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The problem of drifting charge-induced currents is considered in order to predict the pulsed operational characteristics in photo- and particle-detectors with a junction controlled active area. The direct analysis of the field changes induced by drifting charge in the abrupt junction devices with a plane-parallel geometry of finite area electrodes is presented. The problem is solved using the one-dimensional approach. The models of the formation of the induced pulsed currents have been analyzed for the regimes of partial and full depletion. The obtained solutions for the current density contain expressions of a velocity field dependence on the applied voltage, location of the injected surface charge domain and carrier capture parameters. The drift component of this current coincides with Ramo's expression. It has been illustrated, that the synchronous action of carrier drift, trapping, generation and diffusion can lead to a vast variety of possible current pulse waveforms. Experimental illustrations of the current pulse variations determined by either the rather small or large carrier density within the photo-injected charge domain are presented, based on a study of Si detectors.

The drifting charge-induced current is often a prevailing component in detector signals. An analysis of this injected charge drift current is employed for the detection of photons or particles and for reconstruction of the electric field distribution over the charge drift area in detectors. However, the problem of the charge-induced currents remains a sophisticated issue in more complicated situations of photo- and particle-detectors when devices operate in partial or full depletion regimes with a complex field distribution in the presence of carrier capture-generation. Then, the induced current due to the charge trapping/de-trapping should be considered. The effects of the acting electric field screening or the gain caused by the interplay of the injected carrier charge and the bulk charge of ions in the depletion volume should also be involved. As usual, the principles of interpretation of the current transients in detectors under injected charge are based on the Shockley-Ramo theorem [

In this work, the fixed area abrupt junctions with a plane-parallel geometry of electrodes are studied by considering the partial and over full-depletion regimes. The vectorial nature of the employed quantities of the surface charge domain, of the electric field and of the charge drift velocity is always kept in mind. The scalar relations are analyzed along the unite ortho-vector with proper signs ascribed to directions. To simplify the understanding of applied models, the single type of induced surface charge domain motion is initially considered. This simplification can be sufficient to model current transients due to highly absorbed photons or alpha-particles in Si detectors, while the generalization for a bipolar charge domain is also discussed. The specific features of the induced charge domain drift currents (ICDC) are revealed within analysis of the simulated ICDC transients and highlighted in illustrations of the experimental characteristics, measured on Si pin diodes.

The same electrical circuit as in Ramo's theorem derivation, is considered: one electrode is grounded and the high potential is kept on the other one. These two electrodes are connected to the external voltage source in series with a load resistor to register a current transient within external circuit, as illustrated in _{L}_{L}_{L}

As usual, a quasi-neutral domain of excess carriers is initially generated in particle detectors. It is accepted the domains are flat surface vector quantities. These domains injected by light or ionizing radiation are characterized by the surface charge density and the direction (vector) of the surface normal. Thereby, these domains are directly represented by the electric field of the surface charge. The sign (polarity) of the injected charge and the direction of the drift velocity vector are also included. For the grounded circuit, the single-side surface charge (and field) is ascribed to the voltage source. The drifting domain is also considered as the one-side surface charge vector correlated with drift velocity vector direction. The surface charge electric field vectors are initially considered, like the first Poisson equation. Then, the scalar equations for an instantaneous field distribution are analyzed. By applying an external field source, the injected carriers (by light or ionizing radiation) can be separated into oppositely moving surface charge sub-domains _{e}_{h}

External voltage source _{0}_{r}_{D}^{+}) in the n-base active layer is obtained through respective depletion (_{p}_{+}_{A}^{−}) of a junction: _{D}^{+}_{0}_{A}^{-}w_{p}_{+} = 0. In the abrupt junction of a pin type diode, it is valid _{p}_{+} ≪ _{0}_{p}_{+}_{0}_{e}_{0}

For the partial depletion regime, this leads to changes of the depletion width _{q}_{e}

The junction based detectors contain rather complicated field distribution, and, thereby, need specific analysis. The instantaneous field distribution in n-type conductivity region of the p^{+}n abrupt junction structure during the monopolar drift of a negative charge in partially and over-depleted base region is sketched in

The analysis can be easily applied to the consideration of the barrier capacitance changes for unit area _{b}_{0}/w_{q}(t)_{q}(t)_{e}_{0}_{0}(U_{bi})/eN_{Def}^{1/2} serves as an equivalent of the inter-electrode spacing _{b}_{i} denotes the built-in potential barrier, and _{def}_{0}_{Def}(w_{q}(t)_{0})_{Sq}_{FD}

