A simple model of charge transfer by loss-less quantum-mechanical tunneling between two solids is proposed. The model is applicable to electron transport and contact electrification between e.g. a metal and a dielectric solid. Based on a one-dimensional effective-mass Hamiltonian, the tunneling transmission coefficient of electrons through a barrier from one solid to another solid is calculated analytically. The transport rate (current) of electrons is found using the Tsu-Esaki equation and accounting for different Fermi functions of the two solids. We show that the tunneling dynamics is very sensitive to the vacuum potential versus the two solids conduction-band edges and the thickness of the vacuum gap. The relevant time constants for tunneling and contact electrification, relevant for triboelectricity, can vary over several orders of magnitude when the vacuum gap changes by one order of magnitude, say, Å to Å. Coulomb repulsion between electrons on the left and right material surfaces is accounted for in the tunneling dynamics.
Contact electrification has been known since ancient times but the underlying fundamental mechanism is still not known [1–10]. The role of electron transport versus ion transport for the charge transfer has been under dispute [3, 7, 11–14], and only recently did work in the group of Zhong Lin Wang  assert that electron transport is the dominant triboelectricity mechanism for metal-metal, metal-semiconductor, and metal-insulator systems in contact.
Triboelectric charging is important to understand and control. On the one hand, it may have severe negative implications and is known to lead to explosion/fire , damage microelectronic components [17, 18], disturb fluid flow , and be able to increased friction and energy dissipation [20, 21] etc. On the other hand, since energy associated with low-frequency wave motion is abundant in nature as well as due to human activity, there is significant benefit to society in harvesting the energy in an effective way. Triboelectric nanogenerators (TENG’s) are expected to be the primary candidates for energy generation since they can be easily implemented in a vast number of electronic devices. TENGs out-compete electromagnetic generators as the latter are ineffective at harvesting low-frequency ( Hz) [22–26] wave energy from, e.g. ocean, wind, and human motion.
While a classic macroscopic understanding of energy generation due to relative motion of systems of dielectric materials, through the concept of Maxwell’s electric displacement, exists to a certain extent, the physical mechanisms and properties that play a role for the electron transfer processes in nanosystems are much less explored. Electron transfer involving photons (photoelectron emission) in nanoscale contact electrification has been demonstrated experimentally [14, 27–31]. The authors recently presented a mechanism for contact electrification and photon emission in atomic systems  based on Einstein’s two-level system rate equations [33, 34], and in solid systems  using standard theory of electronic structure and optical properties of solids [36–38]. In the present work, we describe a simple loss-less electron transfer mechanism based on the Tsu-Esaki tunneling theory [39, 40] between dissimilar solids and show that the tunneling rate and contact electrification are strongly sensitive to the conduction-band edges of the two materials, the vacuum potential and the vacuum-gap thickness, and the Fermi levels. For simplicity, non-ideal processes such as charge escape from surfaces due to electric conduction of non-vacuum environment, or inside triboelectric materials, which are nonideal insulators, are neglected.
2. Carrier Concentrations and Fermi Levels
Consider a structure (Figure 1) consisting of two materials denoted by subscripts (left) and (right) with conduction-band edges and , respectively. A vacuum gap of thickness exists between them and acts as a barrier (of potential ) for electron transfer. For simplicity, we shall assume an effective mass approximation for electrons and consider the effective mass to be the same, , everywhere in the structure; i.e.,where is the electron energy and
For a general form of the potential , a simple and frequently applied approximation to obtain the tunneling coefficient is to solve the Schrödinger equation (1) using the Wentzel-Kramers-Brillouin (WKB) approximation [41–43]. In the important case of a linear barrier potential, Gundlach  derived a simple expression for the tunneling coefficient. In the following, we present a matrix scheme that can be extended rather easily to any barrier potential function . We give the full details for the most important cases of a flat and a linear barrier potential.
3. Tunneling and Coulomb Effects
The electron number in the left material iswhere prefactor is due to the spin degeneracy, so the number of electrons per area isHere, we have considered the interface to vacuum to be perpendicular to the axis and denote the thickness of the left material . The left material’s total volume is then and is the Fermi functionwhere is the Fermi energy of the left material, is the electron energy contribution in the presence of an electric potential in the left material, is the absolute temperature, and is the Boltzmann constant. Further, parabolic dispersion of the conduction electrons is assumed:This allows us to rewrite the electron number per area in the left material aswhere
Similarly, we can write the electron number per area in the right material aswhere is the electron energy contribution in the presence of an electric potential in the right material, is the Fermi energy of the right materialand the dispersion isHere, it is tacitly assumed the thickness (), effective mass, and temperature to be the same in the two materials. Note that the electric potentials and are different due to charge separation across the vaccum gap (will be shown in the following). Further, we shall assume that the electric potential is constant () in the left (right) material.
