Electrokinematics 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; A 0 , E = − ∇ A 0 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 ∇ ( ε F ) = 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 = ∇ ⋅ ( γ a ) − γ ∇ ⋅ a , the above-mentioned constraint and the fact that ∇ ⋅ J = 0 , we obtain the electrokinematics theorem in the first form

− ∫ S Φ J ⋅ d S = ∫ Ω J q ⋅ F d 3 r − ∫ S ε ∂ A 0 ∂ t F ⋅ d S ,

which, taking into account the corpuscular nature of the current J q = ∑ j = 1 N ( t ) q j δ ( r − r j ) v j , where δ ( r − r j ) is the Dirac delta function and N(t) is the carrier number in Ω at the time t, becomes

− ∫ S Φ J ⋅ d S = ∑ j = 1 N ( t ) q j v j ⋅ F ( r j ) − ∫ S ε ∂ A 0 ∂ t F ⋅ d S .

A component A V k [ r , V k ( t ) ] = V k ( t ) ψ k ( r ) of the total electric potential A 0 = A V k + A q j 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 q j [ r , r j ( t ) ] is due to the jth charge carrier q_j , being A q j [ 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 = ∑ 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

− ∫ S E Φ J q ⋅ d S = ∑ j = 1 N ( t ) q j v j ⋅ F ( r j ) + ∑ j = 1 M ( t ) ∫ S R ε ( Φ ∂ E j ∂ t − ∂ A q j ∂ t F ) ⋅ d S , − ∫ S E Φ J V ⋅ d S = ∑ k = 1 n ∫ S R ε Φ ∂ E k ∂ t ⋅ d S − ∑ k = 1 n ∫ S ε ∂ A V k ∂ t F ⋅ d S ,

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 q j and E k = − ∇ A V k . 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 ≡ − ∫ S h J ⋅ d S = i q h + i V h ,

through an hth electrode of interest corresponding to the surface S h , i q h and i V h 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 q h = ∑ j = 1 N ( t ) q j v j ⋅ F h ( r j ) + ∑ j = 1 M ( t ) ∫ S R ε ( Φ h ∂ E j ∂ t − ∂ A q j ∂ t F h ) ⋅ d S = i d h = ∑ k = 1 n C h k d V k d t ,

where F = F h ( r j ) is relative to the above-mentioned boundary conditions and C h k is a capacitive coefficient of the hth electrode given by

C h k = − ( ∫ S k ε F h ⋅ d S + ∫ S R ε ( Φ h ∇ ψ k + ψ k F h ) ⋅ d S ) .

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 q h = ∑ j = 1 N ( t ) q j v j ⋅ F h ( r j ) − ∑ j = 1 M ( t ) ∫ S R ε ∂ A 0 j ∂ t F h ⋅ d S , C h k = − ( ∫ S k ε F h ⋅ d S + ∫ S R ε Ψ k F h ⋅ d S ) .

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 q h = ∑ j = 1 N ( t ) q j v j ⋅ F h ( r j ) − ∑ j = 1 M ( t ) ∫ S R ε Φ h ∂ E j ∂ t ⋅ d S , C h k = − ( ∫ S k ε F h ⋅ d S + ∫ S R ε ∇ Ψ h ⋅ d S ) .

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

∑ h = 1 n i h − ∫ S R ε ( ∑ j = 1 M ( t ) ∂ E j ∂ t + ∑ k = 1 n ∂ E k ∂ t ) ⋅ d S = 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 = ∑ j = 1 N ( t ) q j v j ⋅ F h ( r j ) + ∑ k = 1 n C h k d V h d t ,

with

C h k = − ∫ S k ε F h ⋅ d S ,

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 = − ∑ k = 1 n V k ( t ) ∇ ψ 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

− ∫ S Φ J ⋅ d S = ∫ Ω J ⋅ F d 3 r ,

from which we have

∑ h = 1 n V h i h = ∫ Ω J ⋅ F V d 3 r ≡ W ,

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

∫ Ω ( E ⋅ J q + E ⋅ ε ∂ E ∂ t ) d 3 r = ∫ Ω E ⋅ J d 3 r ≡ d Ξ d t ,

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 ∫ Ω E q ⋅ J d 3 r = 0 , so that, according to such equations, we also verify the energy balance W = d Ξ / d t 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 q h 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 / ( 2 π ) ≪ 1 / ( 2 π t 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 j e x p ( − j ω t ) ( ∂ Q / ∂ t ) d t ≈ Q ( t j ) − Q ( 0 ) , in which Q ( t j ) = Q ( 0 ) = 0 . Therefore, for the PSD computation we can exploit the relationships

i q h = ∑ j = 1 N ( t ) q j v j ⋅ F h ( r j ) = − ∑ j = 1 N ( t ) q j d Φ h [ r j ( t ) ] d t = ∫ Ω J q ⋅ F d 3 r

Moreover, as it can be shown, this happens also for f ≫ 1 / ( 2 π t 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 E 1 and S E 2 located at a distance L as electrodes, and with S E 1 → u → S E 2 . Indeed, choosing F h = F = − u / L , from the above equation we finally obtain the current i = i 1 = i q 1 = − i 2 ,

i = 1 L ∑ j = 1 N ( t ) q j v j u = 1 L ∫ Ω J q u d 3 r ,

where v j u and J q u 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.

