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Model of electric charge distribution in the trap of a close-contact TENG system

  • SeongMin Kim EMAIL logo
Published/Copyright: January 31, 2020
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Open Physics
From the journal Open Physics Volume 18 Issue 1

Abstract

Electron propagation in a trapped state between an insulator and a metal during very close contact in a triboelectric nanogenerator (TENG) system was considered in this study. A single energy level (E0) was assumed for the trap and wave function inside the trap, which is related to the ground state energy. The phase of the waveform in the metal (neglecting the rebound effect at the wall) was assumed very small (δ′ ≪ 1) because of the large size of the metal. The contact distance between the trap and metal is very small, which allows us to ignore the vacuum potential. Based on our results, the probability of finding an electron inside the trap as a function of time was found to be in oscillation (i.e., back-and-forth propagation of the electron between the trap and metal leads to an equilibrium state). These results can be used to understand the quantum mechanisms of continuous contact, particularly in sliding-mode TENG systems.

Keywords: wave function; trap; oscillation; triboelectric nanogenerator

1 Introduction

The charge transfer mechanism [1, 2, 3, 4] is typically described by two factors: (i) thermionic emission, i.e., temperature-induced electron flow [5, 6] in which penetrating charges take longer to reach thermodynamic equilibrium; (ii) tunneling between the metal and trap sites in the insulator [7, 8, 9]. However, in a triboelectric nanogenerator (TENG) system, an external circuit is connected to the triboelectric part, where charge transfer occurs at the interface between the metal and the insulator, and an additional charge transfer to the bottom substrate through the circuit occurs [10]. The sliding TENG system uses a continuous frictional mode, which is different from the contact-separation mode. Charge transfer can occur between the two triboelectric materials when very close contact is maintained. In the contact-separation mode, an electric field is formed between the two surfaces upon separation, leading to the formation of a vacuum potential through which electrons can tunnel [11]. During contact, the driving force for a charge transfer can be described based on the work function difference between the materials. However, this study describes quantum-mechanical electron propagation from a band-diagram point of view, neglecting the vacuum potential between the two materials. The close-contact model used in this work does not use a potential barrier through which electrons can tunnel owing to the charge neutrality on the contact interface (i.e., zero sum charge). In this model, the probability of an electron transfer has been described by a quantum wave function. The surface state of the insulator, wherein the transferred electrons can accumulate, has been described by a localized state and a trap, which was approximated by a square potential well with a single-electron energy (E0). As the charge transfer between the metal and the trap can occur at the same energy level, a metal-to-trap and a trap-to-metal transfer have the same probability. To simplify the calculation, the initial wave function was described by the trapped electron, and the electron transfer to the empty metal was subsequently calculated. However, the converse probability process could also be applied to the model system. Mathematically, the one-dimensional time-dependent wave function was calculated by an overlap integral. The result obtained by this model showed that the charge transfer was more similar to a direct propagation, resulting in oscillation, rather than tunneling through the potential barrier of the very-close-contact environment.

2 Results: Calculation of the time-dependent wave function in a close-contact TENG system

2.1 Wave functions of the system consisting of trap of an insulator and a metal

The wave function u of the potential of the system shown in Figure 1 (a) was considered as follows:

Figure 1 (a) Electron potential energy along the x-axis from the center of the trap potential (i.e., a square potential with a depth of V and a width of 2a) in the insulator. The potential is assumed to be closely defined by the trap and the metal (a contact line is shown). The shape of the potential energy wave functions is shown. (b) The isolated potential energy of an electron in the trap is shown as a function of position x.
Figure 1

(a) Electron potential energy along the x-axis from the center of the trap potential (i.e., a square potential with a depth of V and a width of 2a) in the insulator. The potential is assumed to be closely defined by the trap and the metal (a contact line is shown). The shape of the potential energy wave functions is shown. (b) The isolated potential energy of an electron in the trap is shown as a function of position x.

