Theories of field and thermionic electron emissions from carbon nanotubes

Transcription

1 Theories of field and thermionic electron emissions from carbon nanotubes Shi-Dong Liang a State Key Laboratory of Optoelectronic Material and Technology, School of Physics and Engineering, Sun Yat-Sen University, Guangzhou , People s Republic of China Lu Chen State Key Laboratory of Optoelectronic Material and Technology, School of Physics and Engineering, Sun Yat-Sen University, Guangzhou , People s Republic of China and Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, People s Republic of China Received 10 September 2009; accepted 4 January 2010; published 31 March 2010 Taking into account the effect of the low-energy band structure of carbon nanotubes CN, we develop the theories of CN field and thermionic emissions. We give the analytic field and thermionic emission equations for both metal and semiconducting CNs. These theories modify the conventional Fowler Nordheim FN and Murphy Good MG theories. For large-diameter CNs and high fields, the field-emission equation reduces the FN-type field-emission equation. For small-diameter CNs and low fields, the field-emission equation goes beyond the FN-type behavior, which provides a possible way to understand the non-fn behavior observed in experimental results. Based on these theories, we give the electron-emission phase diagram on the field, thermionic, and intermediate emissions in the field-temperature space, whose boundaries have a slight shift to the corresponding boundaries of MG s theory. These differences come from the energy-band structure difference between CN and conventional emitters. This theory provides an understanding of field and thermionic emissions for nanoscale materials American Vacuum Society. DOI: / a Author to whom correspondence should be addressed; electronic mail: stslsd@mail.sysu.edu.cn I. INTRODUCTION The electron emission from metals can be driven by temperature and applied electric field. For high field and low temperature, the high field narrows the vacuum potential barrier such that electrons with energies below the Fermi level tunnel through the barrier and predominate the emission current. The emission current mainly depends on the field. This process is called field emission. On the other hand, for high temperature and low field, the electron emission over the vacuum potential barrier predominates and the temperature dependence of the distribution function is mainly responsible for variations in the emitted current, which is called thermionic emission. The field emission as a cold electron source promises a potential application in a high monochromatic electron beam and new types of displays. 1 Theoretically, Fowler and Nordheim 2 developed the field-emission theory based on the free-electron model with the triangle vacuum potential barrier. The early thermionic emission theory was proposed by Richardson 3 and developed by Schottky. Later, Murphy and Good, Jr. 4 studied the entire electron emission from a unified perspective and gave analytical field and thermionic emission equations as well as their boundaries in the temperature-field space. These theories were long successful for conventional metallic emitters. However, in recent years, the rapid progress of nanotechnology allows us to successfully synthesize many low-dimensional and nanoscale materials that provide us with many new choices for fieldemission materials. 5 In particular, carbon nanotubes CNs exhibit experimentally excellent field-emission features, low threshold field, high current density, and high thermal stability. 1 The reduced brightness of individual multiwall carbon nanotubes is more than ten times sharper than that of the Schottky emitter and the cold field-emission gun. 6 Nevertheless, the current-voltage I-V characteristic deviates from the Fowler Nordheim FN-type behavior at high current density. 1 This phenomenon is usually attributed to the spacecharge effect, 7 field-penetration effect, 8 and energy-bandstructure effects. 