High-E-Field NanoScience RECENT IMPROVEMENTS IN FOWLER-NORDHEIM THEORY [A SUMMARY FOR NON-EXPERTS] Richard G Forbes Advanced Technology Institute & Department of Electrical and Electronic Engineering, Faculty of Engineering and Physical Sciences, University of Surrey, Guildford GU2 7XH, UK Permanent e-mail alias: [email protected] SUMMARY This poster provides an overview of attempts made by the present author and collaborators, from about 2006 onwards, to improve "mainstream" Fowler-Nordheim theory and theory relating to the interpretation of Fowler-Nordheim plots. 1. INTRODUCTION 3. (cont.) Fowler-Nordheim (FN) tunnelling is electron tunnelling through an exact or rounded triangular barrier. Deep tunnelling is tunnelling well below the top of the barrier, at a level where simplified tunnelling theory applies. Cold field electron emission (CFE) is a statistical emission regime where most electrons escape by deep tunnelling from states near the emitter Fermi level. Fowler-Nordheim-type (FN-type) equations are a large family of approximate equations used to describe CFE. The original FN-type equation described CFE from the conduction band of a bulk freeelectron metal with a smooth, flat planar surface. Slightly more general situations can be adequately described by slightly more sophisticated FN-type equations. In practice, FN-type equations are also often used to describe CFE from non-metals, particularly when analyzing FN plots (see below), but this is not strictly valid. The theory of how to develop and use FN-type equations to describe CFE and analyze related experimental results is often called Fowler-Nordheim theory. This poster provides an overview of recent (since about 2006) attempts to improve FN theory, made by the author and EMISSION collaborators, attempts to BARRIER put FN theory 2. FIELD THROUGHincluding A SCHOTTKY-NORDHEIM onto an improved scientific basis. The theory given here covers For a planar or quasi-planar field (STFEs) emitter, and the large-area best tunnellingboth single-tip field emitters field barrier (LAFEs) model is the Schottky-Nordheim (SN) barrier. emitters comprising many individual emission sites. A SN barrier with zero-field height equal to the local work function is described by the motive energy MSN(z) = where e eFLz e2/160z , is the elementary positive (1) charge, 0 the electric constant, FL the local barrier field, and z distance measured from the emitter's electrical surface. The reference field FR needed to reduce this barrier to zero is FR = cS2 2 = (40/e3) 2 , (2) where cS is the Schottky constant [1]. The scaled barrier field f is defined by f FL/FR = cS2 FL 2 (1.439965 eV V nm1) FL 2 . (3) This parameter f, introduced in 2006 [2,3], plays an important role in modern FN theory. In the simple-JWKB approach, the probability D of tunnelling through this SN barrier is written physically in the form D exp[FSNb3/2/FL] , (4) where b is the second FN constant [ 6.830890 eV3/2 V nm1] [1], and is the physical barrier form correction factor for this SN barrier. FSN When deriving tunnelling theory for SN barriers, it is found that SN F = v(f) , x(1x)d2W/dx2 Several alternative mathematical definitions of v(x) exist; 3. MATHEMATICS OF THE PRINCIPAL FUNCTION v(x) probably the most convenient is the SN-BARRIER integral definition Improvements understanding 2 v(x) = (3 in 23/2 ) b'a' (a'22over )1/2(the b'2last )1/2 dten , years include (6)the following. where a' and b' are given by (1) In older work a different mathematical argument was used, 1/2 1/2 1/2 1/2 a' = [1+(1-x) ] ; b' = [1(1-x) ] .by y=x1/2 . Using (7) namely the Nordheim parameter y, given x is better, mathematically because the exact series expansion for v(x) (below) contains no half-integral terms in x, physically because the relationship between f and F is linear. (2) A good simple approximation has been found for v(x), namely [2]: (8) This formula has an accuracy of better than 0.33% over the range 0x1, which is better than other approximations of equivalent = (3/16)W . (9) This equation is a special case of the Gauss hypergeometric differential equation; v(x) is the particular solution that meets the boundary conditions: v(0) = 1; lim(x0){dv/dx(3/16)lnx} = (9/8)ln2 . (10) (4) An exact series expansion has 3been found [4] for 2 v(x). The 9 27 2 3 3 3 9 v(x)=1 - ln2+ x - ln2+ x - O(x ) + xlnx + x + O(x ) lowest few 8terms 16are: 16 512 16 256 (11) (5) Numerical expressions have been found [3] that give both v(x) (using 9 terms) and dv/dx (using 11 terms) to an absolute error | 81010 . 