Analytic study on cold field electron emission from metallic nanowall array *

Transcription

1 Analytic study on cold field electron emission from metallic nanowall array * Xizou Qin, Weiliang Wang and Zibing Li a) State Key Laboratory of Optoelectronic aterials and Tecnologies, Scool of Pysics and Engineering, Sun Yat-Sen University, Guangzou 575, Cina Te cold field electron emission from metallic nanowall array is investigated teoretically. Via conformal mapping metod, analytic formulas of tunneling barrier, edge field enancement factor, transmission coefficient, and area emission current density are derived in te semi-classical approximation. Te influence of field enancement effects and screening effects on cold field emission are discussed and te optimal spacing distance between nanowalls for te largest emission current density is found to be a linear function of te eigt of nanowalls and depends on te applied field. Keywords: metallic nanowall array, cold field electron emission, field enancement effects, screening effects PACS: q, Vj. Introduction Cold field electron emission (CFE) as an important vacuum electron source as applications in flat-panel display, electronic olograpy, e-beam litograpy and so on. Emitters of various nanostructures, especially nanotubes and nanowires, ave been extensively studied. Te major merit of nano-emitters is tat an applied electric field can be greatly enanced around te nano-scaled tips or edges. Since te experimental realization of free-standing grapene [], tis two-dimensional (D) atomic crystal as aroused great experimental and teoretical interest. Several groups ave demonstrated tat grapene does sow promising CFE properties suc as a low emission tresold field and large emission current density. [-8] Recent studies sow tat te CFE from te D nanostructures would ave a current-field caracteristic different from tat of te conventional Fowler Nordeim * Te Project was supported by te National Basic Researc Program of Cina (Grant No. 7CB9355) and te National Natural Science Foundation of Cina (Grant No. 4358). a) Corresponding autor. stslzb@mail.sysu.edu.cn

2 (FN) law wic was derived for planar emitters [9]. It is obvious because bot electron supply function and edge electric field of te D nanostructures are dramatically different from tose of te tree-dimensional systems. Te long edge of D emitters wit atomic tickness may lead to a unique line-electron source tat would be useful in electronic olograpy and parallel e-beam litograpy. Te present paper is interested in te aligned array of D metallic nanowalls tat is defined as an array of parallel blade-like conducting structures mounted vertically on a planar catode. It can be anticipated tat te spacing distance between emitters and te eigt of te nanowall are two crucial parameters for te total CFE current density. A narrow spacing will reduce te edge electric field due to te strong screening effect, and ten reduce te CFE current of eac nanowall. On te oter and, larger distance between te emitters means less number of nanowalls in a given area of catode. Obviously tere is an optimistic spacing tat gives te largest emission current density. Te aim of tis paper is to find out tis optimistic spacing. Section describes our model and introduces te conformal mapping metod, wic is usually used to find te solution of te Laplace equation wit D boundary condition [-5]. Section 3 is divided into tree subsections. Subsection 3. solves te D Laplace equation and gives te explicit expression of te vacuum electric potential for te forward emission; subsection 3. derives te enancement factor and discusses te screening effects; subsection 3.3 gives te transmission coefficient and te forward emission current density via te semi-classical (JWKB) approximation. Te angle-dependent transmission coefficient is also calculated to support tat te forward emission is dominant. Te main results will be summarized in Section 4.. odel and metod Let us consider an array of nanowalls mounted on a planar catode vertically. Te sketc and cross section of te set up are sown in Fig., were te bottom and te top planes are te catode and anode respectively. Te nanowalls ave te same widt r, te same eigt, and infinite lengt in te z direction. Te spacing distance between two neigboring nanowalls (NWSD) is denoted by d. To simplify te analysis, we assume tat nanowalls are all metallic. Te catode and nanowalls are grounded and teir electric potential is pinned at zero. To make te problem