9 The thermionic emission from CN also provides some new phenomena. 10,11 These new phenomena allow us to reconsider the validity of the field and thermionic emission theories for nanoscale materials. In principle, the energy-band structure should play an important role in the electronic transport and the field emission for lowdimensional materials. The low-energy band structure of CNs exhibits the Luttinger liquid behavior that has been observed experimentally and attracted much attention. 12 The feature of the CN energy-band structure should also help us to explain the field-emission characteristics of CNs. 9 Based on the low-energy band structure of CNs, we have developed a generalized FN theory of field emission of CNs, in which we give analytically the I-V characteristics for different field and tube parameters, 9 and we provide a physical understanding of the non-fn-type characteristic. Actually, the thermionic emission of CNs has not been understood clearly, even though there has been some computer simulation based on classical theory. 10,11 C2A50 J. Vac. Sci. Technol. B 28 2, Mar/Apr /2010/28 2 /C2A50/8/$ American Vacuum Society C2A50

2 C2A51 S.-D. Liang and L. Chen: Theories of field and thermionic electron emissions from carbon nanotubes C2A51 In this article, we will develop the electron-emission theory of CNs that includes both field and thermionic emissions along our previous concept. 9 Using the similar Murphy Good MG procedure 4 combined with the lowenergy band structure of CN, we will give analytic field and thermionic emission equations and their boundaries that indicate different electron-emission mechanisms in different field-temperature regions. In Sec. II, we will propose the field-emission model of CNs and give a basic formalism of CN field emission. From Secs III V, we give the field and thermionic electron-emission equations and their boundaries, and we then discuss their parameter regions in Sec. VI. Finally, we provide some discussion and conclusions. II. FIELD EMISSION In general, field emission occurs when electrons in the conduction band tunnel through the vacuum potential barrier under the applied field. Two basic physical factors dominate the field-emission behavior. One is the energy-band structure of emitter, including the defect and impurity effects at the tip of emitter. The other factor is the shape of the vacuum potential barrier. The field-emission model developed by Fowler and Nordheim and Murphy and Good, Jr. includes some basic physical assumptions: 1 the energy-dispersion relation is assumed to be a parabola; 2 the vacuum potential barrier is approximated by a triangular potential; 3 the defect and impurity effects at the tip of the emitter are negligible. 2,3 For the conventional bulk emitter, this fieldemission model is quite successful. However, for CNs, the above assumptions should be reconsidered. First, the energydispersion relation of CN is linear, 12 which is a so-called Luttinger liquid or Dirac electrons. 12 Moreover, the energy along the direction of the tube circumference is discrete due to the quantum confinement of electrons along this direction. Second, the shape of the vacuum potential could deviate from the triangular potential. Third, the defect and impurity effects could also play some role in the CN field emission. We believe that the energy-dispersion relation still dominates the field-emission behavior. Therefore, we develop the electron-emission theory based on the linear energydispersion relation of CNs. We still use the triangular potential barrier as an approximation and neglect the defect and impurity effects at the tip of the CN. These effects will be discussed elsewhere. Because the energy-dispersion relation of CN is discrete in the direction of the tube circumference, the emission current is given by 9,14,15 jf,t = q NE q k,tde q k,fdk, BZ where NE q k,t=e/e q k/kfe q k0 is the supply function, fe q k is the Fermi Dirac distribution function, and DE q k,f is the transmission coefficient of emitted electrons through the vacuum potential barrier. In general, the transmission coefficient can be written as 4, DE q k,f = 1 + expqe q k. 2 The derivation of the field-emission equations basically follows the Murphy and Good s approach. 