Low field2

Low-field electron emission from nano-carbons: 

Low-field electron emission from nano-carbons Al. A. Zakhidov*, A. N. Obraztsov, A. P. Volkov, D. A. Lyashenko Moscow State University, Physics Department Moscow 119899, Russia http://carbon.phys.msu.ru This work has been supported in part by INTAS grant No. 01-0254. Electron field emission (FE) from surface of conductive materials induced by of strong electric field has great importance for fundamental and applied science. Various carbon related materials show low-field electron emission and are therefore highly attractive as replacement for metals and semiconductors in FE cold cathodes. For example nano-carbon cathodes can be extensively used in various types of cathodoluminescent lamps [1-2]. One of the main advantages of carbon materials for FE applications is in extremely strong interatomic  covalent forces bonding providing highest mechanical strength and chemical inertness appropriate for cathode material operating in vacuum devices under intense electric field in order of 107-108 V/m and in the harsh conditions of residual gas ion bombardment. FE site model for nC materials FE site model for nanostructured graphite-like carbon with curved graphite sheets: bended part of crystallite contains chain of atoms with de-overlaped  bonds, providing local reducing of WF. Energy band diagram Schematic energy band diagram presentation of vacuum-cathode interface without (a) and with (b) external field. The straight lines show simplified two-barrier structure including rectangular barrier between two carbon phases and outer potential barrier. The dashed lines show quantum wells corresponding to separate carbon atoms in the presurface layer with graphite properties and surface cluster with diamond-like properties similar to WBG semiconductor having gap in density of electron states between highest occupied state (HOS) and lowest unoccupied states (LUS). The electron states in the quantum wells are shown by gray and resonance electron states (RES) in HOS-LUS gap are shown by black. The height of rectangular barrier and work function of surface cluster (Evac-EF) is about 4.5 eV. When external field is applied to the cathode the electron states in quantum well corresponding to the surface cluster are shifted down due to voltage drop. Motivation Electron tunneling trough sp2-sp3 heterogeneous surface layer. A general expression for FE current of electrons tunneling through the potential barriers is expressed by formula: where k is a wave vector of electrons, Еz – energy of electrons, F – external electric field near the cathode surface, e – electron charge, h – Plank constant vz - electronic group velocity normal to surface, and T(Ez,F) – transparency of potential barrier There are some uncertainties in the determination of the shape of the first barrier between sp2 and sp3 carbon phases in the cathode surface. For the simplest approximation with a barrier height The Wentzel-Kramers-Brillioun (WKB) approximation may be used for the second barrier on cathode-to-vacuum interface assumption on its simplest triangle shape. The corresponding transmission coefficient has a form of [3] where P2 is tunneling probability, Е0 – energy of electrons in vacuum near the surface. Following to [4] an expression for two-barrier transmission coefficient may be obtained combining formulas for T1(Ez,F)and T2(Ez,F) as: where in the brackets we have phase ratios for every term. Taking into account that thickness of an internal barrier is much smaller than that for outer one, the transmission coefficient T1 should be much larger than T2. It allows us to neglect in formula (4) by items other than second and forth. 3. L.D. Landau, E.M. Lifshic, Quantum mechanics. Nonrelyativistic theory. “Nauka”, Moscow, 1989. 4. E.O. Kane, Theory of Tunneling. J. of Applied Physics, 32 83-91 (1961) Discussion and Conclusions Expression for the phase ratios in the brackets has simple analytical form for a case of two rectangular barriers. And the brackets in second term is equal to zero when Fermi level coincident with energy levels in the quantum well between the potential barriers. These conditions can naturally be called “resonance tunneling”. The main input into transmission coefficient value at the resonance conditions is going from forth item in (4) which contains multiplication of tunneling probabilities for two separate barriers. In this case we may omit other items except forth in the expression for the transmission coefficient. Let us analyze this special resonance case for our two-barrier model. By assuming 4-'4=2n (n= 0, ±1, ±2, ±3….) we obtain for the transparency coefficient which is determined mainly by second term in (4).After some simple simple analysis we have obtained formula for FE current: Results (1) (2) (4) (3) This formula is different