Introduction

Electron emission from a solid is an important phenomenon that has wide applications and has played irreplaceable roles in the progresses of electronics and physics. The laws governing the behaviors of electron emission include Richardson's law for thermionic emission and Fowler–Nordheim law for field emission. The two laws were derived on the basis of Sommerfeld free electron model1,2,where the Fermi-Dirac statistics is applied to an atmosphere of three-dimensional (3D) free electrons in metals. So they were originally proposed to describe electron emission from macroscopic bulk metals. In recent years, thermionic electron emission and field electron emission from low-dimensional nanostructures, e.g. carbon nanotubes (CNTs), graphenes, etc., have been widely studied and found to exhibit outstanding performances3,4,5,6,7,8,9. To interpret the measured emission current from low-dimensional nanostructures, the macroscopic Richardson's law and Fowler–Nordheim law were usually directly adopted in previous reports3,4,6,7,8. Therefore, it is important to probe the validity of the macroscopic laws at the nanoscale.

The validity of the macroscopic Richardson's law and Fowler–Nordheim law at the nanoscale is highly controversial up to date. On the one hand, since Richardson's law and Fowler–Nordheim law were derived on the basis of 3D Sommerfeld free electron model as mentioned above1,2, they cannot be applied to electron emission from low-dimensional nanostructures that deviate much from 3D free electron systems. On the other hand, previous experimental results showed quite conflicting conclusions on their validity at the nanoscale. While thermionic emission from the yarns of CNTs was observed to obey Richardson's law very well4,10,11 and field emission from both CNT/graphene arrays and individual CNTs was observed to follow Fowler–Nordheim law in some reports6,7,8,12,13, field emission from CNT arrays and individual single-layer graphenes were also observed to deviate from Fowler–Nordheim law in a few other reports9,14. Therefore, it is highly imperative to end the controversy through well-designed experimental measurements, where emission current density of individual low-dimensional nanostructures and emitter temperature for thermionic emission or local pulling electric field for field emission should be both accurately determined. To date, such measurements have never been achieved due to great challenges. Even though field emission from individual nanostructures were studied in several reports6,7,9,13, local electric field at the sharp emitting apex of nanostructures in those reports were determined not by direct measurements, but by fitting experimental data with Fowler–Nordheim law through introducing a field enhancement factor.

Herein, by using a delicate in-situ multi-probe measurement technique which ingeniously combines three electrical probes and one optical probe installed in a scanning electron microscope (SEM), we succeed in simultaneously determining both the electron emission density (J) and the temperature (T) of individual hot CNTs heated by self-Joule heating. It is found that experimental ln(J/T2)-1/T plots exhibit a upward bending feature obviously deviating from the straight lines in Richardson plots and that the measured electron emission density is more than one order of magnitude higher than that predicted by Richardson's law. We present conclusive experimental evidence that electron emission from individual self-Joule-heated CNTs does not obey classical macroscopic Richardson's law, indicating the breakdown of the macroscopic law of electron emission at the nanoscale.

Results

As schematically shown in Fig. 1a, an individual CNT was firstly suspended between two electrical probes through delicate nanomanipulations. After good electrical contacts between CNT and electrical probes were established (the contact resistance between CNT and electrical probes was measured to be ~4.8 kΩ as shown in Supplementary Fig. S1 online), a bias voltage (Vbias), thus an electrical current (Ibias), was applied to the CNT to heat it up through self-Joule heating and induce electron emission from its side surface. The third electrical probe with a collecting voltage (Vcollect) applied was placed close to the middle of the CNT to collect emitted electrons and emission current (Icollect) was measured. At the same time, thermal light emission from the CNT was collected by an optical probe and thermal emission spectra were recorded. The ingenious combination of three electrical probes and one optical probe enables us to achieve simultaneous measurements of electron emission current and thermal emission spectra from the same individual CNTs. SEM images of a CNT under measurements are shown in Fig. 1b and 1c. After measurements in SEM were completed, the CNTs were transferred to a transmission electron microscope (TEM) for diameter measurement15, which enables us to accurately determine the side surface area of a CNT and calculate emission density from measured emission current.

