ξ ) dξ. (C2)For strip-contacts , the dimensionless form give1 = f ls ( ξ ) (cid:112) f ls ( ξ ) + 1 1(1 − ξ ) / (cid:90) − (cid:32) Φ l f s ( ξ (cid:48) ) (1 − ξ (cid:48) ) / ξ − ξ (cid:48) + 1 π (cid:112) f ls ( ξ (cid:48) ) + 1 f ls ( ξ (cid:48) ) (cid:33) dξ (cid:48) , (C3)and V l = 1Φ l (cid:90) − (cid:112) f ls ( ξ ) + 1 f ls ( ξ ) dξ. (C4)The dimensionless parameters are defined as f s ≡ C l n c /l n s ( ξ ) , (C5)Φ l ≡ γe J(cid:15) (cid:18) n c C l (cid:19) /l √ n c , (C6)and V l ≡ (cid:15)V eL (cid:18) C l n c (cid:19) /l , (C7)3where Φ l is current-and material-dependent parameter. ∗ yeesin [email protected] (Corresponding author) † ricky [email protected] (Corresponding author) M. A. Lampert and P. Mark,
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