3.1. Symmetry of the Transmission Coefficient
Consider the general case with plane waves propagating to the left and right of the barrier
The probability current is defined asthus the current density to the left of the barrier iswhile the current density to the right of the barrier isParticle conservation implies that the number of particles entering the material to the left of the barrier is equal to the number of particles leaving the material to the right of the barrier; i.e.,soorThe latter expression can be written is matrix form aswhere the matrix couples outgoing waves to incoming waves (scattering matrix form). DefiningandEquation (18) impliesi.e.,soSince the barrier potential is real, the system possess time-reversal symmetry, i.e., is a solution if is a solution. The time-reversed solution of Eq. (12) isThe left and right solutions are related by the same matrix, i.e.,andThus, the relationstogether yield the conditionHence, combining with (24),andand symmetry in the transmission coefficient is proved.
4. The Effect of Coulomb Repulsion
As electrons are transferred from the left to the right material, the surface charge on the left material piles up according to the dynamic expressionand the electric displacement obeys the Maxwell-Poisson equationsoIn obtaining this we have tacitly assumed that charges are distributed on the surfaces in close proximity and not in the bulk part of the two materials. This assumption implies, strictly speaking, that the two materials are perfect conductors. If one or both materials are semiconductors or dielectrics, the transfered charge will likely be distributed near (but not necessarily on) the material surfaces and the electric displacement result above will be changed. We will, however, neglect this complication in the following.
Since , where is the electric potential, we have in the vacuum region (),setting . This electric potential is repulsive as it opposes electron transfer between the two materials. The electron energy contribution in the presence of the Coulomb repulsion potential is therefore
The effect of the Coulomb repulsion potential is that the tunneling barrier in the vacuum region shifts from a constant potential to a linearly increasing barrier potential with position . Hence, the electron transfer rate decreases as the surface charge density increases and the net transfer of electrons will stop before the Fermi levels of the two materials become equal assuming a constant temperature everywhere in the system. If the Coulomb repulsion potential is negligibly small, net transfer of electrons first stops when the Fermi levels become equal.
5. Tunneling Transmission Probability
Let us derive expressions for the tunneling transmission probabilities in the absence and presence of Coulomb repulsion.
5.1. Tunneling Transmission Probability - Absence of a Coulomb Repulsion Potential
Continuity of and at and yieldThe transmission probability iswhere and .
5.2. Tunneling Transmission Probability - Presence of a Coulomb Repulsion Potential
The Schrödinger equation in the vacuum region accounting for Coulomb repulsion has the formwhere, with reference to Eq. (36),We note that here, and in the following, the surface charge density is a time-dependent function.
This equation can be recast aswhereSettingallows rewriting Eq. (48) as Airy’s equation:for which the solution iswhere are Airy functions, and and are constants. Hence, with reference to Figure 2, the electron envelope function in the whole domain can be written aswhereLet us consider, as shown in Figure 2, that . Transport left-to-right and right-to-left can then only take place for energies ().
Continuity of and at and yields
The transmission probability is againwhere , . By setting and solving the matrix Eq. (56) for , the transmission probability can be determined.
For the reverse transmission probability, we already know that .
6. Tunneling Current
We can now write down an expression for the tunneling current. Consider electrons in the left material moving to the right. The velocity component perpendicular to the interface with vacuum is so the electron (particle) current iswhere is the tunneling transmission coefficient assumed to only depend on the momentum perpendicular to the interface, is the electron density (plane wave normalized over ), and . Here, we have used that current is given by charge times velocity times density.
There is also a flow from the right material to the left (of opposite sign):where so the net particle flow isThe transmission coefficients obey the conditionAbove we proved that the transmission coefficient is symmetric, i.e.,
The particle current density per area can then be written as
Usingthe particle current density per area can be written asThen, sincewhereit follows thatand Eq. (66) can be written aswhere we introduced as the lower integration limit to reflect that tunneling is only possible above the energy of the highest conduction band edge of the two materials. Similarly, as upper integration limit, the vacuum barrier edge is introduced.
This expression can be further simplified by use of
the integral expressionto getwhere
7. Equation Framework
We are now in a position to formulate a mathematical modeling framework for the transfer of electrons between two distinct solids separated by a vacuum gap. The charge transfer between the two solids is described by the following equations where , and are determined by use of Equations (4), (9), (57), and (71)-(72). Obviously, electron conservation in the whole structure is automatically guaranteed.
7.1. Numerical Scheme
Given initial values and at time , expressions for are determined. At timewhere is chosen sufficiently small to guarantee accuracy and stability in the numerical scheme, new values are found fromfrom which updated values and are determined. The new electron tunneling current is then given by Equations (71)-(72). Further:etc. This concludes the equation framework to determine the temporal evolution of electron tunneling transfer through a vacuum gap.