4,13 The key problem is how to analytically integrate the integration of emission current in Eq. 1 with some approximations based on the physical considerations. The conditions of the approximation give the constraint of the emission current equations, which leads to the boundaries of field emissions. For field emission, the emitted electrons are mainly near the Fermi energy. In the typical field-emission region, expqe q k1, 4,13 we have DE q k,fexp QE q k. The main features of the vacuum potential barrier for field emission can be described approximately by a triangular-shape barrier Ux,F= efx, where x=0 is set at the end of CN; is the work function and F is the local electric field. This simple form of the vacuum potential barrier has been valuable to explain some physical properties, 14,16 such as the chiral effect, 14 the quantum size effect, 14 and the Aharonov Bohm phase effect in field emission. 14 Since the image potential of CN emitters is not well defined, we neglect the image potential without losing the qualitative properties of field emission. The QE q k can be expanded near the Fermi energy, 4,13 QE q k b 0 c 0 E q k + f 0 E q 2 k +, where b 0 =4/3 2m/ 2 3/2 /ef, c 0 =2 2m/ 2 1/2 /ef, f 0 = 2m/ 2 1 e 3 F/ 2 1 /2eF 1/2, and we have assumed the Fermi energy at E F =0. Since the emitted electrons come out mainly near the Fermi energy we can neglect the third term in Eq. 3 as long as 4,13 F 4 /m 2 e 5 1/4 1 E q k/e 3 F 1/2 1 and f 0 E 2 q k 1/2 as an approximation. 4,13 Thus the transmission coefficient can be written as DE q k,f exp b 0 + c 0 E q k for E q k 1 for E q k, where E q k is the energy-dispersion relation of CN, which is measured relative to the Fermi level. k and q are quantum numbers labeling the wave vector of electrons along the tube and the transversal modes of the tube. For the single-wall CN SWCN, the energy-dispersion relation can be obtained from that of graphene. In the low-energy approximation, the energy-dispersion relation of graphene can be obtained, 17 Ek= 3atk kf /2, where a is the lattice constant of the hexagon and t is the electron hopping amplitude. The energydispersion relation is linear and radial symmetric around point K. This is a typical characteristic of Dirac electrons. 12 These states near the Fermi level play the main role in field emission. Because of quantum confinement of electrons in the direction of the tube circumference, the energydispersion relation is discrete in this direction. It is noted that JVST B-Microelectronics and Nanometer Structures

3 C2A52 S.-D. Liang and L. Chen: Theories of field and thermionic electron emissions from carbon nanotubes C2A52 one of the discrete lines passes K point in the first Brillouin zone for the metallic SWCNs, and no discrete lines passes K point for the semiconducting SWCNs. 9,16 The electron-state energies at the discrete lines perpendicularly to the Fermi level can be written as 16 q min + q max e c 0 E q 1+e E/k B TdE e c 0 E de e + c 0 1/k B TE de. q 11 q = q E for metallic tubes 7 q /3 E for semiconducting tubes, where q=0,1,2,...,n for metallic tubes and q =1,2,...,N and =1,2 for semiconducting tubes, labeling the parallel lines in the first Brillouin zone, where N is the number of hexagons in the CN unit cell. E= 3at/d is the energy distance between the two parallel lines, 16 where d is the diameter of the tube. Taking the energy-band structure of SWNTs into account and using Eqs. 2 and 3, we can write the emission current as 9 jf,t = e 0 e b q q min + q max e c 0 E 1+e E/k BTdE, 8 We can extend the energy integral region to infinity as an approximation without losing the main physical features. 