This regime is partially discussed in [_{bi}_{FD}^{+}-type layer. This leads to a synchronous change of the depletion widths in the n- and p- type conductivity layers to keep the junction system electrically neutral behind the depletion _{0,n}_{p}_{+} width boundaries. To simplify the analysis, an assumption of the asymmetric doping of n- and p-layers, ^{+}-layer. It is also assumed that the external metallic electrodes are in a rapid dynamic balance with neutral n- and p^{+}- layer material. Thus, a rate of the processes within an n-layer region is the slowest one. The latter processes determine the current transient caused by a drift of the injected electron domain.

Using the methodology described in [_{e}

Here, a vector of the electric field is directed towards the junction, while a surface charge domain of electrons can drift towards high potential electrode. To find a depletion width _{q}_{1}_{e}_{r}

Here, a common depletion boundary condition (_{q})_{r}_{bi}_{Mq,w0}_{TOF,w0}_{0}_{Mq,w0}_{TOF,w0}

These characteristic times, namely, their equality (_{Mq,w0}_{TOF,w0}_{TOF,w0}_{Mq,w0}

A steady-state depletion width _{0}_{0}_{0}U/eN_{D})^{1/2}. Consequently, a barrier capacitance is obtained as:
_{b0}_{0}/w_{0}_{q}_{0}

It can be inferred from _{q}

For electrodes of surface unite area

Here, _{M,Ndef}_{0}/eμ_{e}N_{Def}_{FD}

The rearranged (by these differentiation procedures) expression of a module of the current density of the injected charge domain (ICD) drift can be represented as follows:

The obtained scalar form of the current density within a coordinate system at rest (_{0}_{τ}_{e}/w_{0}_{0}, and it is composed of the characteristic times as:

The appearance of the coefficient _{τ}_{q}_{0}_{q}_{M,Ndef}_{0}/eμN_{Def}_{0}/eμn_{ENR}_{ENR}_{Def}

The additional scalar equation (with properly accepted vector direction sign) for a velocity of the charge domain drift is now expressed as follows:

The rearranged equation into the dimensionless _{e}/w_{0}_{0}_{dr}

Extraction of the _{TOF,w0}/τ_{Mq,w0})^{1/2} − _{dr}

Then, the obtained _{ICD,F}(t)_{ICD,R}(t)_{Mq,w0}_{e}

For _{e}_{e}_{e}_{e}/S_{e}_{TOF,w0}_{Mq,w0}

Thus, for the partially depleted junction, both components ^{2}/S_{e}_{FD})_{TOF,w0}/τ_{Mq,w0}_{TOF,w0}/τ_{Mq,w0}_{τ}

Generally, variation of an initial component _{ICD,F}(t)_{dr}

The rearward component (_{ICD,R}(t)_{b,Sq}_{b,S0}_{0}/_{0}_{0}_{e}/εε_{0}_{D}(w_{q}_{0})/εε_{0}_{e}/εε_{0}_{0}_{0}_{dr}_{e}/εε_{0}_{q}_{0}_{dr}

This current component may be responsible for the appearance of the offset within a current transient, inherent for the partially depleted detector. Duration of this process is determined by a dielectric relaxation time of the material, namely, _{M,Ndef}_{Cw0}_{0}/w_{0}_{dr}_{Cw0}_{w0}_{b,S0}S

As a result, the current density within a pulsed ICD transient is expressed as follows:

The offset current relaxation to zero is additionally governed by the parameters of the external circuit. The relaxation component of the current pulse is determined by the relaxation processes within an RC chain consisting of the system capacitance _{b,Sq}_{L}_{ICD,F}_{ICD,R}

The surface charge on a metallic (or heavily doped layer) electrode changes together with the space charge bar width due to a moving surface charge domain, when external voltage _{FD}_{FD}

Again, using the methodology described above, a field distribution for the negative drifting charge is obtained by taking the first Poisson equation:
_{FD}_{Def}d^{2}/_{0}

The solution for a scalar surface charge density on the high potential electrode is expressed as either:

The instantaneous field distribution can then be represented as follows:

It can be deduced from _{e}(t)

This _{e}_{SNq}_{C}_{g}_{0}/d

A module of the current density, derived from

However, the considered situation for a fully-depleted junction is more complicated relative to those discussed above. The reason is a degenerated point _{FD}_{FD}_{FD}_{e}

