Electrokinematics theorem

The electrokinematics theorem connects the velocity and the charge of carriers moving inside an arbitrary volume to the currents, voltages and power on its surface through an arbitrary irrotational vector. Since it contains, as a particular application, the Ramo-Shockley theorem, the electrokinematics theorem is also known as Ramo-Shockly-Pellegrini theorem.

Statement

To introduce the electrokinematics theorem let us first list a few definitions: q_j, r_j and v_j are the electric charge, position and velocity, respectively, at the time t of the jth charge carrier; A₀, E =  − ∇A₀ and ε are the electric potential, field, and permittivity, respectively, J_q, J_d = ε∂E/∂t and J = J_q + J_d are the conduction, displacement and, in a 'quasi-electrostatic' assumption, total current density, respectively; F =  − ∇Φ is an arbitrary irrotational vector in an arbitrary volume Ω enclosed by the surface S, with the constraint that ∇(εF) = 0. Now let us integrate over Ω the scalar product of the vector F by the two members of the above-mentioned current equation. Indeed, by applying the divergence theorem, the vector identity a ⋅ ∇γ = ∇ ⋅ (γa) − γ∇ ⋅ a, the above-mentioned constraint and the fact that ∇ ⋅ J = 0, we obtain the electrokinematics theorem in the first form

$$-\int_{S} \Phi J\cdot dS=\int_{\Omega}J_{q}\cdot Fd^{3}r-\int_{S}\varepsilon\frac{\partial A_{0}}{\partial t}F\cdot dS$$ ,

which, taking into account the corpuscular nature of the current $J_{q}=\sum_{j=1}^{N(t)} q_{j}\delta(r-r_{j})v_{j}$, where δ(r−r_j) is the Dirac's delta function and N(t) is the carrier number in Ω at the time t, becomes

$$-\int_{S} \Phi J\cdot dS=\sum_{j=1}^{N(t)} q_{j}v_{j}\cdot F(r_{j})-\int_{S}\varepsilon\frac{\partial A_{0}}{\partial t}F\cdot dS$$ .

A component A_Vk[r,V_k(t)] = V_k(t)ψ_k(r) of the total electric potential A₀ = A_Vk + A_qj is due to the voltage V_k(t) applied to the kth electrode on S, on which ψ_k(r) = 1 (and with the other boundary conditions ψ_k(r) = ψ_k(∞) = 0 on the other electrodes and for r → ∞), and each component A_qj[r,r_j(t)] is due to the jth charge carrier q_j , being A_qj[r,r_j(t)] = 0 for r and r_j(t) over any electrode and for r → ∞. Moreover, let the surface S enclosing the volume Ω consist of a part $S_{E}=\sum_{k=1}^{n}S_{k}$ covered by n electrodes and an uncovered part S_R.

According to the above definitions and boundary conditions, and to the superposition theorem, the second equation can be split into the contributions

$$-\int_{S_{E}} \Phi J_{q}\cdot dS=\sum_{j=1}^{N(t)} q_{j}v_{j}\cdot F(r_{j})+\sum_{j=1}^{M(t)}\int_{S_{R}}\varepsilon(\Phi\frac{\partial E_{j}}{\partial t}-\frac{\partial A_{qj}}{\partial t}F)\cdot dS$$ ,

$$-\int_{S_{E}} \Phi J_{V}\cdot dS=\sum_{k=1}^n \int_{S_{R}}\varepsilon \Phi \frac{\partial E_{k}}{\partial t} \cdot dS-\sum_{k=1}^n \int_{S}\varepsilon \frac{\partial A_{Vk}}{\partial t} F\cdot dS$$,

relative to the carriers and to the electrode voltages, respectively, M(t) being the total number of carriers in the space, inside and outside Ω, at time t, E_j =  − ∇A_qj and E_k =  − ∇A_Vk. The integrals of the above equations account for the displacement current, in particular across S_R.

Current and capacitance

One of the more meaningful application of the above equations is to compute the current

i_h ≡  − ∫_ShJ ⋅ dS = i_qh + i_Vh ,

through an hth electrode of interest corresponding to the surface S_h, i_qh and i_Vh being the current due to the carriers and to the electrode voltages, to be computed through third and fourth equations, respectively.

