
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: "qj", "rj" and "vj" are the electric charge, position and velocity, respectively, at the time t of the "j"th charge carrier; formula_1, formula_2 and formula_3 are the electric potential, field, and permittivity, respectively, formula_4, formula_5 and formula_6 are the conduction, displacement and, in a 'quasi-electrostatic' assumption, total current density, respectively; formula_7 is an arbitrary irrotational vector in an arbitrary volume formula_8 enclosed by the surface S, with the constraint that formula_9. Now let us integrate over formula_8 the scalar product of the vector formula_11 by the two members of the above-mentioned current equation. Indeed, by applying the divergence theorem, the vector identity formula_12, the above-mentioned constraint and the fact that formula_13, we obtain the electrokinematics theorem in the first form
which, taking into account the corpuscular nature of the current formula_15, where formula_16 is the Dirac delta function and "N(t)" is the carrier number in formula_8 at the time "t", becomes
A component formula_19 of the total electric potential formula_20 is due to the voltage formula_21 applied to the "k"th electrode on "S", on which formula_22 (and with the other boundary conditions formula_23 on the other electrodes and for formula_24), and each component formula_25 is due to the "j"th charge carrier "qj" , being formula_26 for formula_27 and formula_28 over any electrode and for formula_24. Moreover, let the surface "S" enclosing the volume formula_8 consist of a part formula_31 covered by "n" electrodes and an uncovered part formula_32.
According to the above definitions and boundary conditions, and to the superposition theorem, the second equation can be split into the contributions
relative to the carriers and to the electrode voltages, respectively, formula_35 being the total number of carriers in the space, inside and outside formula_8, at time "t", formula_37 and formula_38. 
The integrals of the above equations account for the displacement current, in particular across formula_32.
Current and capacitance.
One of the more meaningful application of the above equations is to compute the current
through an "h"th electrode of interest corresponding to the surface formula_41, formula_42 and formula_43 being the current due to the carriers and to the electrode voltages, to be computed through third and fourth equations, respectively.
Open devices.
Consider as a first example, the case of a surface "S" that is not completely covered by electrodes, i.e., formula_44, and let us choose Dirichlet boundary conditions formula_45 on the "h"th electrode of interest and of formula_46 on the other electrodes so that, from the above equations we have
where formula_49 is relative to the above-mentioned boundary conditions and formula_50 is a capacitive coefficient of the "h"th electrode given by
formula_52 is the voltage difference between the "h"th 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 formula_53 and for any other choice of boundary conditions on formula_32.
A second case can be that in which formula_46 also on formula_32 so that such equations reduce to
As a third case, exploiting also to the arbitrariness of formula_32 , we can choose a Neumann boundary condition of formula_60 tangent to formula_32 in any point. Then the equations become
In particular, this case is useful when the device is a right parallelepiped, being formula_32 and formula_65 the lateral surface and the bases, respectively.
As a fourth application let us assume formula_66 in the whole the volume formula_8, i.e., formula_68 in it, so that from the first equation of Section 1 we have
which recover the Kirchhoff law with the inclusion the displacement current across the surface formula_32 that is not covered by electrodes.
Enclosed devices.
A fifth case, historically significant, is that of electrodes that completely enclose the volume formula_8 of the device, i.e. formula_72 . Indeed, choosing again the Dirichlet boundary conditions of formula_73 on formula_74 and formula_46 on the other electrodes, from the equations for the open device we get the relationships
with
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 formula_78 depends on time, we can have a sixty application if we select as formula_79 the electric field generated by the electrode voltages when there is no charge in formula_8. Indeed, as the first equation can be written in the form
from which we have
where formula_83 corresponds to the power entering the device formula_8 across the electrodes (enclosing it). On the other side
gives the increment of the internal energy formula_86 in formula_8 in a unit of time, formula_88 being the total electric field of which formula_89 is due to the electrodes and formula_90 is due to the whole charge density in formula_8 with formula_92 over "S". Then it is formula_93, so that, according to such equations, we also verify the energy balance formula_94 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 formula_32.
Fluctuations.