This _{displ}_{Def}dX_{e}/dt_{OFD}

On the other hand, the displacement current (_{De}_{f}_{e}/dt_{0}dX_{e}/dt_{Def}Ψ_{e}d_{0}dX_{e}/dt_{0}_{0}d_{Def}Ψ_{e}d_{e}_{e}_{0}d_{FD}_{e}_{e}_{0}∂E/∂t_{0}∂E/∂ t

Thus, the complete current within an external circuit is again determined by the displacement current due to the injected charge, as obtained in

A dimensionless velocity field can be considered by using the accelerating electric field component for a geometric width

The coefficients in _{e}

These equations (_{dr}_{dr}_{dr})

These different regimes can be realized by varying the applied voltage _{TOF}_{Def}_{M,Ndef}_{e}_{Mq}_{dr}_{Mq}_{Mq}_{M,Ndef}_{dr}_{TOF}_{TOF}≅τ_{Mq}_{M,Ndef}_{dr}_{e}_{C}≅C_{g}(1 − U_{FD}/U)U_{e}_{C}_{g}U_{e}_{C}

The current density ascribed to different regimes can be modelled by using the relevant expressions for

The current density changes within a pulse vertex acquire a relaxation curve shape for the regime A, when screening of electrons (drifting charge domain) by ion charge within the depletion width prevails. For the correlated screening regime B, a square-wave shape current density pulse appears with a flat vertex. While for the correlated (Ramo's type) drift regime C (_{TOF}≅τ_{Mq}

The monopolar drift of holes can be expressed using methodology described above for the case of electrons drift. The positive charge _{h}^{+}-layer. Then, the field for _{0}, which accelerates _{h}

The current density, for

The drift velocity field is described by a differential equation:

Assuming the proper boundary conditions:

The monopolar drift time is then evaluated as:

The current density of the hole drift is expressed (by inserting

In the case of the positive charge drift within n-base material, holes are always accelerated due to acting space charge field. The transient is then observed with the current density increasing with time.

The moving charge inside the over depleted space charge layer induces a displacement current component, which exactly compensates the conductivity current component, arisen due to a proximate contacting of the depleted layer with external electrode (outside layer). As can be inferred for the regime B (_{Mq}_{M,Ndef}_{Def}_{dr}_{TOF}/_{TOF}/_{M,Ndef}_{TOF}_{0}

However, for the regime A of the non-correlated relaxation times of the space charge _{Def}_{e}/d_{Def}

Then, a surface density of the drifting charge _{e,ef}_{Mq}_{e}_{M, Ndef}_{e}_{Def}X_{e}

The large injected charge is able to locally screen the depletion space charge of ions, for the regime C. Then, a drift of the injected domain proceeds similarly to that in a capacitor-type device.

As usual in detectors, a quasi-neutral domain of the excess carriers is initially generated. Then, owing to a steady-state applied field, these carriers can be separated into the oppositely moving surface charge sub-domains _{e}_{h}_{e}_{h}

An instantaneous electric field distribution along the

A field discontinuity at the instantaneous location of surface charge domains is expressed as:

Then, the relation between the surface charge +_{FD}_{D}d^{2}/2_{0}_{e,h}_{e,h}/d_{D}_{Def}

It is worth to point out, that in the case of the bipolar drift, the charge _{e}, Ψ_{h})_{2}_{1}

The induced charge current density, due to a bipolar drift, is expressed as follows:

It can be noticed that, owing to v_{e} = −v_{h}, the scalar current density can be represented by a sum of Ramo's-type components:

The bipolar drift velocities are correlated during the bipolar drift time _{b}_{b}

Here, the drift directions are included by accepting the relevant sign for the scalar velocity. There exist several situations of the pure bipolar and the mixed drift regimes. These regimes can be separated as:

The regime (_{0}^{+}-layer or it becomes the monopolar drift of the hole domain after electrons reach the high potential electrode. These latter situations depend on the injection location _{0}_{0h}_{0e}^{+}-layer of the abrupt junction to exactly account for the bipolar and monopolar regimes.