Open devices

Let us consider as a first example, the case of a surface S that is not completely covered by electrodes, i.e., S_R ≠ 0, and let us choose Dirichlet boundary conditions Φ = Φ_h = 1 on the hth electrode of interest and of Φ_h = 0 on the other electrodes so that, from the above equations we have

$$i_{qh}=\sum_{j=1}^{N(t)}q_{j}v_{j}\cdot F_{h}(r_{j})+\sum_{j=1}^{M(t)}\int_{S_{R}}\varepsilon(\Phi_{h}\frac{\partial E_{j}}{\partial t}-\frac{\partial A_{qj}}{\partial t}F_{h})\cdot dS=$$

$$i_{dh}=\sum_{k=1}^{n}C_{hk}\frac{dV_{k}}{dt}$$ ,

where F = F_h(r_j) is relative to the above-mentioned boundary conditions and C_hk is a capacitive coefficient of the hth electrode given by

C_hk =  − (∫_SkεF_h⋅dS+∫_SRε(Φ_h∇ψ_k+ψ_kF_h)⋅dS) .

V_h is the voltage difference between the hth electrode and an electrode held to a constant voltage (DC), for instance, directly connected to ground or through a DC voltage source. The above equations hold true for the above Dirichlet conditions for Φ_h and for any other choice of boundary conditions on S_R.

A second case can be that in which Φ_h = 0 also on S_R so that such equations reduce to

$$i_{qh}=\sum_{j=1}^{N(t)}q_{j}v_{j}\cdot F_h(r_{j})-\sum_{j=1}^{M(t)}\int_{S_{R}}\varepsilon\frac{\partial A_{0j}}{\partial t}F_{h}\cdot dS$$ ,

C_hk =  − (∫_SkεF_h⋅dS+∫_SRεΨ_kF_h⋅dS) .

As a third case, exploiting also to the arbitrariness of S_R , we can choose a Neumann boundary condition of F_h tangent to S_R in any point. Then the equations become

$$i_{qh}=\sum_{j=1}^{N(t)}q_{j}v_{j}\cdot F_{h}(r_{j})-\sum_{j=1}^{M(t)}\int_{S_{R}}\varepsilon\Phi_{h}\frac{\partial E_{j}}{\partial t}\cdot dS$$ ,

C_hk =  − (∫_SkεF_h⋅dS+∫_SRε∇Ψ_h⋅dS) .

In particular, this case is useful when the device is a right parallelepiped, being S_R and S_E the lateral surface and the bases, respectively.

As a fourth application let us assume Φ = 1 in the whole the volume Ω, i.e., F = 0 in it, so that from the first equation of Section 1 we have

$$\sum_{h=1}^{n}i_{h}-\int_{S_{R}}\varepsilon(\sum_{j=1}^{M(t)}\frac{\partial E_{j}}{\partial t}+\sum_{k=1}^{n}\frac{\partial E_{k}}{\partial t})\cdot dS=0$$ ,

which recover the Kirchhoff law with the inclusion the displacement current across the surface S_R that is not covered by electrodes.

Enclosed devices

A fifth case, historically significant, is that of electrodes that completely enclose the volume Ω of the device, i.e. S_R = 0 . Indeed, choosing again the Dirichlet boundary conditions of Φ_h = 1 on S_h and Φ_h = 0 on the other electrodes, from the equations for the open device we get the relationships

$$i_{h}=\sum_{j=1}^{N(t)}q_{j}v_{j}\cdot F_h(r_{j})+\sum_{k=1}^{n}C_{hk}\frac{dV_h}{dt}$$ ,

with

C_hk =  − ∫_SkεF_h ⋅ dS ,

thus obtaining the Ramo-Shockly theorem as a particular application of the electrokinematics theorem, extended from the vacuum devices to any electrical component and material.

As the above relationships hold true also when F(t) depends on time, we can have a sixty application if we select as $F=F_{V}=-\sum_{k=1}^{n}V_{k}(t)\nabla\psi_{k}(r)$ the electric field generated by the electrode voltages when there is no charge in Ω. Indeed, as the first equation can be written in the form