4,13 For metallic tubes, since q =0 for q=0, the integrand decreases with e Ec 0 1/k B T and e c 0E above below the Fermi energy The main electronic states are in the range 4,13 c 0 1 E k B T1 c 0 k B T 1. Combining Eqs. 4, 5, and 9, we have e 3 F 1/2 4 /m 2 e 2 1/4 e 3 F 1/2 / + k B T/1 c 0 k B T 1 c 0 k B T 2f 0 1/2 k B T. For q0, the integral can be expressed approximately to 9 10 The validity of the approximation can be constrained roughly by c 0 1 E= 3at/d and kb T1 c 0 k B T 1 E= 3at/d. Thus, we obtain the field-emission region for metallic tubes for Eqs. 8 and 11, e 3 F 1/2 4 /m 2 e 2 1/4 e 3 F 1/2 / + k B T/1 c 0 k B T, 12a 1 c 0 k B T 2f 0 1/2 k B T, 12b 3at/d 1/c0 and 3at/d kb T/1 c 0 k B T. 12c Notice that there is no q=0 term for semiconducting tubes, the field-emission boundary becomes e 3 F 1/2 4 /m 2 e 2 1/4 e 3 F 1/2 / + k B T, 1 c 0 k B T 2f 0 1/2 k B T, 13a 13b 3at/d 1/c0 and 3at/d kb T/1 c 0 k B T. 13c Carrying out the integrations in Eqs. 8 and 11 with some mathematical techniques and summing q, we can obtain the field-emission current 9 see Appendix A, jf,t = F 1/2e b3/2 /F c0kbt sinc 0 k B T + 2e 1/2 /Fd 1 e 1/2 /Fd + 2e 1/2 /Fd 1/2 /Fd 1 e for metallic tubes e 1/2 /3Fd + e 21/2 /3Fd e 1/2 /3Fd + e 21/2 /3Fd 14 + for semiconducting tubes, 1 e 1/2 /3Fd 1 e 1/2 /Fd where =e 2 /2 2m=7.558 AeV 1/2 enm, b=4/3e 2m/ 2 =6.83 ev 3/2 Vnm 1, and =at 6m/e =5.988 ev 1/2 e 1 ; =1/c 0 k B T 1. Equation 14 gives a general formula of the CN electron emission for a given temperature, which is valid in the regions of Eqs. 12 and 13 for metallic and semiconducting CNs. In the field-emission region, c 0 k B T1, especially for room temperature, c 0 k B T1, the last terms in Eq. 14 can be negligible, and we can simplify the emission current in Eq. 14 to yield 9 jf = F 1/2e b3/2 /F coth1/2/fd coth 1/2 /3Fd sinh 1/2 /Fd for metallic tubes for semiconducting tubes. 15 J. Vac. Sci. Technol. B, Vol. 28, No. 2, Mar/Apr 2010

4 C2A53 S.-D. Liang and L. Chen: Theories of field and thermionic electron emissions from carbon nanotubes C2A53 FIG. 1.Color online Current-field characteristics of field emission in a for metallic CN and in c for semiconducting CN. The FN plots of metallic CN in b and semiconducting CN in d. The FN theory curves are used for comparison with CN theory. This field-emission equation gives a modified FN I-V characteristic of the CN field emission. For two special cases, large-diameter tubes and high fields 1/2 /Fd1 and smalldiameter tubes and low fields 1/2 /Fd1, the I-V characteristic of the emission current in Eq. 5 reduces 9 to F 2 d j m F = e b3/2 /F 1/2 for Fd 1 F 1/2e b3/2 /F for 1/2 Fd 1 16 for metallic tubes. Similarly, for semiconducting tubes, the emission current becomes j s F = F2 d e b3/2 /F for 1/2 Fd 1 F 1/2e b3/2 /F e 23/2 /3Fd 1/2 for Fd It can be seen that for the case of 1/2 /Fd1, the emission current in Eq. 15 reduces the FN-type I-V characteristic, but for 1/2 /Fd1, the emission current goes beyond the FN type, which is a typical characteristic of Dirac electrons. 9 For semiconducting tubes, the I-V characteristic has an extra factor e 2/3Fd that describes the energy gap at the Fermi level to be inversely proportional to the diameter of the tubes. The field-emission equations in Eqs. 16 and 17 modify the conventional FN field-emission equation, 2 which provide a physical mechanism from the CN energyband structure perspectives to understand the non-fn behavior of CN field emission in experiments. It should be remarked that the field-emission equations in Eqs. 16 and 17 are obtained in the field-emission condition 4 that leads to inequalities 12 and 13. In other words, these inequalities give the field-emission region of metallic and semiconducting CNs in the parameter space, which modify the fieldemission region of the conventional MG theory. 4,13 Interestingly, Forbs 18 proposed a phenomenological fieldemission formula to cover various field-emission behaviors, J=AV e B/V, where A and B are the material parameters, V is the applied electric field, and is a parameter within 1 3. describes different mechanisms. For example, = 1 describes the electron emitted from a single atom and =2 corresponds the FN-type field emission for metallic emitters. 