Figure 1
figure 1

Multi-probe measurement setup.

(a) A schematic drawing of the experimental setup. (b) A SEM image showing the simultaneous measurements of electron emission current and thermal emission spectra from individual Joule-heated CNTs by combining three electrical probes (W tips) and one optical probe (optical fiber tip). (c) An enlarged SEM image of the framed area in (b).

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Figure 2 shows the results of electrical measurements on a CNT with a diameter of 17 nm and a length of 5.7 μm. The Ibias-Vbias plot exhibits a typical feature for a multiwalled CNT with a saturation region at high bias voltage range (Fig. 2a)16. Emission current (Icollect) started to become measurable when Vbias located at the saturation region, namely larger than about 2.8 V for the CNT. We measured Icollect-Vcollect plots when Vbias increased from 2.87 to 3.21 V at intervals of 0.01 V and the plots with Vbias from 2.97 to 3.21 V at 0.04 V intervals are shown in Fig. 2b. It can be seen that, similar to thermionic emission characteristics from bulk metals, each Icollect-Vcollect plot exhibits a fast increasing region followed by a slow increasing one. The local electric field at the surface of a CNT corresponding to the Vcollect we applied was calculated to be less than ~ 108 V m−1, which is much lower than that needed for field emission (~ 109 V m−1)17. So tunneling dominated field emission is excluded here. The slow increasing region in Fig. 2b is attributed to accelerating field regime as a result of the Schottky effect and the fast increasing one is attributed to space charge regime. In accelerating field regime, Icollect is observed to increase exponentially with Vcollect, which is different from the exponential dependence of Icollect on for the Schottky effect in bulk thermionic emitters18. The difference is attributed to the breakdown of classical image potential, which is responsible for the exponential dependence of Icollect on in bulk thermionic emitters18, in the nanoscale curved surface of a CNT. Therefore, a modified formula for the Schottky effect was adopted to describe the accelerating field regime of the Icollect-Vcollect plots, with being the emission current corresponding to zero collecting voltage and β being a fitting parameter.

Figure 2
figure 2

Electrical measurements on a CNT.

(a) Ibias-Vbias plot of a CNT with a diameter of 17 nm and a length of 5.7 μm. The bottom-right inset is a SEM image of the CNT under measurement and the up-left inset is its TEM image after measurements. (b) Icollect-Vcollect plots of the CNT with Vbias ranging from 2.97 to 3.21 V at 0.04 V intervals. (c) Icollect-Vbias and J-Vbias plots of the CNT with Vcollect = 0 and 15 V.

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Emission current corresponding to zero collecting voltage () of the CNT shown in Fig. 2a was obtained by extrapolating the accelerating field regime of its Icollect-Vcollect plots with the above modified formula and is shown in Fig. 2c. It can be seen that, increases for nearly two orders from 24.9 pA to 1.9 nA, corresponding to an emission density from 82 to 6245 A m−2, when Vbias increases only for 0.34 V from 2.87 to 3.21 V. Emission current density (J) was calculated from Icollect through J = Icollect/πdl with l being the length and d being the diameter of the CNTs measured from TEM observation. We also directly measured an Icollect-Vbias plot when Vcollect was set to be 15 V (Fig. 2c). For each Vbias, J at Vcollect = 15 V is about several times larger than that at Vcollect = 0 V, which is attributed to the lowering of surface barrier due to the Schottky effect18. Figure 2c shows that Icollect or J increases approximately exponentially with Vbias for both Vcollect = 0 and 15 V.

To probe the validity of Richardson's law, in addition to the measurement of electron emission density, the temperature of a CNT during electron emission should be accurately determined as well. We determined the temperature of a CNT during electron emission by measuring its thermal emission spectra. When we measured Icollect-Vcollect plot at a Vbias, the thermal emission spectrum corresponding to the bias voltage was simultaneously recorded. Figure 3a shows the thermal emission spectra of the same CNT as that shown in Fig. 2 when Vbias increases from 2.93 to 3.21 V at intervals of 0.04 V. The thermal emission spectra were carefully calibrated by taking into account the quantum efficiency of the optical detector and the transmission coefficient of the optical fiber (see Supplementary Fig. S2 online). Featureless blackbody emission was observed in the wavelength range of 350–750 nm, in good agreement with thermal emission spectra recorded from multiwalled CNT bundles19. It can be seen that spectrum intensity increases with Vbias and a blue shift is observed as Vbias increases, indicating the increase of CNT temperature with Vbias.