from usual Fowler-Nordheim law by second additional term in exponent function. This new term ( 1/2w) may lead to significant increase in the current density.Using for estimation =4.5eV and w =4 Å, as it was proposed above, we will have current density increase on 4 orders for the same fields in comparison with values predicted by FN for metal emitters. Fig on the right shows for the same values of =4.5eV and w =4 Å and for =4.5 eV current-versus-field dependences obtained using usual FN theory and using formula (5) for the two-barrier model. The dependences clearly show reduction of the threshold field and dramatic increase of the current density for the two-barrier emitters with resonance electron states. Current-vs.-field dependencies for electron emission calculated in accord with FN low (dots line) and with respect of two-barrier model using formula (5) (straight line) (5) 1.The presence of the resonance electronic states (RES) in the quantum well corresponding to the WBG layer is an important feature which provides low-field and intensive electron emission. 2.The efficient electron tunneling requires the special energetic position of these RES near Fermi level and slightly above it to explain the existence of nonzero FE thresholds. 3.FE current increases in order of 4 in case of tunneling though RES at the same electric field. 4.FE resonance current exponentially depends on width of WBG layer w. It should be mentioned that the two-barrier mechanisms are well known in the analysis of electron emission phenomena. For example, two-barrier structure of potential barrier on cathode surface may be formed due to presence of additional levels in density of electronic states related to adsorbed gas molecules or structural defects. But in this case emission current intensity and stability are very low due to strong localization of such states in contrast to carbon cathodes where according to our model it is assumed that two-barrier structure is formed on essentially sp2 highly conductive carbon surface due to presence of extended sp3 insulator carbon clusters. These clusters provide delocalized electron states but remain transparent for tunneling electrons in contrast to other two-barrier models of FE, which are proposed for interfaces of two materials with different energy band structure. The last one may be realized, for instance, on interfaces of well conductive materials (metals, graphite) and wide band gap (WBG) semiconductors. But even in the case of very thin WBG material layers their thicknesses and interface region dimensions are much large in comparison to sp3 clusters on sp2 carbon surface and therefore corresponding efficiency of electron emission is much lower than experimentally observed for nano-carbon cathodes. In contrast to formal speculation about resonance tunneling via electronic states in WBG material with unknown structure our model allows self consistent explanation of the mechanism of low-field emission and of the nature of two-barrier structure and the resonance states. An phenomenological model of FE sites and corresponding mechanism of FE has been proposed in our recent publications for nano-carbon materials allowing explanation of the experimental observations mentioned above. This explanation assumes that two kinds of FE may exist in carbon emitters: (i) first type takes place at high electric fields similar to that for usual metallic emitters (FN emission) and (ii) second type takes at low electric fields with threshold values much lower than those predicted by the FN theory. The latest type abnormal low-field emission is specific for some nanostructured carbon materials only. This specificity, in fact, originates from the possibility of carbon atoms being arranged both sp3 (diamond-like) and sp2 (graphite-like) hybridization. These two different carbon phases may be combined with a very narrow interface providing immediately high electrical conductivity and significant reduction of the potential barrier for electrons escaping into vacuum. Our phenomenological model and mechanism of cold emission have universal character since the discussed nanostructured species do exist in various carbon materials. However this explanation of low-field electron emission requires additional theoretical and experimental confirmations. In this work we performed analysis of a probability for electrons to be emitted from nanocarbon cathode surface through heterogeneous sp2-sp3 surface layer. V and width w the corresponding transmission coefficient (and tunneling probability P1) is [3] 1. A.N. Obraztsov, Catholuminescent light source, PCT/RU 02/00175 2. A.N. Obraztsov, A.P. Volkov, Al.A. Zakhidov, D.A. Lyashenko, Yu.V. Petrushenko, O.P. Satanovskaya, Field emission characteristics of nanostructured thin film carbon materials, Appl. Surf. Sci. 215(2003)214-221