18 The formula in Eq. 15 actually includes different values. This is consistent with the Forbs phenomenological theory. 18 To compare our field-emission equation with FN theory we plot I-V characteristic and FN plot in Fig. 1. It can be seen that the emission currents of the FN theory are larger than that of our theory. However, it should be pointed out that the emission currents of our theory in Eqs are total current, we used Eq. 15 in Fig. 1, while the emission current of the FN formula is the current density. We assume that the current moves only on the layer of the tube to estimate the total emission current of the FN theory. Actually it is not significant to compare directly the quantities of the emission currents of two theories at given fields. The important of our theory is that we give different I-V characteristic from the FN theory, which can be also seen numerically from Figs. 1b and 1d. III. THERMIONIC EMISSION For thermionic emission, the typical field F5 V/nm, temperature excites electrons overcoming the vacuum potential barrier and coming out. The approach to obtain the ther- JVST B-Microelectronics and Nanometer Structures

5 C2A54 S.-D. Liang and L. Chen: Theories of field and thermionic electron emissions from carbon nanotubes C2A54 mionic emission equation basically follows the MG method. 4,13 Because the emitted electron energy is mainly near the value of the work function, the QE q k function in Eq. 2 can be expanded at the work function, QE q kf 4 /m 2 e 5 1/ F4 /m 2 e 5 1/4 2 +, 18 where =1 E q k/e 3 F 1/2. We neglect the second term in Eq. 18 approximately, which is valid within F 4 /m 2 e 5 1/8 19 as an estimation. The transmission coefficient can be expressed approximately as 4,13 DE q k,f1+exp Fh 4 0 /m 2 e 5 1/4 1, where =1 E q K/e 3 F 1/2 and h 0 =F 4 /m 2 e 5 1/4 e 3 F 1/2 /k B T. The Fermi function can be expressed approximately to fe q ke E qk/k B T for the thermionic emission. Consequently the thermionic emission current can be written as jf,t = e q q max min + q e E/k B T 1 + exp F 4 /m 2 e 5 1/4 de. 20 For larger energy the integrand behaves roughly like exp E/k B T and the approximation condition can be estimated by Ek B T. For small energy the integrand decreased as fast as exp 1 h 0 E e 3 F 1/2 /h 0 k B T, and the emitted electron energy is within mainly E h 0 k B T+e 3 F 1/2 /1 h 0, where h 0 =F 4 /m 2 e 5 1/4 e 3 F 1/2 /k B T. 4,13 Combining the above two inequalities we obtain k B T e 3 F 1/2 /1 h Combining Eqs. 19 and 21, we obtain the thermionic emission boundary, h 0/1 h 0 F 4 /m 2 e 5 1/8, 22 k B T e 3 F 1/2 /1 h 0. Notice that the emitted electron with the energy at the top of the vacuum potential barrier can be assumed that emitted electrons in different transverse channels have an equal contribution to emission current. Using the infinite integration technique 4 see Appendix B we can obtain jt,f = en k BT h 0 sinh 0 exp e3 F 1/2. 23 k B T The constraint condition in Eq. 22 implies that h 0 is bounded below one for the convergence of the integral in Eq. 20 and is limited by the melting temperature of the metal. It should be noted that the prefactor in the formula in Eq. 23 is linear to temperature T, which modifies the MG thermionic emission equation. 4,13 For high temperatures h 0 /sinh 0 1, we obtain jt,f = en k BT exp e3 F 1/2. 24 k B T This becomes the Richardson Schottky-like formula. Theoretically, we can plot lnj/t1/t to compare the conventional Richardson Schottky RS formula with the experimental results. However, the difference between the RSlike formula in Eq. 24 and the conventional RS formula originates from different energy-band structures, and the parabola and linear energy-dispersion relations. If the energyband structure of the emitter still plays a key role in thermionic electron emission, we should expect to observe experimentally new thermionic emission behavior in Eq. 22. The thermionic emission boundary, inequality 23, gives the CN thermionic emission regions in the parameter space for our thermionic emission equations, which modifies the MG theory of conventional metals. 