Figure 3
figure 3

Thermal emission spectra and temperature of a CNT.

(a) Thermal emission spectra of the same CNT as that shown in Fig. 2a with Vbias ranging from 2.93 to 3.21 V at 0.04 V intervals. The solid lines are the fitting of experimental spectra according to Planck's law. (b) T-Vbias and P-Vbias plots of the CNT.

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Fitting thermal emission spectra of individual CNTs with Planck's law has been demonstrated to be a reliable method for obtaining CNT temperature20. According to Planck's law, we have

where I is spectrum intensity, C is a constant, λ is wavelength, h is Planck constant, c is the speed of light in vacuum, kB is Boltzmann constant and T is temperature. It can be seen from Fig. 3a that, our measured thermal emission spectra from individual multiwalled CNTs can be well described by Planck's law. By fitting thermal emission spectra with equation (1) (solid lines in Fig. 3a), the temperature of the CNT at each Vbias was obtained. Figure 3b shows the obtained temperature of the CNT in Fig. 2a at different Vbias. The temperature increases from 1622 to 2264 K when Vbias increases from 2.70 to 3.21 V. The slope of T-Vbias plot slightly decreases at high bias voltage, indicating CNT temperature increases slightly slower with Vbias there. The increase of the CNT temperature is found to follow the increase of Joule-heating power (P = IbiasVbias) very well (Fig. 3b).

After obtaining the dependences of both J and T on Vbias of the same CNT (Fig. 2c and 3b), we could immediately get the dependence of J on T after correlating J to T corresponding to the same Vbias. Figure 4a shows the J-T plots of the CNT as shown in Fig. 2a. It can be seen that emission density increases approximately exponentially with temperature for both Vcollect = 0 V and 15 V. When T increases for about 500 K from 1739 to 2255 K, J increases for about three orders from 57 to 2.1 × 104 A m−2 when Vcollect = 15 V. The successful determination of the dependence of emission density of a CNT on its temperature enables us to probe the validity of Richardson's law at the nanoscale.

Figure 4
figure 4

Comparison of experimental data with Richardson's law.

(a) J-T plots of the same CNT as that shown in Fig. 2a with Vcollect = 0 and 15 V. (b) ln(J/T2)−1/T plots of three CNTs with J in the unit of A m−2 and T in the unit of K. Solid symbols correspond to Vcollect = 0 V while hollow symbols correspond to Vcollect = 15 V. The length and diameter of CNTs are shown in (b). The dashed lines are Richardson plots with W = 3.9, 4.1 and 4.7 eV.

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Discussion

Richardson's law, which was firstly proposed by O. W. Richardson based on thermodynamics arguments in 191121 and later by R. H. Fowler based on the quantum theory of statistics in 19281, is a classical law for describing thermionic electron emission from Joule-heated macroscopic metallic filaments. According to Richardson's law, thermionic emission density (J) of a hot filament is determined by its temperature and work function (W) through the following formula

where A = (1 − r)A0 with r being the reflection coefficient of electrons incident on emission surface and A0 being a constant with the value of 120.2 A cm−2 K−2. Since r ≪1 is expected22, we have A ~ A0. The formula can be rewritten as . The plot of ln(J/T2) − 1/T is named as Richardson plot, which is a straight line with a constant intercept and a slope solely determined by the work function of a filament. To compare our measured electron emission from individual CNTs with Richardson's law, the data shown in Fig. 4a was replotted in the form of ln(J/T2) versus 1/T (Fig. 4b), together with the experimental data corresponding to Vcollect = 0 V of another two CNTs. The experimental data corresponding to Vcollect = 15 V of the three CNTs is shown in Supplementary Fig. S3 online. It can be seen from Fig. 4b and S3 that measurement results from different CNTs coincide very well, implying good replicability of our measurements.