 − ∫_SΦJ ⋅ dS = ∫_ΩJ ⋅ Fd³r ,

from which we have

$$\sum_{h=1}^{n}V_{h}i_{h}=\int_{\Omega}J\cdot F_{V}d^{3}r\equiv W$$,

where W corresponds to the power entering the device Ω across the electrodes (enclosing it). On the other side

$$\int_{\Omega}(E\cdot J_{q}+E\cdot \frac{\varepsilon \partial E}{\partial t})d^{3}r=\int_{\Omega}E\cdot Jd^{3}r\equiv \frac{d\Xi}{dt}$$ ,

gives the increment of the internal energy Ξ in Ω in a unit of time, E = F_V + E_q being the total electric field of which F_V is due the electrodes and E_q =  − ∇ψ_q(r,t) is due to the whole charge density in Ω with ψ_q(r,t) = 0 over S. Then it is ∫_ΩE_q ⋅ Jd³r = 0, so that, according to such equations, we also verify the energy balance W = dΞ/dt by means of the electrokinematics theorem. With the above relationships the balance can be extended also to the open devices by taking into account the displacement current across S_R.

Fluctuations

A meaningful application of the above results is also the computation of the fluctuations of the current i_h = i_qh when the electrode voltages is constant, because this is useful for the evaluation of the device noise. To this end, we can exploit the first equation of section Open devices, because it concerns the more general case of an open device and it can be reduced to a more simply relationship. This happens for frequencies f = ω/(2π) ≪ 1/(2πt_j), (t_j being the transit time of the jth carrier across the device) because the in time integral of the above equation of the Fourier transform to be performed to compute the power spectral density (PSD) of the noise, the time derivatives provides no contribution. Indeed, according to the Fourier transform, this result derives from integrals such as ∫₀^t_jexp(−jωt)(∂Q/∂t)dt ≈ Q(t_j) − Q(0) , in which Q(t_j) = Q(0) = 0. Therefore for the PSD computation we can exploit the relationships

$$i_{qh}=\sum_{j=1}^{N(t)}q_jv_j\cdot F_h(r_j)=-\sum_{j=1}^{N(t)}q_{j}\frac{d\Phi_{h}[r_{j}(t)]}{dt}=\int_{\Omega}J_{q}\cdot Fd^{3}r$$

Moreover, as it can be shown, this happens also for f ≫ 1/(2πt_j), for instance when the jth carrier is stored for a long time τ_j in a trap if the screening length due to the other carriers is small in comparison to Ω size. All the above considerations hold true for any size of Ω, including nanodevices. In particular we have a meaningful case when the device is a right parallelepiped or cylinder with S_R as lateral surface and u as the unit vector along its axis, with the bases S_E1 and S_E2 located at a distance L as electrodes, and with S_E1 → u → S_E2. Indeed, choosing F_h = F =  − u/L, from the above equation we finally obtain the current i = i₁ = i_q1 =  − i₂,

$$i=\frac{1}{L}\sum_{j=1}^{N(t)}q_{j}v_{ju}=\frac{1}{L}\int_{\Omega}J_{qu}d^{3}r$$ ,

where v_ju and J_qu are the components of v and J_q along u. The above equations in their corpuscular form are particularly suitable for the investigation of transport and noise phenomena from the microscopic point of view, with the application of both the analytical approaches and numerical statistical methods, such as the Monte Carlo techniques. On the other side, in their collective form of the last terms, they are useful to connect, with a general and new method, the local variations of continuous quantities to the current fluctuation at the device terminals. This will be shown in the next sections.

Noise

Shot noise

Let us first evaluate the PSD S_S of the shot noise of the current i = i_qh for short circuited device terminals, i.e. when the V_h's are constant, by applying the third member of the first equation of the above Section. To this end, let us exploit the Fourier coefficient

$$D(\omega_{l})\equiv\frac{1}{T^'}\int_{-T^'/2}^{T^'/2}\Delta i(t)exp(-j\omega_{l}t)dt$$

and the relationship

$$S_{S}(\omega_{l})\equiv\lim_{\Delta f\to 0}\frac{\left \langle D(\omega_l)D^*(\omega_l)\right \rangle}{\Delta f}=\lim_{T^' \to \infty}(2T^'\left \langle D(\omega_l)D^*(\omega_l)\right \rangle)$$

where ω_l = l(2π/T^′), l = ...,  − 2,  − 1, 1, 2, ... in the second term and l = 1, 2, ... in the third. If we define with t_bj and (t_bj+t_j) the beginning and the end of the jth carrier motion inside Ω, we have either Φ_h[r_j(t_bj)] = 1 and Φ_h[r_j(t_bj+t_j)] = 0 or vice versa (the case of Φ_h[r_j(t_bj)] = Φ_h[r_j(t_bj+t_j)] give no contribution), so that from the first equations of the above and of this Section, we get