4,13 We may compare numerically the thermionic emission current in Eq. 24 with the RS formula in Fig. 2. We can see that the thermionic emission currents we predict are larger than that from the RS formula. However, it is really meaningful that the difference in temperature dependence between our thermionic emission and the RS formula, namely, the temperature factor, is linear for CN emitters instead of nonlinear square of temperature for conventional metallic emitters. Because of the logarithm function of J/T 2, the difference between two behaviors is not easy to be distinguished from the numerical result in Figs. 2b and 2d. IV. ELECTRON EMISSION IN THE INTERMEDIATE REGION In the intermediate region, electron emission is driven by both temperature and field. We can take the temperature effect into account approximately in field emission in this region. Thus, we have no longer c 0 k B T1, we return to Eq. 14 and use the approximation c 0 k B T/sinc 0 k B T1 +c 0 k B T 2 /6 to give the electron-emission equation, jt,f = F 1/2e b3/2 /Fcoth 1/2 + c 0k B T 2 Fd 6 + c 0k B T 2e 1/2 /Fd 25 c 0 k B T 1e E/k B T e /Fd 1/2 for metallic tubes and jt,f = F 1/2e b3/2 /F cosh1/2 /3Fd sinh 1/2 /Fd + c 0k B T c 0 k B T 1 + coshe/6k BT + 1/2 /3Fd sinhe/2k B T + 1/2 26 /Fd for semiconducting tubes. Physically, the electron emission in the intermediate region mixes both field and thermionic emissions. That means that the tunneling electrons are comparative with the thermionic electrons. V. PHASE DIAGRAM OF ELECTRON EMISSION To clarify the electron-emission mechanism in different parameter regions is helpful to understand theoretically the electron-emission behavior. The field and thermionic emis- J. Vac. Sci. Technol. B, Vol. 28, No. 2, Mar/Apr 2010

6 C2A55 S.-D. Liang and L. Chen: Theories of field and thermionic electron emissions from carbon nanotubes C2A55 FIG. 2.Color online Current-temperature characteristics of thermionic emission in a for metallic CN and in c for semiconducting CN. The lnj/t 2 1/T plots of metallic CN in b and semiconducting CN in d. The RS theory curves are used for comparison with CN theory. sion equations that we give in Eqs. 15 and 24 describe the behaviors of the CN field and thermionic emissions, and inequalities 12, 13, and 22 correspond to their parameter regions. To compare the conventional metals with CNs, we choose 10,10 metallic and 20,0 semiconducting CNs as examples and we plot the boundaries of the field and thermionic emissions in the temperature-field space in Fig. 3, in which the black curve is the boundary of field emission based on the MG formula for comparison with our theory. 4,13 The red and green curves represent the boundaries of CN field emission based on Eq. 15. The yellow curve is the boundary of thermionic emission for conventional metals based on the MG theory. 4,13 The pink and blue curves are the boundaries of CN thermionic emission based on Eq. 24. It can be seen that the boundary of the CN field has a slight shift for the conventional field-emission boundary and there is a slight difference between metallic and semiconducting CNs in the right-upper region. The thermionic emission boundary of CNs shifts slightly right to the boundary of conventional metals, but there is no difference between metallic and semiconducting CNs. The field-emission boundary of CNs is independent of the helicity of CN and depends slightly on the diameter of the CNs. The thermionic emission boundary of CNs is independent of the helicity and diameter of CNs. The differences in the boundaries between CN and conventional metals originate from the difference in the energy-band structures between CN and conventional metals. The regions of field and thermionic emissions in Fig. 3 can be regarded as a phase diagram, which gives the fieldtemperature regions for