Interestingly, experimental ln(J/T2) − 1/T plots of individual CNTs are found to exhibit an upward bending feature (Fig. 4b), obviously deviating from the straight lines in Richardson plots. The experimental plots corresponding to Vcollect = 0 V coincides well with the Richardson plot of W = 4.1 eV at high temperature range, but starts to bend upward and deviates from the Richardson plot at lower temperature range. The deviation from Richardson plot is also verified by the experimental plot corresponding to Vcollect = 15 V. It coincides with the Richardson plot of W = 3.9 eV at mediate temperature range but starts to bend upward at both high and low temperature range.

In addition to the upward bending feature in the experimental ln(J/T2) − 1/T plots, our measured emission density is found to be much higher than that predicted by Richardson's law. The work function of the side surface of CNTs was measured to be 4.7–4.9 eV11. If we assume a work function of 4.7 eV for CNT side surface, our measured emission density is more than one order of magnitude higher than that predicted by Richardson's law and the difference between experimental data and theoretical prediction becomes more and more significant as temperature decreases due to the upward bending feature (Fig. 4b). Considering that macroscopic graphite has a similar work function as that of CNTs23,24, our results imply that individual CNTs exhibit much better electron emission performance as compared to its macroscopic allotrope, of which thermionic emission is assumed to follow Richardson's law.

As shown in Fig. 2, our studied CNTs were heated by self-Joule heating and suspended between two relatively large metal probes that act as heat sinks. CNTs heated by such a setup are expected to have a non-uniform temperature profile25, which may maximize in the middle of a tube and thus a non-uniform emission density distribution along the tube axis. Since the tube section with a higher temperature induces a stronger thermal light emission, our collected thermal light emission was mainly from the hottest tube section in the middle. So our determined temperature can be considered as the peak temperature along a CNT, while the determined emission density is the averaged density of the whole tube. If we consider electron emission from the tube section exhibiting a temperature as we determined, then its emission density is underestimated here. Therefore, the actual deviation of the electron emission from self-Joule-heated CNTs from Richardson's law is expected to be more significant than that shown in Fig. 4b.

Five CNTs were studied in our experiments in total (see Supplementary Fig. S3 for the results of another two CNTs). The electron emission performances of all the five CNTs deviate from Richardson's law in the same manner. Therefore, we conclude that Richardson's law fails to describe electron emission from individual self-Joule-heated CNTs. The breakdown of Richardson's law can be attributed to the following reasons. First, Richardson's law is based on a 3D atmosphere of free electrons in metals1, but quasi-one-dimensional (quasi-1D) CNTs obviously deviate from 3D free electron systems. Second, while Richardson's law was derived by applying the Fermi-Dirac statistics to a 3D free electron atmosphere1, the distribution of electrons in an electrically-biased CNT was found to deviate much from Fermi-Dirac distribution, especially at high energy states close to the vacuum level, duo to quasi-1D character of CNTs and their special characteristics of electron-phonon coupling26. The significant effect of bias voltage (or internal electric field) on the distribution of electrons in a CNT makes internal electric field along a CNT, together with thermal effect, co-contribute to electron emission from it17. Third, nanoscale thickness of our studied CNTs makes their electrons be confined in a 10–20 nm wide cylindrical quantum well formed by surface energy barrier along emission direction. Such quantum confinement is absent in macroscopic thermionic emitters and may also make our measured electron emission deviate from Richardson's law.

Our conclusion on the validity of Richardson' law in the electron emission from self-Joule-heated CNTs is in contrary to that in a previous report, where electron emission from self-Joule-heated CNT yarns was found to follow the straight Richardson plot with W ~ 4.6 eV and A ~ 120 A cm−2 K−2 very well in a temperature range similar to that in our measurements10. The contrary conclusions are thought to be caused by the different dimensions of the samples studied in the report and our experiments. While individual CNTs with a diameter of 10–20 nm and a length of several microns were studied in our experiments, 20 μm thick and 8 mm long CNT yarns consisting of a large number of individual CNTs were studied in the report. Such a large size of the CNT yarns is believed to degrade the above discussed characters of CNTs responsible for the breakdown of Richardson's law and makes CNT yarns behave like macroscopic metallic filaments and follow Richardson's law well. Therefore, electron emission performance of individual CNTs, of which emission current is more than one order of magnitude higher than that predicted by Richardson's law as shown above, is seriously degraded in forming CNT yarns.