4. FORMAL STRUCTURING OF FOWLER-NORDHEIM EQUATION SYSTEM Formal structuring of the system of FN-type equations has been introduced, in order to: (a) describe the relationships between the many different FN-type equations found in the literature; (b) allow high-level formulae relating to FN plots (see below); and (c) clarify issues relating to what parameters can be predicted and/or measured reliably. The general (or "universal") form for a FN-type equation is Y = CYXX2 exp[BX/X] , (12) where X is any CFE independent variable (usually a field or voltage), Y is any CFE dependent variable (usually a current or current density), and BX and CYX are related parameters. Both BX and CYX depend on the choices of X and Y and on barrier form, and CYX also depends on various other physical factors. Both often exhibit weak to moderate functional dependences on the chosen variables. The choices of X and Y determine the form Y(X) of a FN-type equation. The core theoretical form (the form initially derived from theory) is the JL(FL) form, which gives the local emission current density (ECD) JL in terms of the local work function and the local barrier field FL. The characteristic local barrier field FC is the value of FL at some point "C" (in the emitter's electrical surface) that is considered characteristic of the emitter. In modelling, "C" is often taken at the emitter apex (or, in the case of a large-area field emitter, at the apex of the most strongly emitting individual emitter). For a general barrier (GB), the Y(FC)-form FN-type equation can be written formally as the linked equations (13a) and (13b), and then expanded into a Y(X)-form equation via eq. (13c). Y where v(x) is a defined mathematical function, sometimes called the principal SN barrier function, and x is a purely mathematical variable. In the course of the derivation, x is set equal to the SN-barrier modelling parameter f. (cont.) (3) It has been shown [4] that there is a defining equation for v(x), namely (5) v(x) = 1 x + (1/6)xlnx . SN-BARRIER MATHEMATICS JkCGB = cY JkCGB , a1FC2 exp[FGBb3/2/FC] , = (13a) (13b) a1(cXX)2 exp[FGBb3/2/(cXX)] . (13c) Eq. (13a) is an auxiliary equation, wherein cY is the auxiliary parameter linking Y to the characteristic kernel current density JkCGB for the general barrier. JkCGB is defined by eq. (12b), in which a is the first FN constant [1], and FGB correction factor for the general barrier. the barrier-form In eq. (13c), the auxiliary equation FC=cXX has been used to obtain a FN-type equation containing the independent variable X of interest (often a voltage). The merit of these linked forms is that, in suitable emission situations (including orthodox emission situations), for any given choices of barrier form and related barrier-defining parameters (often and FC), the kernel current density JkCGB can be calculated exactly. Thus, in suitable emission situations, High-E-Field NanoScience RECENT IMPROVEMENTS IN FOWLER-NORDHEIM THEORY [A SUMMARY FOR NON-EXPERTS]: 5. THE SERIES-RESISTANCE PROBLEM Figure 1. When resistance is present in the measuring circuit, then careful distinction is needed between emission variables and measured variables. This need has grown increasingly apparent in recent years. Figure 1 is a schematic circuit diagram for CFE measurements. Although the parallel resistance Rp can usually be eliminated by careful system design, series resistance often cannot. In this case, the measured voltage Vm is not equal to the emission voltage Ve, defined by = Ve/Vm and = a (current-dependent) Re/(Re+Rs) , voltage ratio can be (14) The im(Vm)-form FN-type equation (giving measured current as a function of measured voltage, for a general barrier) thus becomes im A a (C Vm) exp[ b C/Vm] , (15) 6. AUXILIARY PARAMETERS & EQNS FOR INDEPENDENT VARIABLES where C [ Ve/FC] is the characteristic local conversion length Table 1that shows the "independent" used in (LCL) relates FC to Ve (seevariables below). (X) The currently presence of is FN a theory, and the main related auxiliary parameters and equations. recently introduced feature. Some points arising are: 1 (1)It is convenient to "theoretical variables", variables". 