different emission mechanisms and their corresponding emission equations. It should be remarked that exactly speaking, there is no exact physical standard to define the boundaries of the field and thermionic emissions. In practice, these boundaries may be quite blurry. It is nonsignificant to distinguish the field and thermionic emissions in the practical sense. However, from the theoretical perspective, the field and thermionic emission formulas should have a valid region in the space of some physical parameters. The phase diagram provides the meaningful understanding of the mechanisms of the field and thermionic emissions. VI. CONCLUSION In summary, taking into account the effect of the lowenergy band structure of CNs, we develop the theories of CN field and thermionic electron emissions. We give analytic FIG. 3. Color online Electron-emission phase diagram of the field, thermionic and intermediate emissions in the field-temperature space. The black curve is the boundary of field emission based on MG s formula for comparison with our theory. The red and green curves represent the boundaries of CN field emission. The yellow curve is the boundary of thermionic emission for conventional metals based on MG s theory. The pink and blue curves are the boundaries of CN thermionic emission. JVST B-Microelectronics and Nanometer Structures

7 C2A56 S.-D. Liang and L. Chen: Theories of field and thermionic electron emissions from carbon nanotubes C2A56 field and thermionic emission equations of the metal and semiconducting CNs. These theories modify the conventional FN and MG theories and provide the electronemission phase diagram of the field, thermionic, and intermediate emissions in different parameter regions to understand different mechanisms of CN electron emission. The different behaviors of the field and thermionic emissions of CN come from the energy-band structure difference between CN and conventional emitters, namely, a linear energy-dispersion relation of Dirac electrons and parabolic energy-dispersion relation of the Fermi liquid. A typical quasi-one-dimensional material CN belongs to Dirac electrons and shows Luttinger liquid behavior from the basic many-electron-system perspective. 17 These theories give theoretical understanding of the non-fn-type behavior of CN field and thermionic emissions from the energy-band structure perspective for a new class of materials. In practice, the effects of impurities and defects at the tip of CN could play some role in electron emission for low-dimensional and nanoscale materials. Nevertheless, if the energy-band structure of emitters still dominates the key behavior of electron emission, we should expect to observe new electronemission behavior and verify our theory experimentally. Actually, there has been experimental evidence indicating a new field-emission I-V characteristic. 1 Thus, we have a reason to believe that our theory reveals a new physical story on electron emission for low-dimensional materials. ACKNOWLEDGMENTS This work was supported financially by the National Natural Science Foundation of China Grant No , the National Basic Research Program of China 973 Program No. 2007CB935501, and the Advanced Academic Center of Sun Yat-Sen University. APPENDIX A: DERIVATION OF THE INTEGRAL OF THE FIELD-EMISSION CURRENT jf,t = e 0 e b q q min + q max e c 0 E 1+e E/k BTdE. A1 Let the integral be I q = q min + max q e c0e /1+e E/KBT de. Extending the integral range to infinity approximately, q e c 0 E 1+e E/k BTdE. A2 I q + q For q=0, Let x=e E/k BT and extend the range of integration to infinity approximately, 4,13 e c 0 E I q 1+e E/k B TdE = k dx, A3 BT0 1+x where n=c 0 k B T. Making the integral transform x=1 t/t, the integration in Eq. A2 can be solved analytically, I q = k B t T n 1 1 t n n dt = k B T sin n = c 0 1 c 0 k B T sinck B T. A4 For q0, I q qe x n e c 0 E de e +qe c 0 1/k B TE de = c 0 1e c 0 qe + 1 e c 0 qe, A5 where =1/c 0 k B T 1. Notice that q=1,2... e qx =e x / 1 e x, we have e c 0 E I q =2c 0 1 e c 0 E q=1,2,... 