The breakdown of Richardson's law at the nanoscale implies an urgent need of developing new theoretical descriptions of electron emission from individual low-dimensional nanostructures, where all characters associated with the nanoscale size and low dimensionality of nanostructures, e.g. low-dimensional electron atmosphere, non-equilibrium electron distribution, quantum well confinement, modified image potential, etc., should be well taken into account27. Those characters peculiar to low-dimensional nanostructures are expected to induce new properties of electron emission and potential applications.

In summary, we report on experimental evidence that electron emission density (J) from individual self-Joule-heated CNTs at different temperature (T) does not obey classical Richardson's law of thermionic emission. Experimental ln(J/T2) − 1/T plots exhibit a upward bending feature deviating much from the straight lines in Richardson plots and the measured electron emission density is more than one order of magnitude higher than that predicted by Richardson's law. The breakdown of Richardson's law implies much better electron emission performance of individual CNTs as compared to their macroscopic allotropes and clusters and an urgent need of new theoretical descriptions of electron emission from individual low-dimensional nanostructures.

Methods

Our experiments were performed in situ in a FEI Quanta 600F scanning electron microscope (SEM) with four Kleindiek MM3A nanomanipulators installed in the SEM chamber. The vacuum level in SEM chamber was ~5 × 10−4 Pa. Three of the nanomanipulators were each equipped with a 100–500 nm wide W tip working as an electrical probe and the fourth one was equipped with an 10–30 μm wide optical fiber tip working as an optical probe. Both W tips and optical fiber tip were made by chemically etching and a 40 nm Pd or 10 nm Ag film was coated on the optical fiber tip to avoid electric charging under electron beam irradiation. Under the control of nanomanipulators, each probe can move in three directions with a step size as small as a few nanometers. The three electrical probes were connected to a Keithley 4200 semiconductor characterizing system for electrical measurements and the optical probe was connected to a Horiba Jobin Yvon iHR320 spectrometer equipped with a Synapse CCD detector for optical spectrum recording. Electron beam of SEM was blanked during the measurements, so the effects of electron beam irradiation on the measurements are absent.

Arc-discharge grown multiwalled CNTs were used in our experiments28. They were introduced into SEM chamber through a Pt wire, which was dipped into the carbon deposit grown on the negative carbon electrode of arc-discharge to make some CNTs attached to it. To obtain a CNT for measurements, an individual CNT protruding from the edge of the Pt wire was firstly connected by a W tip (see Supplementary Fig. S4 online), then it was separated from the Pt wire by applying a gradually increasing voltage between the Pt wire and W tip until it broke down. After a CNT adhered to a W tip was obtained, it was manipulated to make contact with the second W tip. The third W tip and an optical fiber tip were then manipulated to approach the CNT to set up the multi-probe measurement configuration as shown in Fig. 1a. TEM images indicate that CNTs used in our experiments have uniform diameter and very good tubular structure without visible defects (see Supplementary Fig. S4). Their near-perfect structure ensures that our measured electron emission was from a homogeneous surface of honeycomb carbon lattice. After measurements in SEM were completed, CNTs were placed onto the carbon film of a TEM grid and were then transferred into a TEM (FEI Tecnai G20) for structure characterization (see Supplementary Fig. S5 online). TEM images indicate that the tubular structure of a CNT was well preserved after measurements (inset of Fig. 2a and Supplementary Fig. S5).

The extrapolation of the accelerating field regime of Icollect-Vcollect plots and the fitting of thermal emission spectra with the Planck's law were performed by using the non-linear least-square fitting function of MATLAB software (lsqcurvefit), with and β, T and C as fitting parameters, respectively. The error of the fitting parameter was evaluated through , where χ2 is the squared 2-norm of the residual of least-square fitting, Cii is the diagonal elements of the matrix C = (FT × F)−1 with F being the Jacobian matrix of the fitting function.