In the case of the pure bipolar drift regime (

These solutions should satisfy the boundary conditions:

However, in the more precise approximation, the monopolar drift of holes within p^{+}-region included by _{0}_{0n}_{0p}_{+}, should be analyzed. Therefore, in rigorous consideration, the pure bipolar drift can be assumed as an idealization. Nevertheless, for the case of _{tr,hp}_{+} ≪ _{tr,hn}

Then, the inherent time _{b}

Inserting these solutions (

Thus, the pure bipolar drift leads to an invariable current density with pulse duration of _{P}≅Ψ_{0}τ_{TOF,h}_{0}_{P}_{TOF,h}_{TOF,e}_{0})

In the case of the hole drift time _{tr,h}_{bB}_{tr,h}

These solutions should satisfy the boundary conditions:

Here, _{e}^{0}_{bB}

Here, the step-like change of field and current density would have obtained for an instant of hole arrival to the grounded electrode. To validate the charge, charge momentum, and energy conservation, the coordinate transform should be performed, to stitch the solutions obtained in the moving (_{bB}

These transforms relate the “new” ^{+} coordinate in the system at rest with that _{Mq,e}_{M,Ndef}

Here, _{e1}_{e2}_{0,e,mon}

This gives a coincidence of _{0,e,mon}_{Σbip}_{Ψ}_{e}*^{0} values at the position _{e}^{0}

These solutions are given as:

The entire duration _{P}_{bB}_{e}_{mon}) drift of electrons. Inserting these solutions (_{e}^{0}

The current transient (for the analyzed case of _{Mq,e}_{M,Ndef}_{Mq,e}_{M,Ndef}_{0,e,mon}_{Mq,e}_{M,Ndef}

In the case the electron drift time _{tr,e}_{bC}_{tr,e}^{+} layer of the junction. After performing analogous (as described in previous section) coordinate transformations and solving drift velocity equations for the bipolar (during drift time of electrons (_{P}

Here, _{h}^{0}_{0,h,mon}

In the case of the proceeded hole monopolar drift within n-base material (after the phase of the bipolar drift is finished), holes are always accelerated due to the acting space charge field. The hole drift with an increasing velocity determines an inherent shape of the increasing current density within a transient, during the monopolar drift phase.

The injected charge current can also be changed by carrier trapping and generation. The surface charge density dependence on time for the simple traps can be expressed as:

Introducing a trapping dependent dielectric relaxation time as:

Here, the time dependent quantities of _{Def}(t)_{e}(t)_{Def}_{Def}(t)_{Def0}_{C}

This equation should be properly matched with the drift kinetic equation:

The latter equation can only be solved numerically, although the general solution [

A few aspects of the impact of carrier trapping on the injected charge transients for a partially depleted junction layer have been mentioned in [_{dr}

No drift exists (_{e0}_{e}(t)

Carrier trapping, associated with a drifting surface charge domain, may determine the immediate (during time significantly shorter than other characteristic time parameters) and local changes of the effective charge. A decrease of _{Def}

Here, _{0}_{g,ef}

This simplified approach enables one to include into consideration the local charge generation. Unfortunately, the solutions can be obtained only by numerical analysis.

The discussed above simplified analysis of the components of carrier drift, trapping and thermal release enables one to make the rough estimations of the impact of different components. However, the rigorous consideration of processes should be based on the causality principle. The current changes can only appear during or after injection of _{e}_{e}_{e}(t

This specification leads to the integral-differential equations. These equations can only be analyzed numerically. Then, the above presented simplified models can be employed for the initial and qualitative prediction of the numerical solutions.

The drift velocity is varied through the electrostatic interaction of injected charge _{e}_{C}_{e}_{C}_{g}U

In the partially depleted junction layer, for _{FD}_{q}(t)_{e-h}(t)_{e}−X_{h}_{q}_{0,n&p}_{+}) in both layers of the junction, _{q},_{n}_{q},_{p}_{+}. The extracted excess holes are located at p^{+}-side producing the same value of the surface field. Thus, the overall charge balance _{p}^{-}_{A,p}_{+} = _{n}N^{+}_{D,n}_{h,p}_{0,p}_{+} = _{e,n}X_{0,n}

In the general case of the local injection of excess carrier pairs, separation of counter-partners depends on their densities. The external source induced charge _{e}_{Mq}_{e}_{Def}d_{e}/ε_{0}ε)d_{Sq}_{e}_{0}_{e}/ε_{0}ε)d_{Sq}

The reason is the excess carrier diffusion and appearance of the diffusion induced inner field [