1 e c 0 E e c E. A6 0 For semiconducting tubes, there is no q=0 term. Once we replace q q 1/3 and q q 2/3, we get q=1,2,... I q =2c 0 1e c 0 E/3 + e 2c 0 E/3 + e c 0 E/3 + e 2c 0 E/3 1 E e c 0 E 1 e c 0 e c 0 E 1 e c E. 0 A7 Substituting Eqs. A3, A6, and A7 to Eq. A1, we obtain jf,t = F 1/2e b3/2 /F c0kbt sinc 0 k B T + 2e 1/2 /Fd 1 e 1/2 /Fd + 2e 1/2 /Fd 1/2 /Fd 1 e for metallic tubes e 1/2 /3Fd + e 21/2 /3Fd e 1/2 /3Fd + e 21/2 /3Fd A8 + for semiconducting tubes. 1 e 1/2 /3Fd 1 e 1/2 /Fd For room temperature c 0 k B T/sinc 0 k B T1 and is larger enough such that we can ignore the last term in Eq. A8. The field-emission current simplifies to jf = F 1/2e b3/2 /F coth1/2/fd coth 1/2 /3Fd sinh 1/2 /Fd for metallic tubes for semiconducting tubes. A9 J. Vac. Sci. Technol. B, Vol. 28, No. 2, Mar/Apr 2010

8 C2A57 S.-D. Liang and L. Chen: Theories of field and thermionic electron emissions from carbon nanotubes C2A57 APPENDIX B: DERIVATION OF THE INTEGRAL OF THE THERMIONIC EMISSION CURRENT jf,t = e q q min + q e E/k B T 1 + exp F4 m 2 e 5 max de. B1 1/4 For thermionic emission most of the emitted electron lies with the energy at the top of the vacuum potential barrier. Different transverse channels may be equivalent to emission current approximately and the integral range can be extended to infinity. The emission current can be written as approximately jf,t = en e E/k B T de, B2 1 + exp 1/4 F2 m 2 e 5 where N is the number of hexagons in the CN unit cell, namely, the number of transverse modes. Let x=exp/h 0 +E/h 0 k B T, where =e 3 F 1/2 /k B T, integral B2 becomes I = e x h 0 h 0 k B T0 1+x dx = e k B T h 0 sinh 0. Thus, the thermionic emission current can be obtained as jt,f = en k BT h 0 sinh 0 exp e3 F 1/2. k B T 1 J. M. Bonard, H. Kind, T. Stockli, and L.-O. Nilsson, Solid-State Electron. 45, ; J. M. Bonard, J. P. Salvetat, T. Stockli, L. Forre, and A. Chatelain, Appl. Phys. A: Mater. Sci. Process. 69, ; J.M. Bonard, K. A. Dean, B. F. Coll, and C. Klinke, Phys. Rev. Lett. 89, R. H. Fowler and L. W. Nordheim, Proc. R. Soc. London, Ser. A 119, ; R. G. Forbes, J. Vac. Sci. Technol. B 17, O. W. Richardson, The Emission of Electricity from Hot Bodies Longmans, Green, New York, 1921; S. Dushman, Rev. Mod. Phys. 2, E. L. Murphy and R. H. Good, Jr., Phys. Rev. 102, W. A. de Heer, A. Châtelain, and D. Ugarte Science, ;W. A. de Heer, W. Bacsa, A. Chatelain, T. Gerfin, R. Humphrey-Baker, L. Forro, and D. Ugarte, ibid. 268, ; A.G. Rinzler et al., ibid. 269, N. de Jonge, Y. Lamy, K. Schoots, and T. H. Oosterkamp Nature London 420, ; N. de Jonge, J. Appl. Phys. 95, J. P. Barbour, W. W. Dolan, J. K. Trolan, E. E. Martin, and W. P. Dyke, Phys. Rev. 92, X. Zheng, G. H. Chen, Z. B. Li, S. Z. Deng, and N. S. Xu, Phys. Rev. Lett. 92, S. D. Liang and L. Chen, Phys. Rev. Lett. 101, N. Y. Huang et al., Phys. Rev. Lett. 93, S. T. Purcell, P. Vincent, C. Journet, and V. ThienBinh, Phys. Rev. Lett. 88, ; P. Vincent, S. T. Purcell, C. Journet, and V. Thien- Binh, Phys. Rev. B 66, M. Bockrath, D. H. Cobden, A. G. Rinzler, R. E. Smalley, L. Balents, and P. L. McEuen Nature London 397, ; R. Egger, Phys. Rev. Lett. 83, ; L. S. Levitov and A. M. Tsvelik, ibid. 90, A. Modinos, Field, Thermionic, and Secondary Emission: Emission Spectroscopy Plenum, New York, 2002; J. W. Gadzuk and E. W. Plummer, Rev. Mod. Phys. 45, S. D. Liang, N. Y. Huang, S. Z. Deng, and N. S. Xu, Appl. Phys. Lett. 85, ; S. D. Liang, N.Y. Huang, S. Z. Deng, and N. S. Xu, J. Vac. Sci. Technol. B 24, V. Filip, D. Nicolaescu, and O. Okuyama, J. Vac. Sci. Technol. B 19, J. W. Mintmire and C. T. White, Phys. Rev. Lett. 81, ; R. Saito, G. Dresselhaus, and M.S. Dresselhaus, Physical Properties of Carbon Nanotubes Imperial College Press, London, A. Komnik and A. O. Gogolin Phys. Rev. B 66, ; A.O. Gogolin and A. Komnik, Phys. Rev. Lett. 87, R. Forbs, Technical Digest of 22nd International Vacuum and Nanoelectronics Conference, Hamamatsu, Japan, 2009 unpublished, p.13. JVST B-Microelectronics and Nanometer Structures