(1) u = D ⋅ e μ x + a x ≤ − a , A ⋅ cos k x ( x ≤ a ) , C ⋅ sin ⁡ k x − a + δ ′ x ≥ a ,

where the wave function inside the trap was assumed an even function based on the ground state energy. Here, δ′ ≪ 1 because of the large size of the metal (L ≫ 1) (i.e., the rebound effect was neglected). E was assumed a single energy of a specific wave function, which resulted in k = 2 m E / ℏ a n d μ = 2 m ( V − E ) / ℏ , where V is the depth of the potential well. The coefficients D, A, C, and δ′ were determined from the boundary conditions, whereby u and ∂ u ∂ x are continuous at x = ±a.

To calculate the overlap integral, an initial wave function ψ(0)inside an isolated trap was first considered [5], as shown in Figure 1 (b), where the trap is far from the metal.

(2) ψ ( 0 ) = F ⋅ cos k 0 x x ≤ a , G e − μ 0 | x − a | x ≥ a ,

where

(3) F = μ 0 1 + μ 0 a , G = k 0 ⋅ F k 0 2 + μ 0 2 , k 0 = 2 m E 0 ℏ , μ 0 = 2 m ( V − E 0 ) ℏ .

Applying the boundary conditions for the electron wave function inside the trap that is very close to the metal, resulted in the following relations:

(4) D = A cos a k , μ D = k ⋅ A sin a k , A cos ⁡ k a = C ⋅ sin ⁡ k δ ′ , − k ⋅ A sin ⁡ k a = k ⋅ C cos ⁡ k δ ′ .

Assuming δ′ ≪ 1 in Eq. (4) gave the following relations:

(5) μ = k ⋅ tan a k , δ ′ = − 1 k ⋅ tan k a = − 1 μ ,
A C = − 1 sin k a = − 1 sin 2 m E ℏ ⋅ a

when C was set to 1,

(6) A = − 1 sin 2 m E ℏ ⋅ a .

2.2 Overlap integral a(E)

An overlap integral was defined as follows:

(7) a ( E ) = ∫ − ∞ ∞ ψ ( 0 ) ⋅ u ∗ E d x = ∫ − ∞ ∞ F cos ⁡ k 0 x ⋅ A ⋅ cos k x d x = F ⋅ A ∫ − ∞ ∞ cos 2 k 0 x d x = F ⋅ A E a + sin 2 k 0 ⋅ a 2 k 0 ,

where the resonance condition at k ∽ k0 was used.

2.3 Time-dependent wave functions ψ(x, t)

To calculate the time-dependent wave function inside the trap, the following formula was used:

(8) ψ ( x , t ) = ∫ − ∞ ∞ a ( E ) ⋅ u E e − i E t / ℏ ⋅ N E d E = ∫ − ∞ ∞ N F ⋅ A ( E ) ⋅ a + sin 2 k 0 a 2 k 0 ⋅ A ( E ) cos ⁡ k x ⋅ e − i E t / ℏ ⋅ d E = N F ⋅ a + sin 2 k 0 a 2 k 0 ⋅ ∫ − ∞ ∞ A 2 E ⋅ cos ⁡ k x ⋅ e − i E t ℏ d E .

Considering the resonance to occur at k ≈ k0, then

(9) ψ x , t ≅ N F ⋅ a + sin 2 k 0 a 2 k 0 cos ⁡ k 0 x ⋅ ∫ − ∞ ∞ e − i E t ℏ sin 2 2 m E ℏ ⋅ a d E .

When E was changed to −E in the integral, then

(10) ψ x , t = N F ⋅ a + sin 2 k 0 a 2 k 0 cos ⁡ k 0 x
⋅ ∫ − ∞ ∞ e i E t / ℏ sin 2 2 m E ℏ ⋅ a i d E = N F ⋅ a + sin 2 k 0 a 2 k 0 cos ⁡ k 0 x ⋅ ∫ − ∞ ∞ e i E t ℏ − sin ⁡ h 2 2 m E ℏ ⋅ a d E

As a ≪ 1, the denominator in the integral could be expanded by a power series, such that

(11) s i n h 2 2 m E ℏ ⋅ a ≈ 2 E m a 2 ℏ 2 + 4 E 2 ⋅ m 2 ⋅ a 4 3 ℏ 4 + ⋯

The integral in Eq. (10) could then be rewritten as

(12) ∫ − ∞ ∞ − e i E t / ℏ 2 E m a 2 ℏ 2 + 4 E 2 ⋅ m 2 ⋅ a 4 3 ℏ 4 d E .