8. Numerical Test Cases for Electron Transfer by Tunneling
In this section, we will calculate electron transfer dynamics by tunneling through a barrier defined by a vacuum gap. The tunneling efficiency depends strongly on the value of the vacuum potential vs. the conduction band edges of the left and right solids. It also depends strongly on the thickness of the vacuum gap.
8.1. Case 1: Varying the Thickness of the Vacuum Gap and Fixing the Vacuum Potential to 2 eV
We consider the following parameter values: where is the free-electron mass, Å, eV, eV, eV, the inital Fermi levels eV, eV, K, and the vacuum gap thickness is varied between 4 Å and 20 Å.
In Figure 3, we show the temporal variation of the electron number (arbitrary unit) for a set of different vacuum-gap thicknesses. It’s clear that the electron transfer drops strongly with the increase of the vacuum gap thickness. Observe the small time scale. Later, we will demonstrate electron transfer on a much slower time scale for the case of a larger vacuum potential. Note that is a constant as a function of time, i.e., overall charge conservation is guaranteed. If the left and right solids initially are uncharged, the left solid becomes increasingly charged with time according toSimilarly the charge on the right solid is as a function of time . The build-up of surface charges opposes the net transfer of electrons until the net electron transfer until a quasi-equilibrium situation is established whereafter the electron densities in the left and right materials are constants.
8.2. Case 2: Varying the Potential of the Vacuum Barrier and Fixing the Distance between the Materials to 10 Å.
Next, we examine the effect of changing the vacuum potential while fixing the vacuum-gap thickness to 10 Å. The initial Fermi levels are eV and eV. The other parameters are the same as in case 1. Again, we see (Figure 4), that the electron transfer rate drops by several orders of magnitude as the vacuum potential is increased from 3 eV to 7 eV. The initial carrier density is approximately for all vacuum potential values. The upper (lower) plots correspond to cases with and without the influence of Coulomb repulsion, respectively. The effect of Coulomb repulsion is most pronounced for the smallest potential barrier eV. Evidently, the carrier density is higher in the case with Coulomb repulsion as compared to the case without Coulomb repulsion and most noticeably at times above approximately 30 psec.
8.3. Case 3: Varying the Potential of the Vacuum Barrier and Fixing the Distance between the Materials to 20 Å.
We then examine the effect of changing the vacuum potential between 2 eV and 5 eV while fixing the vacuum-gap thickness to 20 Å. The initial Fermi levels are again eV and eV. The other parameters are the same as in case . We see in Figure 5 that the electron transfer rate drops strongly with the increase of the vacuum potential but the time scale is considerably slower than for Cases due to the much larger vacuum-gap thickness. In this case, the final simulation time value is 0.1μsec!
The above case studies reveal the influence of parameters such as distance between contacting materials, the Fermi levels, temperature, and effective masses. In particular, it is shown that the charge transfer rate is critically dependent on the distance (exponential decrease with increasing distance), a high Fermi level of the electron donor material and a low Fermi level of the electron acceptor material. Furthermore, a small effective mass increases the electron mobility and the tunneling transmission probability.
We point to that the general trends and influence of distance and Fermi levels that our model displays are captured by experimental results . Recent theoretical models for contact electrification [15, 45] also compare well to the present more detailed modeling results.
It must be emphasized that the present model describes the first process of contact electrification and not the subsequent operation of a TENG mode. We refer the reader to recent work addressing TENG mode operation and power generation [46–48].
It is important to notice that our model demonstrates that the tunneling time may vary by several orders of magnitude (from picosecs to microsecs or millisecs) depending on model parameters. Therefore, for large separation of contacting materials, tunneling times can be of the same order of magnitude as typical triboelectric time constants.
A simple model is proposed for electron transfer between two solids. The model effectively demonstrates tunneling of electrons through the vacuum barrier and is applicable to both metals, semiconductors, and dielectrics. We use a simple effective-mass Hamiltonian for the electronic wavefunctions to calculate the electron transmission coefficient between two solids separated by a vacuum gap. The Tsu-Esaki equation is then used to compute the dynamics of loss-less electron tunneling and contact electrification. We demonstrate in three case studies that the strength of the tunneling process is very sensitive to the relative positions of the conduction-band edges of the two solids, the vacuum gap and potential, the Fermi levels and temperature of the solids, and the electron effective mass. Coulomb repulsion between electrons on the left and right material surfaces is accounted for in the tunneling dynamics.
Conflicts of Interest
Authors declare no conflicts of interest with regard to the publishing of this paper.
Morten Willatzen acknowledges financial support from a Talent 1000 Program for Foreign Experts, China.
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