3 analytically tractable, we assume te anode is remote from te catode ( Xa ), and te macroscopic applied electric field F Vapp Xa as a fixed value wile Vapp, wit V app te voltage applied to te anode. Fig.. Setup for field emission from metallic nanowall array. All nanowalls are vertically mounted on te catode (bottom plane). (a) Tree-dimensional view; (b) projection onto te x-y plane. Electric potential in te vacuum gap (VGEP) between te catode and anode is denoted by V( x, y ), wic satisfies te D Laplace equation and te boundary conditions tat V is zero on te surfaces of catode and nanowalls, and te partial derivative V y is F wen y. Te complex coordinate z x iy will be used to specify points of te xy - plane. Two successive conformal transformations (CTs) are introduced to find te VGEP. As sown in Fig., te CT from (b) to (a) maps a virtual space ( ) into te pysical space ( z ), and te CT from (c) to (b) maps te target space ( ), were te potential is easy to obtain, into te virtual space ( ). Te representative points W, W, W, and W in te pysical space correspond to points,, and on te real axis of te virtual space. Te parameters and will be specified in te following (Eqs. (3) and (4)). We ave required for te latter convenience.

4 Fig.. (a) Te pysical space (z); (b) te virtual space ( ); and (c) te target space ( ). Tese spaces are connected by two conformal transformations G and G. Denote te CT from Fig. (b) to Fig. (a) by z G( ). It is given by Scwarz Cristoffel formula [6], r B(, ),() / / / / G ( ) i ( ) ( ) ( ) d were B( t, t ) is a real function of t and t defined by. () t / / / / B t t t t t t dt (, ) ( ) ( ) ( ) t Tis CT maps te representative points,, and in te virtual space to te points W, W, W, and W in te pysical space, respectively. Te requirements of W i, W r i, W parameters and satisfy te relations r, and W d can be fulfilled if te B(, ) r, (3) B(,) B(, ) d. (4) B(, ) r Equations (3) and (4) specify te parameters and. Te values of B( t, t ) in Eqs. (3) and (4) can be reduced to complete elliptic integrals (Appendix A) tat depend only on te parameters and, B(, ) c( ) (, m) K( m), (5)

5 B(,) c (, m) K( m), (6) B(, ) c( ) ( b, m), (7) were we ave defined b ( ) ( ), m b, m m, and c / ( ). Te second CT G( ) tat maps te target space to te virtual space is G, (8) ( ) sin ( ) were is te complex coordinate of te target space. 3. Results and discussion 3.. Electric potential Te electric potential Vtarget( ) in te target space (Fig. (c)) is a uniform field. Te solution reads, V ( ) F r B(, ) Im( ). (9) target and Denote te inversed function of z G( ) and G( ) as G ( ) G () z, respectively. Ten te electric potential in te pysical space as a formal expression, V( x, y) V ( G ( G ( x iy))). () target Because G () z is very complicated, it is difficult to use Eq. () to calculate V( x, y ) directly even wen te parameters and are known. But as will be sown below, one can make a good estimation for te emission current density via just knowing te potential along te y- axis as te forward emission is dominant. Te function G( ) can be expressed in elliptic integrals at te real axis of te virtual space. Te point z iy on te y- axis of te pysical space for y is corresponding to te point i on te - axis of te virtual space for

6 , wile te latter is corresponding to te point i on te -axis of te target space for. From Eqs. () and (8), one as, ( b, m) y ( d r), () ( bm, ) sin ( ), () were we ave defined b ( ), and arcsin( [ b( )]). Denote te inverse function of u ( b,arcsin t m) as t, () u. Ten Eqs. () and () read bm arcsin b, m b, m b k( y ) b k( y ), (3) were we ave defined k ( ) ( b, m) ( ( d r)). Wit tis and Eq. (9) and (), it follows tat, V, ( ) (, y ) F r bb m k y arcsin B (, ) b k b, m ( y ). (4) Tis is te expression for electric potential along te y- axis. Figure 3 sows te electric potentials along te y- axis wit various d. Te inset sows tat te electric potential is approximately linear wit ( y ) Te d- dependence sows te screening effect. / wen.3 nm y. nm.