This integral could be solved using a contour integral and the residue theorem with two branch poles.

(13) Poles = 0 − 3 ℏ 2 2 a 2 ⋅ m , Residues = − ℏ 2 2 a 2 ⋅ m ℏ 2 e − 3 i t ℏ / 2 a 2 m 2 a 2 ⋅ m .

The value of the integral in Eq. (10) was then obtained as

(14) π i − ℏ 2 2 a 2 ⋅ m + ℏ 2 e − 3 i t ℏ / 2 a 2 m 2 a 2 ⋅ m = i − 1 + e − 3 i t ℏ / 2 a 2 m π ℏ 2 2 a 2 ⋅ m .

Therefore, ψ(x, t) could be restated under the condition a ≪ 1, as

(15) ψ ( x , t ) ≅ N μ o ⋅ a cos ⁡ k 0 x ⋅ i e − 3 i t ℏ 2 a 2 m − 1 π ℏ 2 2 a 2 ⋅ m .

The probability of finding an electron inside the trap was then obtained as

(16) ψ x , t 2 ≅ N 2 ⋅ μ 0 ⋅ a 2 ⋅ cos 2 k 0 x ⋅ π 2 ℏ 4 sin 2 3 t ℏ 4 a 2 ⋅ m a 4 ⋅ m 2 ,

where N is the density of states in the metal and is given by mL/πħ2k0 [5].

Substituting N, μ0, and k0 into Eq. (16) led to the following relation:

(17) ψ x , t 2 ≅ L 2 m ( V − E 0 ) ⋅ ℏ ⋅ cos 2 2 m E 0 ℏ ⋅ x ⋅ sin 2 3 t ℏ 4 a 2 ⋅ m 16 ⋅ 2 ⋅ a 2 ⋅ E 0 ⋅ m ⋅ π 4 .

Selecting the time and space information from Eq. (17), we obtained

(18) ψ x , t 2 ∼ cos 2 2 m E 0 ℏ ⋅ x ⋅ sin 2 3 t ℏ 4 a 2 ⋅ m

where E0 is the initial single-electron energy inside the isolated trap.

Therefore, the triboelectric charge density formed on the interface could be described as

(19) σ = ψ x , t 2 ⋅ n ⋅ e ,

where e is the elementary charge and n is the surface density of the traps contained on the surface of the insulator (in units of #/m2).

Furthermore, V- E0 was the work function of the insulator øI0 , and Eq. (17) could be restated as

(20) ψ x , t 2 ≅ L 2 m ∅ I 0 ⋅ ℏ ⋅ cos 2 2 m E 0 ℏ ⋅ x ⋅ sin 2 3 t ℏ 4 a 2 ⋅ m 16 ⋅ 2 ⋅ a 2 ⋅ E 0 ⋅ m ⋅ π 4 ∼ ∅ I 0

3 Discussion

In a close-contact model, the trap present on the surface of an insulator was assumed to be defined by the surface density of states, which was determined by the quantum square well potential. The electronic potential energy as a function of position for this close-contact model is shown in Figure 1 (a). As the distance between the trap and the metal is approximately zero, the image potential on the metal side was not considered, and there was no potential barrier (i.e., there is no decay of the wave function). However, the potential changes discontinuously as the distance approaches zero (i.e., close contact). A calculation of the wave function considering the distance between the insulator and metal was provided by Lowell [5]. Here, the wave functions were calculated for a complete system consisting of a metal and trap on an insulator. Subsequently, the time dependent method was used to calculate the time-evolution wave functions [5, 12]. The probability of a charge transfer can be applied equally to both cases, i.e., metal-to-trap and trap-to-metal. The initial electron wave function is described by the electron trapped inside the isolated quantum well,which propagates towards the metal with time. It is assumed that the Fermi energy of the trap is greater than that of the metal. The wave function inside the trap is not influenced by the metal, which is located far from it. This structure can lead to the formation of a localized wave function with ground state energy (i.e., an even function), as shown in Figure 1 (b).