Shot noise

Let us first evaluate the PSD S S of the shot noise of the current i = i q h 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 ( ω l ) ≡ 1 T ′ ∫ − T ′ / 2 T ′ / 2 Δ i ( t ) exp ⁡ ( − j ω l t ) d t

and the relationship

S S ( ω l ) ≡ lim Δ f → 0 ⟨ D ( ω l ) D ∗ ( ω l ) ⟩ Δ f = lim T ′ → ∞ ( 2 T ′ ⟨ D ( ω l ) D ∗ ( ω l ) ⟩ )

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

D ( ω l ) ≡ q T ′ ( Δ N + − Δ 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 j m i n , τ c being the correlation time, and for carriers with a motion that is statistically independent and a Poisson process we have ⟨ Δ N + Δ N − ⟩ = 0 , ⟨ Δ N + Δ N + ⟩ = ⟨ N + ⟩ and ⟨ Δ N − Δ N − ⟩ = ⟨ N − ⟩ so that we obtain

S S = 2 q ( 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 + = I 0 exp ⁡ ( q v / k B T ) , I − = I 0 , 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 = ( d I / d v ) = q I 0 / ( k B T ) and the above equation gives

S S = 4 k B T g ,

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 ( 2 q I ) ,

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 V 1 = V 2 = 0 (i.e., at thermal equilibrium) which implies N ( t ) = N ¯ , becomes

⟨ i ( t ) i ( t + θ ) ⟩ = q 2 L 2 ∑ j = 1 N ¯ ⟨ v j u 2 ( t ) ⟩ t e x p ( − | θ | / τ c ) = q 2 N ¯ k B T L 2 m exp ⁡ ( − | θ | / τ c ) ,

where m is the carrier effective mass and τ c ≪ τ j m i n . As μ = q τ c / [ m ( 1 + j ω ) ] and G = q μ 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 = 4 k B T ℜ { G ( j ω ) } ,

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 q u = q μ n q E , ( E ≡ E u ) , that is by neglecting the above velocity fluctuations of thermal origin, from the mentioned equation we get

i = 1 L ∫ Ω q μ n q E d 3 r ,

in which n q is the carrier density, and its steady state value is i ¯ ≡ I = q μ n q E A , 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

Δ i I = 1 Ω ( 1 n q ∫ Ω Δ n q d 3 r + 1 E ∫ Ω Δ E d 3 r + 1 μ ∫ Ω Δ μ d 3 r ) ,

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 Δ i 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 q Δ χ in the trap generates variations of n q and of E . However, the variation Δ E does not contribute to Δ i because it is odd in the u direction, so that we get

Δ i I = 1 Ω n q ∫ Ω Δ n q d 3 r ,

from which we obtain

Δ i I = 1 Ω n q ∫ δ Ω Δ n q a d 3 r = − 1 Ω n q Δ χ ,

where the reduction of the integration volume from Ω to the much smaller one δ Ω around the defect is justified by the fact that the effects of Δ n q and Δ E 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 ∫ δ Ω Δ n q d 3 r = − Δ χ and the r.h.s. of the equation. In it the variation Δ χ occurs around the average value χ ¯ given by the Fermi-Dirac factor χ ¯ ≡ ϕ = { [ 1 + exp ⁡ [ ( ε t − ε f ) / k B T ] } − 1 , ε f being the Fermi level. The PSD S t of the fluctuation Δ i due to a single trap then becomes S t / I 2 = [ 1 / ( Ω n q ) ] 2 S χ , where S χ = 4 ϕ ( 1 − ϕ ) τ / [ 1 + ( ω τ ) 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 g r of the g-r noise given by

S g r = 4 I 2 n t ϕ ( 1 − ϕ ) τ Ω n q 2 [ 1 + ( ω τ ) 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 t Ω (statistically independent) traps of the device, becomes

S f = n t B Ω n q 2 I 2 f γ ,

where 0.85 < γ < 1.15 down to the frequency 1 / 2 π τ M , τ M being the largest τ and B ( ε f / k B T ) a proper coefficient. In particular, for unipolar conducting materials (e.g., for electrons as carriers) it can be n q ∝ e x p ( ε f / k B T ) and, for trap energy levels ε t > ε f , from S χ ∝ ϕ = e x p ( ε f / k B T ) we also have B ( ε f / k B T ) ∝ e x p ( ε f / k B T ) , so that from the above equation we obtain,

S f = α I 2 N q f γ ,

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.

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 to the quantum mechanical expression.