7 Fig. 3. Electric potentials versus Z y wit d 3. μm (das-dot-dot), 6. μm (das-dot),. μm (dot), 3. μm (das), and infinite NWSD (solid). Te Z-axis as origin at te center of te edge of a nanowall of 3. μm in eigt and.6 nm in widt. Te applied / field is 3. V μm. Te inset is electric potentials versus Z. Te curves wit d 3. μm and infinite NWSD coincide wit eac oter. 3.. Field enancement factor Note tat, (). Hence te r..s. of Eq.(4) is at y, as te bm boundary conditions required. Using d, ( u) du dt d ( b,arcsin t m) at y, te electric field at te middle of te nanowall top can be obtained from Eq. (4), bm dv (, y) F(, ) F dy. (5) y Tus it is straigt forward to obtain te field enancement factor, at te middle of te nanowall top, resulting F(, ) ( F ),. (6) In te limit of d, Eq. (4) imply. Substitute te infinite limits of B(, ) and B(,) into Eq. (3), (, ) K( ) r. (7) (, ) K( ) Using te identity ( n, n) E( n) ( n), Eq.(7) can be reduced to r Es ( ), (8) E ( ) s were we ave defined and adopted te notation (Appendix A) E s( ) E(arcsin( ) ). Figure 4 sows tat te field enancement factor increases wit te NWSD, and tends to a constant wen NWSD tends to infinity. Actually (8) is te equation (5.) of Ref. [9]. We see, as expected, te field enancement of a single nanowall is recovered in large NWSD.

8 Fig. 4. Field enancement factor versus d wit.5 μm (das-dot), 3. μm (dot), 3.5 μm(sort-das), 4. μm(das-dot-dot), 4.5 μm(das), and 5. μm (solid). Te nanowall widt is.6 nm and eac curve is obtained wit a fixed and various NWSDs Transmission coefficient and current density Te transmission coefficient of te electron in te Fermi level estimated by te JWKB approximation, E F can be D exp g e U( Z) E F dz, (9) were UZ ( ) is te tunneling barrier wit te interval (, ) as te classical forbidden region were U( Z) E F, te potential barrier widt (BW), g 4 ( m ) te JWKB constant for electron, P te Planck constant, and / e e P m e te mass of an electron. Te BW can be calculated by Eq. (). Te angular dependence of BW (Fig. 5) sows tat te forward emission is dominant. Along te forward direction ( y- axis) a simple expression for BW y can be found (Appendix B) b, m y r, () (, m) K( m) were is te work function of te nanowalls, e te elementary positive carge and we ave defined arcsin ( ) ( sin [ B(, ) ( ef r)]).

9 Fig. 5. (a) Potential barrier tickness versus te angle between te emission direction and te y-axis wit d 3. μm (solid) and infinite NWSD (dotted). (b) Logaritmic plot of te transmission coefficient versus. Eac nanowall as widt.6 nm and eigt 3. μm. Te applied electric field is F 3. V μm and te work function is assumed to be 4.3 ev. (Appendix C) From Eqs. (9) and (), te transmission coefficient along te y- axis is D y ge D exp 3/ ef, () were we ave defined t dt D sin [ B(, ) t ( efr )]. () sin [ B(, ) t ( efr )] Te area emission current density at zero temperature is (Appendix C), 3/ 3/ nws nws F g e D J A ( F ) z ( d) exp 3/ 3/4 C e F, (3) were we ave defined z e ( m ), and nws 5/ 3/4 / /4 P e dt, (4) sin [ (, ) ( )] sin [ B(, ) t ( efr )] C t B t efr Figure 6 are typical Jd - curves. Firstly, J increases wit small d as te screening effect gets weaker, and ten J decrease wit d as te area density of nanowall decreases. Te optimal d is larger for iger nanowall. Yet more

10 complicated DFT calculation sould be adopted wen d is close to te tickness of te nanowall, as te interaction between adjacent nanowall becomes important [7]. Fig. 6. Area current density nws J A versus d wit (a). μm, (b) 3. μm, (c) 4. μm, (d) 5. μm. Te applied field F is 5. V μm, and te nanowall widt is.6 nm. Figure 7 sows te eigt-dependent optimal NWSD for different applied electric fields. Te best fitting to te simulation (Fig. 7) gives, d opt.9.8 F.4.55 F. (5) Were d opt and are in unit of μm, F in V μm. Fig. 7. Optimal NWSD d versus wit F 4. V μm (solid), 45. V μm(das), opt 5. V μm(dot), 55. V μm(das-dot), 6. V μm(das-dot-dot). Te nanowall widt is.6 nm.