2 GB F 3/2 classify independent "emission variables", variables as or "measured (2)To avoid present ambiguities over the meaning of the symbol it is suggested that use of local conversion lengths (LCLs) (C) should replace use of voltage-to-barrier-field conversion factors (VCFs). (3)Usually, theory is clearer if the voltage ratio is shown explicitly. (4)Auxiliary equations of the form FC=cXX (shown in blue in Table 1) have a special role in FN theory, as indicated earlier. 7. AUXILIARY PARAMETERS AND EQNS FOR DEPENDENT VARIABLES (5)For LAFEs, field enhancement factors (FEFs) can be derived from In this the superscript is omitted for macroscopic notational LCLs by section, the formula C = M /C ,"GB" where M is the simplicity; however, the equations apply to a barrier of any conversion length . specific form. Characteristic local current density JkC ECD JC relates to characteristic kernel JC = CJkC , by (16) where C is the characteristic local pre-exponential correction factor. C takes formal account of factors not considered elsewhere, including improved tunnelling theory, temperature, the use of atomic-level wave-functions, and electronic bandstructure. For the SN barrier, our current best guess (in 2015) is that C lies in the range 0.005

= JLdA A nJ C = AnCJkC AfJkC , (17) where the notional emission area An and formal emission area Af [ie/JkC] are defined via eq. (17). Both parameters are needed because, for orthodox emission, it ought to be possible to extract reasonably accurate values of Af from experiment; but An (which cannot be extracted accurately) appears in some existing theory (e.g. [6]) and might in principle be closer to geometrical area estimates. TABLE 2.JMComplexity levels of For8.LAFEs, the macroscopic current density is the average ECD EQUATION COMPLEXITY LEVELS FN-type equations taken over the whole LAFE macroscopic area (or "footprint") AMGB , GB Level name Date Barrier C F In the literature, many choices and can be written have been made for the barrier Elementary 1999? 1 ET 1 GB )J Jform ie/AM hence = (An/A for nJC Original = nCJkC 1928 fJPkC , ET (18)1 (and and M M C FN F ), Fowler-1936 1936 4 ET 1 what physical effects to where the notional area efficiency Extended n and formal 2015 area CET efficiency ET 1 elementary include when modelling the f [ JM/JkC] are defined via eq. (18) . Dyke-Dolan 1956 1 SN vF GB parameter C defined above. Murphy-Good 1956 tF2 SN vF These choices determine the Orthodox 2013 CSN* SN vF New-standard 2015 CSN SN vF complexity level of FN-type "Barrier2013 CGB* GB FGB equations. changes-only" For bulk emitters with planar surfaces, Table 2 shows the main complexity levels historically and currently used. A given complexity level applies to all related Y(X)forms of equation at the given Schematic circuit for measurement of field emission current-voltage characteristics, showing resistances in parallel and in series with the emission resistance Re [=Ve/ie]. Due to the resistance in series with the emission resistance, the emission voltage Ve is less than the measured TABLE 1:Vm . Independent variables, and main related auxiliary parameters and equations voltage Independent variable name and symbol links via auxiliary parameter to name and symbol (symbol ) Formulae Theoretical variables where Re is the emission resistance [ Ve/ie], and Rs [ Rs1+Rs2] is the total series resistance. GB f Sheet 2 General 1999 C GB GB GB F For citations behind dates of introduction, see [7]. ET=Exactly triangular; SN=Schottky-Nordheim; GB= General barrier. vF & tF are SN-barrier functions. PFN is FN pre-exponential (see [1]). * denotes that parameter is to be treated as constant. Characteristic local barrier field FC - - - - Scaled barrier field f FC Reference field FR FC = f F R Emission voltage Ve FC (True) local voltage-to-barrier- V,C field conversion factor (VCF) a FC = V,CVe Emission