$$D(\omega_l)\equiv \frac{q}{T^'}(\Delta N^+-\Delta N^-)$$ ,

where N⁺(N⁻) is the number of the carriers (with equal charge q) that start from (arrive on) the electrode of interest during the time interval  − T^′/2, T^′/2. Finally for τ_c ≪ t_jmin, τ_c being the correlation time, and for carriers with a motion that is statistically independent and a Poisson process we have ⟨ΔN⁺ΔN⁻⟩ = 0, ⟨ΔN⁺ΔN⁺⟩ = ⟨N⁺⟩ and ⟨ΔN⁻ΔN⁻⟩ = ⟨N⁻⟩ so that we obtain

S_S = 2q(I⁺+I⁻) ,

where I⁺(I⁻) is the average current due to the carriers leaving (reaching) the electrode. Therefore we recover and extend the Schottky's theorem on shot noise. For instance for an ideal pn junction, or Schottky barrier diode, it is I⁺ = I₀exp(qv/k_BT), I⁻ = I₀, where k_B is the Boltzmann constant, T the absolute temperature, v the voltage and I = I⁺ − I⁻ the total current. In particular, for v = 0 the conductance becomes g = (dI/dv) = qI₀/(k_BT) and the above equation gives

S_S = 4k_BTg ,

that is the thermal noise at thermal equilibrium given by the Nyquist theorem. If the carrier motions are correlated, the above equation has to be changed to the form (for I⁺ ≫ I⁻)

S_S = F_a(2qI) ,

where F_a is the so called Fano factor that can be both less than 1 (for instance in the case of carrier generation-recombination in nonideal pn junctions), and greater than 1 (as in the negative resistance region of resonant-tunneling diode, as a result of the electron-electron interaction being enhanced by the particular shape of the density of states in the well.)

Thermal noise

Once again from the corpuscular point of view, let us evaluate the thermal noise with the autocorrelation function ⟨i(t)i(t+θ)⟩ of i(t) by means of the second term of the second equation of section Fluctuations, that for the short circuit condition V₁ = V₂ = 0 (i.e., at thermal equilibrium) which implies $N(t)=\overline{N}$, becomes

$$\left \langle i(t)i(t+\theta)\right\rangle=\frac{q^2}{L^2}\sum_{j=1}^{\overline{N}}\left \langle v_{ju}^2(t)\right \rangle_t exp(-\left |\theta\right |/\tau_c)=\frac{q^2\overline{N}k_BT}{L^2m}exp(-\left |\theta\right |/\tau_c)$$ ,

where m is the carrier effective mass and τ_c ≪ τ_jmin. As μ = qτ_c/[m(1+jω)] and $G=q\mu\overline{N}/L^2$ are the carrier mobility and the conductance of the device, from the above equation and the Wiener-Khintchine theorem we recover the result

S_T = 4k_BTRe{G(jω)} ,

obtained by Nyquist from the second principle of the thermodynamics, i.e. by means of a macroscopic approach.

Generation-recombination (g-r) noise

A significant example of application of the macroscopic point of view expressed by the third term of the second equation of section Fluctuations is represented by the g-r noise generated by the carrier trapping-detrapping processes in device defects. In the case of constant voltages and drift current density J_qu = qμn_qE, (E≡E_u), that is by neglecting the above velocity fluctuations of thermal origin, from the mentioned equation we get

$$i=\frac{1}{L}\int_{\Omega}q\mu n_qEd^3r$$ ,

in which n_q is the carrier density, and its steady state value is $\overline{i}\equiv I=q\mu n_qEA$, A being the device cross-section surface; furthermore, we use the same symbols for both the time averaged and the instantaneous quantities. Let us first evaluate the fluctuations of the current i, that from the above equation are

$$\frac{\Delta i}{I}=\frac{1}{\Omega}(\frac{1}{n_q}\int_{\Omega}\Delta n_qd^3r+\frac{1}{E}\int_{\Omega}\Delta Ed^3r+\frac{1}{\mu}\int_{\Omega}\Delta \mu d^3r)$$ ,