When the metal is approaching or is in contact with the trap, electron tunneling through the barrier is not considered because of the instantaneous nature of the contact. During contact, the wave function inside the trap is assumed identical to that of the isolated trap. This resonance condition was used to calculate the overlap integral a(E), which is maximized when the wave function inside the trap is approximately equal to that of an isolated trap (i.e., E ≈ E0). Otherwise, for very small values of a(E), the filled electron state of the trap gradually propagates into the empty metal. Figure 2 shows the time component of the probability of finding an electron, i.e., sin2 3 t ℏ 4 a 2 ⋅ m . This figure implies that the electron probability inside the trap oscillates. Conversely, the electron probability in the metal is given by 1 − |ψ(x, t)| 2, which is also an oscillation pattern. In other words, the electron charge density as a function of time alternates between the metal and the trap. Figure 3 shows the distribution of both time and spatial components inside the trap, i.e., cos 2 2 m E 0 ℏ ⋅ x ⋅ sin 2 3 t ℏ 4 a 2 ⋅ m (the time component shows the oscillation). The results obtained for the close-contact-mode TENG system showed the oscillation of electron propagation, which is driven by the work function difference between the trap and the metal. It is evident from Eq. (20) that the electron charge density inside the trap is proportional to the square root of the work function of the insulator ( ∅ I 0 ) , as shown in Figure 4.

Figure 2 Plot of the time component of |ψ(x, t)| 2
Figure 2

Plot of the time component of |ψ(x, t)| 2

Figure 3 Three-dimensional plot of both time and spatial components of |ψ(x, t)| 2
Figure 3

Three-dimensional plot of both time and spatial components of |ψ(x, t)| 2

Figure 4 Plot of |ψ(x, t)| 2 as a function of ØI0
Figure 4

Plot of |ψ(x, t)| 2 as a function of ØI0

4 Conclusion

The charge transfer between the trap in the insulator and the metal in a close-contact-mode TENG system was described using quantum mechanics. Our physical system was described by close contact model between the metal and the trap in the insulator as in, for example, sliding-mode triboelectric nanogenerator where contact and separation were not repeated, but close contact was maintained. Once the contact and separation occur, there exists a vacuum region between the metal and the insulator that can form a potential barrier [11]. After electron transfer, due to the accumulated charges the electric field can be formed which can lead to the formation of sloped potential barrier. In this model system, the vacuum potential barrier was not considered due to the absence of the vacuum where electron can transfer or tunnel through it. Instead, in this region electron can freely transfer back and forth that was modeled as non-tunneling region. Furthermore, it was assumed that the trap was located close to the surface that was directly connected with the surface of the metal for simple calculation. The trap potential was assumed a square potential. The time-dependent wave function,which is related to the probability of finding electrons (or the electron density) inside the trap, was found to be in oscillation. This implies that, with the metal in close contact, the charge density inside the trap flows back and forth. The results were analytically investigated.

  1. Conflict of Interest

    Conflict of Interests: The authors declare that there are no conflicts of interest.