voltage Ve FC (True) local conversion length (LCL) C FC = V e / C True macroscopic field FM Ve (True) macroscopic conversion lengthb M F M = V e / M True macroscopic field FM FC (True) (electrostatic) macroscopic field enhancement factor (FEF) C F C = C F M Emission variables C = M / C Measured variables Measured voltage Vm Ve Voltage ratio Ve = Vm Measured voltage Vm FC Measured-voltage-defined LCL C/) FC = C1Vm Apparent macroscopic field FA Vm Macroscopic conversion length M F A = V m/ M Apparent macroscopic field FA FM Voltage ratio FM = FA Apparent macroscopic 9. field FA FC Apparent-field-defined FEFc Cafd FC = CafdFA = C FC = CFA INTERPRETATION OF FOWLER-NORDHEIM PLOTS When in form eq. (12) said to be written in FN Future re-written use of the parameter V,C (19), is discouraged: use is C and related formulae instead. bcoordinates, and thegeometry, slope SM is of normally the resulting can plate be In planar-parallel-plate taken as FN equalplot to the separation sep. form (20): written din a c Use of the parameter Cafd is discouraged: use the combination C instead. L(X1) ln{Y/X2} = S ln{CYX} BX/X dL/d(X1) = ln{CYX} FGBb3/2/cXX, b3/2/cX , (19) (20) where is a slope correction factor (taken as 1 in elementary FN plot analysis), defined by eq. predicted, i.e., if dL/d(X1) measurement of S allows extraction characterization parameters, such (FEF). (20). If can be reliably can be reliably evaluated, of cX-values and related emitter as a field enhancement factor The presence of in eq. (15) and related equations massively complicates practical FN plot analysis. When 1, then can be significantly less than 1 but cannot (at present) be reliably predicted. More generally, the usual methods of FN-plot analysis work correctly only if the emission situation is orthodox [5] (which requires that =1, that cX, cY and be constant, and that emission can be treated as taking place through a SN barrier). Many real emitters are not orthodox; thus, it is likely [5] that many published FEF-values are spuriously large. Steps taken to deal with this problem and develop FN-plot theory include the following. (1) A careful definition of emission orthodoxy has been given [5]. (2) A robust test for emission orthodoxy has been given, which can be applied to any form of FN plot [5]. (3) For emitters that fail the orthodoxy test, a process called phenomenological adjustment has been developed that allows cX to be roughly estimated [7]. (4) In SN-barrier theory, new forms of intercept correction factor (r2012) [7] and area extraction parameter (SN) [7] have been defined. (5) For emitters that pass the orthodoxy test, better methods of extracting formal emission area Af and formal area efficiency f have been developed [8]. (6) Attempts have been made [7] to determine by simulating the circuit in Fig. 1, using a constant total series resistance, but have proved unsuccessful. (7) Using the "barrier effects only" approximation [7], values REFERENCES have for Athe generalized intercept 1. R.G. been Forbes &determined J.H. B. Deane,for Proc.R. and Soc. Lond. 467 (2011) 2927. See electronic supplementaryfactor materials details constants. relating to spherical and correction for , for theof barriers 2. R.G. Forbes, Appl. Phys. Lett. 89 (2006) 113122 sphere-on-cone (SOC) Deane, emitter 3. R.G. Forbes and J.H.B. Proc.models. R. Soc. A. 463 (2007) 2907. 4. J.H.B. Deane & R.G. Forbes, J. Phys. A: Math. Theor. 41 (2008) 395301. Unfortunately, has Proc. recently [9] that, for non5. R.G. it Forbes, R. Soc. been Lond. Aconcluded 469 (2013) 20130271 6. R.G. Forbes, J. Vac. Sci. B 27 methods (2009) 1200. of planar emitters, theTechnol. usual finding tunnelling 7. R.G. Forbes, J.H.B. Deane, A. Fischer and M.S. Mousa, "Fowler-Nordheim Plot probabilities may not be strictly valid quantum mechanics, and Analysis: a Progress Report", Jordan J. Phys. (in press, 2015), arXiv:1504.06134v4. that the motive energy may July sometimes be 8. R.G. transformation Forbes, 28th Intern. of Vacuum Nanoelectronics Conf., M(z) Guangzhou, 2015. 9. R.G. Forbes,This arXiv:1412.1821v4. necessary. is an active topic of research. Work continues on these and other attempts to improve mainstream FN theory.