where only the fluctuation terms are time dependent. The mobility fluctuations can be due to the motion or to the change of status of defects that we neglect here. Therefore we ascribe the origin of g-r noise to the trapping-detrapping processes that contribute to Δi through the other two terms via the fluctuation of the electron number χ = 0, 1 in the energy level ε_t of a single trap in the channel or in its neighborhood. Indeed the charge fluctuation qΔχ in the trap generates variations of n_q and of E. However, the variation ΔE does not contribute to Δi because it is odd in the u direction, so that we get

$$\frac{\Delta i}{I}=\frac{1}{\Omega n_q}\int_{\Omega}\Delta n_q d^3r$$ ,

from which we obtain

$$\frac{\Delta i}{I}=\frac{1}{\Omega n_q}\int_{\delta\Omega}\Delta n_q ad^3r=-\frac{1}{\Omega n_q}\Delta\chi$$ ,

where the reduction of the integration volume from Ω to the much smaller one δΩ around the defect is justified by the fact that the effects of Δn_q and ΔE fade within a few multiples of a screening length, which can be small (of the order of nanometers in graphene); from Gauss's theorem, we obtain also ∫_δΩΔn_qd³r =  − Δχ and the r.h.s. of the equation. In it the variation Δχ occurs around the average value $\overline{\chi}$ given by the Fermi-Dirac factor $\overline{\chi}\equiv\phi=\{[1+exp[(\varepsilon_t-\varepsilon_f)/k_B T]\}^{-1}$, ε_f being the Fermi level. The PSD S_t of the fluctuation Δi due to a single trap then becomes S_t/I² = [1/(Ωn_q)]²S_χ, where S_χ = 4ϕ(1−ϕ)τ/[1+(ωτ)²] is the Lorentzian PSD of the random telegraph signal χ and τ is the trap relaxation time. Therefore, for a density n_t of equal and uncorrelated defects we have a total PSD S_gr of the g-r noise given by

$$S_{gr}=\frac{4I^2n_t\phi(1-\phi)\tau}{\Omega n^2_{q}[1+(\omega\tau)^2]}$$ .

Flicker noise

When the defects are not equal, for any distribution of τ (except a sharply peaked one, as in the above case of g-r noise), and even for a very small number of traps with large τ, the total PSD S_f of i, corresponding to the sum of the PSD S_t of all the n_tΩ (statistically independent) traps of the device, becomes

$$S_f=\frac{n_tB}{\Omega n_q^2}\frac{I^2}{f^\gamma}$$ ,

where 0.85 < γ < 1.15 down to the frequency 1/2πτ_M, τ_M being the largest τ and B(ε_f/k_BT) a proper coefficient. In particular, for unipolar conducting materials (e.g., for electrons as carriers) it can be n_q ∝ exp(ε_f/k_BT) and, for trap energy levels ε_t > ε_f, from S_χ ∝ ϕ = exp(ε_f/k_BT) we also have B(ε_f/k_BT) ∝ exp(ε_f/k_BT), so that from the above equation we obtain,

$$S_f=\frac{\alpha I^2}{N_qf^{\gamma}}$$ ,

where N_q is the total number of the carriers and α is a parameters that depends on the material, structure and technology of the device.

Extensions

Electromagnetic field

The shown electrokinetics theorem holds true in the 'quasi electrostatic' condition, that is when the vector potential can be neglected or, in other terms, when the squared maximum size of ω is much smaller than the squared minimum wavelength of the electromagnetic field in the device. However it can be extended to the electromagnetic field in a general form. In this general case, by means of the displacement current across the surface S_R it is possible, for instance, to evaluate the electromagnetic field radiation from an antenna. It holds true also when the electric permittivity and the magnetic permeability depend on the frequency. Moreover the field F(r,t) =  − ∇Φ other than the electric field in 'quasi electrostatic' conditions, can be any other physical irrotational field.

Quantum mechanics

Finally, the electrokinetics theorem holds true in the classical mechanics limit, because it requires the simultaneous knowledge of the position and velocity of the carrier, that is, as a result of the uncertainty principle, when its wave function is essentially non null in a volume smaller than that of device. Such a limit can however be overcome computing the current density according the quantum mechanical expression.

Notes

• Bruno Pellegrini has been the first Electronic Engineering graduate at the University of Pisa, where he is currently Professor Emeritus. He is also author of the cut-insertion theorem, that is at the basis of a novel feedback theory for linear circuits.

See also

• Maxwell's equations
• Transport phenomena