Acknowledgement

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2017R1A2B4010642)

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Received: 2019-02-18
Accepted: 2020-01-07
Published Online: 2020-01-31

© 2020 S. Kim, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2020-0001
Keywords for this article
wave function; trap; oscillation; triboelectric nanogenerator
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  1. Regular Articles
  2. Model of electric charge distribution in the trap of a close-contact TENG system
  3. Dynamics of Online Collective Attention as Hawkes Self-exciting Process
  4. Enhanced Entanglement in Hybrid Cavity Mediated by a Two-way Coupled Quantum Dot
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  28. Stable walking of biped robot based on center of mass trajectory control
  29. Modeling and simulation of dynamic recrystallization behavior for Q890 steel plate based on plane strain compression tests
  30. Edge effect of multi-degree-of-freedom oscillatory actuator driven by vector control
  31. The effect of guide vane type on performance of multistage energy recovery hydraulic turbine (MERHT)
  32. Development of a generic framework for lumped parameter modeling
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  35. 3D bending simulation and mechanical properties of the OLED bending area
  36. Resonance overvoltage control algorithms in long cable frequency conversion drive based on discrete mathematics
  37. The measure of irregularities of nanosheets
  38. The predicted load balancing algorithm based on the dynamic exponential smoothing
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  40. A comparative study of cohesive zone models for predicting delamination fracture behaviors of arterial wall
  41. Analysis on dynamic feature of cross arm light weighting for photovoltaic panel cleaning device in power station based on power correlation
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  44. Simultaneous measurement of ionizing radiation and heart rate using a smartphone camera
  45. On the relations between some well-known methods and the projective Riccati equations
  46. Application of energy dissipation and damping structure in the reinforcement of shear wall in concrete engineering
  47. On-line detection algorithm of ore grade change in grinding grading system
  48. Testing algorithm for heat transfer performance of nanofluid-filled heat pipe based on neural network
  49. New optical solitons of conformable resonant nonlinear Schrödinger’s equation
  50. Numerical investigations of a new singular second-order nonlinear coupled functional Lane–Emden model
  51. Circularly symmetric algorithm for UWB RF signal receiving channel based on noise cancellation
  52. CH4 dissociation on the Pd/Cu(111) surface alloy: A DFT study
  53. On some novel exact solutions to the time fractional (2 + 1) dimensional Konopelchenko–Dubrovsky system arising in physical science
  54. An optimal system of group-invariant solutions and conserved quantities of a nonlinear fifth-order integrable equation
  55. Mining reasonable distance of horizontal concave slope based on variable scale chaotic algorithms
  56. Mathematical models for information classification and recognition of multi-target optical remote sensing images
  57. Hopkinson rod test results and constitutive description of TRIP780 steel resistance spot welding material
  58. Computational exploration for radiative flow of Sutterby nanofluid with variable temperature-dependent thermal conductivity and diffusion coefficient
  59. Analytical solution of one-dimensional Pennes’ bioheat equation
  60. MHD squeezed Darcy–Forchheimer nanofluid flow between two h–distance apart horizontal plates
  61. Analysis of irregularity measures of zigzag, rhombic, and honeycomb benzenoid systems
  62. A clustering algorithm based on nonuniform partition for WSNs
  63. An extension of Gronwall inequality in the theory of bodies with voids
  64. Rheological properties of oil–water Pickering emulsion stabilized by Fe3O4 solid nanoparticles
  65. Review Article
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  67. Review of research, development and application of photovoltaic/thermal water systems
  68. Special Issue on Fundamental Physics of Thermal Transports and Energy Conversions
  69. Numerical analysis of sulfur dioxide absorption in water droplets
  70. Special Issue on Transport phenomena and thermal analysis in micro/nano-scale structure surfaces - Part I
  71. Random pore structure and REV scale flow analysis of engine particulate filter based on LBM
  72. Prediction of capillary suction in porous media based on micro-CT technology and B–C model
  73. Energy equilibrium analysis in the effervescent atomization
  74. Experimental investigation on steam/nitrogen condensation characteristics inside horizontal enhanced condensation channels
  75. Experimental analysis and ANN prediction on performances of finned oval-tube heat exchanger under different air inlet angles with limited experimental data