A meaningful application of the above results is also the computation of the fluctuations of the current formula_96 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 formula_97, (formula_98 being the transit time of the "j"th 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 formula_99 , in which formula_100. Therefore, for the PSD computation we can exploit the relationships
Moreover, as it can be shown, this happens also for formula_102, for instance when the "j"th carrier is stored for a long time formula_103 in a trap if the screening length due to the other carriers is small in comparison to "formula_8" size. All the above considerations hold true for any size of "formula_8", including nanodevices.
In particular we have a meaningful case when the device is a right parallelepiped or cylinder with formula_32 as lateral surface and "u" as the unit vector along its axis, with the bases formula_107 and formula_108 located at a distance "L" as electrodes, and with formula_109. Indeed, choosing formula_110, from the above equation we finally obtain the current formula_111,
where formula_113 and formula_114 are the components of formula_115 and formula_4 along formula_117. 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 formula_118 of the shot noise of the current formula_119 for short circuited device terminals, i.e. when the formula_52'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
and the relationship
where formula_123, formula_124 in the second term and formula_125 in the third. If we define with formula_126 and formula_127 the beginning and the end of the jth carrier motion inside formula_8, we have either formula_129 and formula_130 or vice versa (the case of formula_131 give no contribution), so that from the first equations of the above and of this Section, we get
where formula_133 is the number of the carriers (with equal charge "q") that start from (arrive on) the electrode of interest during the time interval formula_134. Finally for formula_135, formula_136 being the correlation time, and for carriers with a motion that is statistically independent and a Poisson process we have formula_137, formula_138 and formula_139 so that we obtain
where formula_141 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 formula_142, formula_143, where formula_144 is the Boltzmann constant, "T" the absolute temperature, "v" the voltage and formula_145 the total current. In particular, for formula_146 the conductance becomes formula_147 and the above equation gives
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 formula_149)
where formula_151 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 formula_152 of formula_153 by means of the second term of the second equation of section Fluctuations, that for the short circuit condition formula_154 (i.e., at thermal equilibrium) which implies formula_155, becomes
where "m" is the carrier effective mass and formula_157. As formula_158 and formula_159 are the carrier mobility and the conductance of the device, from the above equation and the Wiener-Khintchine theorem we recover the result
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 formula_161, that is by neglecting the above velocity fluctuations of thermal origin, from the mentioned equation we get
in which formula_163 is the carrier density, and its steady state value is formula_164, formula_165 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
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 formula_167 through the other two terms via the fluctuation of the electron number formula_168 in the energy level formula_169 of a single trap in the channel or in its neighborhood. Indeed, the charge fluctuation formula_170 in the trap generates variations of formula_163 and of formula_172. However, the variation formula_173 does not contribute to formula_167 because it is odd in the "u" direction, so that we get
from which we obtain
where the reduction of the integration volume from formula_8 to the much smaller one formula_178 around the defect is justified by the fact that the effects of formula_179 and formula_173 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 formula_181 and the r.h.s. of the equation. In it the variation formula_182 occurs around the average value formula_183 given by the Fermi-Dirac factor formula_184, formula_185 being the Fermi level. The PSD formula_186 of the fluctuation formula_167 due to a single trap then becomes formula_188, where formula_189 is the Lorentzian PSD of the random telegraph signal formula_190 and formula_191 is the trap relaxation time. Therefore, for a density formula_192 of equal and uncorrelated defects we have a total PSD formula_193 of the g-r noise given by
Flicker noise.
When the defects are not equal, for any distribution of formula_191 (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 formula_191, the total PSD formula_197 of "i", corresponding to the sum of the PSD formula_186 of all the formula_199 (statistically independent) traps of the device, becomes
where formula_201 down to the frequency formula_202, formula_203 being the largest formula_191 and formula_205 a proper coefficient. In particular, for unipolar conducting materials (e.g., for electrons as carriers) it can be formula_206 and, for trap energy levels formula_207, from formula_208 we also have formula_209, so that from the above equation we obtain,
where formula_211 is the total number of the carriers and formula_212 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 formula_213 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 formula_32 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 formula_215 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.