11 4. Conclusion Cold field electron emission from metallic nanowall array is analytically investigated wit conformal mapping metod. Te field enancement factor and transmission coefficient are obtained. Te optimal nanowall spacing distance for te largest field emission current density is found to be a linear function of nanowall eigt and depends on te applied field. Te expression of by fitting te numerical results. d opt d opt is obtained Appendices A. Elliptic integrals Te incomplete elliptic integrals of te first, second and tird kinds, F( m), E( m) and ( n, m) respectively, are specified in terms of te elliptic parameters n, m ( m ), and te amplitude ( ) by, (A) / F( m) ( msin ) d, (A) / E( m) ( msin ) d / ( n, m) ( nsin ) ( msin ) d. (A3) At, Eqs. (A), (A) and (A3) become corresponding complete elliptic integrals, i.e., K( m) F( m), E( m) E( m) and ( n, m) ( n, m). B. Barrier widt Te potential barrier widt along te emission direction satisfies U( ) E, (B) y F were te potential energy U( Z) W ev (, Z), and W te top of te tunneling barrier. Tus V W E. (B) F (, y) e e

12 Substitution of Eqs. (9) and () into (B) yields B(, ). (B3) e F r From Eqs. (), (), (4), (5) and (7), ( b, m) y r, (B4) (, m) K( m) were, and sin ( ) sin [ B(, ) ( ef r)] arcsin sin [ B(, ) ( efr )]. (B5) Let, ten one obtains Eq. (). C. Transmission coefficient and current density Via JWKB approximation, te forward emission transmission coefficient is D exp y y ge e V (, Z) dz, (C) Let t e V(, Z) and Z B(, ) t ( efr ), ten B(, ) 3/ y e V (, Z) dz t dt e Fr, (C) dz were, similarly to Eq. (B4), Z r ( ) ( b, m) [ (, m) K( m)], and arcsin( [ b( )]), so Z sin ( Z ) Z Z Z Z dz dz r d B(, ) sin [ B(, ) t ( e F r)]. (C3) Z sin [ B(, ) t ( efr )] Wit Eqs. (C), (C) and (C3), one as Eq. (). According to te equation (4.3) of Ref. [9] te area emission current density at zero temperature is J ( F ) z ( d) d exp[ G ], (C4) nws nw 3/ A were nw z e me P and

13 y G ge e V (, Z) dz, (C5) g dz, (C6) y e ev (, Z) d Using Eqs. (C) and (C3), one as G g 3/ e D, ef (C7) g d, (C8) / e C ef were C is defined as Eq. (4). From (C4), (C7) and (C8) one can obtain 3/ 3/ 3/ nws nw e F ge D A ( ) ( ) exp / ge C ef J F z d. (C9) z z e g e ( m ), one obtains Eq. (3). By denoting 3/ nws nw 5/ 3/4 / /4 e P e References [] Novoselov K S, Jiang D, Scedin F, Boot T J, Kotkevic V V, orozov S V and Geim A K 5 Proc. Nat. Acad. Sci. U. St. A. 45 [] Eda G, Unalan H E, Rupesinge N, Amaratunga G A J and Cowalla 8 Appl. Pys. Lett [3] Lee S W, Lee S S and Yang E H 9 Nanoscale Res. Lett. 4 8 [4] alesevic A, Kemps R, Vanulsel A, Cowdury P, Volodin A and Van Haesendonck C 8 J. Appl. Pys [5] Qian, Feng T, Ding H, Lin L, Li H, Cen Y and Sun Z 9 Nanotecnology 457 [6] Spilman Z, Pilosop B, Kalis R, icaelson S and Hoffman A 6 Appl. Pys. Lett [7] Wu Z S, Pei S F, Ren W C, Tang D, Gao L B, Liu B L, Li F, Liu C and Ceng H 9 Adv. ater. 756

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