  76. Investigation on thermal-hydraulic performance prediction of a new parallel-flow shell and tube heat exchanger with different surrogate models
  77. Comparative study of the thermal performance of four different parallel flow shell and tube heat exchangers with different performance indicators
  78. Optimization of SCR inflow uniformity based on CFD simulation
  79. Kinetics and thermodynamics of SO2 adsorption on metal-loaded multiwalled carbon nanotubes
  80. Effect of the inner-surface baffles on the tangential acoustic mode in the cylindrical combustor
  81. Special Issue on Future challenges of advanced computational modeling on nonlinear physical phenomena - Part I
  82. Conserved vectors with conformable derivative for certain systems of partial differential equations with physical applications
  83. Some new extensions for fractional integral operator having exponential in the kernel and their applications in physical systems
  84. Exact optical solitons of the perturbed nonlinear Schrödinger–Hirota equation with Kerr law nonlinearity in nonlinear fiber optics
  85. Analytical mathematical schemes: Circular rod grounded via transverse Poisson’s effect and extensive wave propagation on the surface of water
  86. Closed-form wave structures of the space-time fractional Hirota–Satsuma coupled KdV equation with nonlinear physical phenomena
  87. Some misinterpretations and lack of understanding in differential operators with no singular kernels
  88. Stable solutions to the nonlinear RLC transmission line equation and the Sinh–Poisson equation arising in mathematical physics
  89. Calculation of focal values for first-order non-autonomous equation with algebraic and trigonometric coefficients
  90. Influence of interfacial electrokinetic on MHD radiative nanofluid flow in a permeable microchannel with Brownian motion and thermophoresis effects
  91. Standard routine techniques of modeling of tick-borne encephalitis
  92. Fractional residual power series method for the analytical and approximate studies of fractional physical phenomena
  93. Exact solutions of space–time fractional KdV–MKdV equation and Konopelchenko–Dubrovsky equation
  94. Approximate analytical fractional view of convection–diffusion equations
  95. Heat and mass transport investigation in radiative and chemically reacting fluid over a differentially heated surface and internal heating
  96. On solitary wave solutions of a peptide group system with higher order saturable nonlinearity
  97. Extension of optimal homotopy asymptotic method with use of Daftardar–Jeffery polynomials to Hirota–Satsuma coupled system of Korteweg–de Vries equations
  98. Unsteady nano-bioconvective channel flow with effect of nth order chemical reaction
  99. On the flow of MHD generalized maxwell fluid via porous rectangular duct
  100. Study on the applications of two analytical methods for the construction of traveling wave solutions of the modified equal width equation
  101. Numerical solution of two-term time-fractional PDE models arising in mathematical physics using local meshless method
  102. A powerful numerical technique for treating twelfth-order boundary value problems
  103. Fundamental solutions for the long–short-wave interaction system
  104. Role of fractal-fractional operators in modeling of rubella epidemic with optimized orders
  105. Exact solutions of the Laplace fractional boundary value problems via natural decomposition method
  106. Special Issue on 19th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering
  107. Joint use of eddy current imaging and fuzzy similarities to assess the integrity of steel plates
  108. Uncertainty quantification in the design of wireless power transfer systems
  109. Influence of unequal stator tooth width on the performance of outer-rotor permanent magnet machines
  110. New elements within finite element modeling of magnetostriction phenomenon in BLDC motor
  111. Evaluation of localized heat transfer coefficient for induction heating apparatus by thermal fluid analysis based on the HSMAC method
  112. Experimental set up for magnetomechanical measurements with a closed flux path sample
  113. Influence of the earth connections of the PWM drive on the voltage constraints endured by the motor insulation
  114. High temperature machine: Characterization of materials for the electrical insulation
  115. Architecture choices for high-temperature synchronous machines
  116. Analytical study of air-gap surface force – application to electrical machines
  117. High-power density induction machines with increased windings temperature
  118. Influence of modern magnetic and insulation materials on dimensions and losses of large induction machines
  119. New emotional model environment for navigation in a virtual reality
  120. Performance comparison of axial-flux switched reluctance machines with non-oriented and grain-oriented electrical steel rotors
  121. Erratum
  122. Erratum to “Conserved vectors with conformable derivative for certain systems of partial differential equations with physical applications”
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