This field is proportional to the excess carrier density gradients. For the locally generated domain of the nearly infinitesimal width (for instance in tracking of hadron path) within significantly wider inter-electrode gap of detector, diffusion due to a sharp gradient (which is also proportional to the carrier density) induces an inner electric field which balances a further widening of the domain. Strength of this field can be sufficient to compensate partially or fully the applied external field (surface charge _{a}_{D}_{D}_{m}

The space frequencies (

Here, _{1}_{D,h}/τ_{TOF,LD,h}_{2}_{D,e}/τ_{TOF,LD,e}_{D,h}, L_{D,e}_{D}_{1}^{2}D

A signal registration circuit (namely, load resistor) inevitably transforms the current transient shape. This appears due to the voltage sharing and the consequent change of a voltage drop on detector depending on current value within the circuit. In more general case, the transients are described by the solutions of the differential equation with variable coefficients, derived as:

This leads to a differential equation:

The changes of a system dynamic capacitance determine the initial delay and the final stage (relaxation) components within the simulated transient. These components are inevitable within the charge drift current transients, recorded in experiments. Also, these components should be included into the evaluation of the charge collection efficiency. Depending on the geometrical capacitance (_{g}_{L}) values, the current pulses are significantly modified.

The simplified models [

Adaptability of the simplified models presented above is additionally limited by several factors. A principal limitation leads to the threshold values of the acquired drift velocity that should be significantly less than those of the electric field (light) propagation velocity in the material under consideration, to ensure the validity of the electrostatic approach. This condition excludes the possibility to detect the primary charged particles (moving with relativistic velocities) within the inter-electrode spacing. Thus, the secondary particle (the electron-hole pairs with a zero initial drift velocity at the injection point) induced currents should be calibrated to the primary particle impact. The specific feature of the prevailing drift current caused by the monopolar charge domain is the increase of the drift current with time within the vertex of a current pulse. To separate the neutral domain (locally generated) into the drifting charge sub-domains, the sensitivity threshold for the applied voltage appears. This limitation leads to a condition of the elevated values of bias voltage, at least _{bi}_{dr}(X_{e})

In the analysis of the junction type detectors, the parabolic approach has been employed, which relates the applied voltage and the width of the depleted region, and it is routinely exploited in device physics [_{0}, w_{q}_{FD}_{D}_{0}_{λ}_{0}_{0,q,FD}_{TOF, Uλ}^{2}/μ_{e}U_{λ}

The transitional layers are actually inherent to the boundaries between the metallic electrodes and dielectric or external heavy doped layers of junctions. Owing to a short dielectric relaxation in the heavy doped layers, the semiconductor junction is preferential relative to a dielectric in between of electrodes.

In the Ramo's derivation of the charge drift current, it was clearly proved the reciprocity principle: the reversibility and equality of the mutual action and reaction of the charged electrode and drifting charge. This is based on the conservation of the charge (_{dr}_{e}_{e}_{e}), —during charge drift. The temporal changes of the surface charge on the electrode gives current density variations dependent on time (for a fixed external voltage), and this current density is generally expressed as _{e}/U)(dΦ/dX_{e})(dX_{e}/dt_{e}_{e}/εε_{0}_{q} and assuming that instantaneous _{q}_{e}_{e}Ψ_{e}_{0}, for its scalar representation, the expression for the current density is rearranged as _{e}(d^{2}/U)(q_{e}/dεε_{0})[d(X_{e}/d)/dt]_{e}(τ_{TOF,e}/τ_{M,q})[d(X_{e}/d)/dt]_{TOF,e}/τ_{M,q}_{TOF,e}_{M,q}_{σ}_{q} ≡ ▽·E_{q}_{q})_{e}_{0}_{q}_{σ}_{q} ≡ ▽·E_{q}_{q})_{e}_{0}_{E}_{q}_{σ}^{−1}. This result validates the equal action and re-action of the surface (_{Mq,e}_{TOF,e}_{Mq,e}_{TOF,e}_{dr}_{e}) = _{e}/

As it has been demonstrated above, the drift velocity _{dr}_{TOF,e}, τ_{M,q}_{M,Ndef}_{0}μU^{2}/^{3}_{dr}_{e} = _{e}/_{0})_{e}_{2}^{2})U_{Mq}_{dr}_{Mq})_{Mq}_{Mq}^{2})_{Mq}_{e}_{h}

In real detectors, the prevailing regime is the detection and collection of a small drifting charge, where _{Mq,e}_{TOF,e}_{Mq,e}_{TOF,e}_{e}_{e}_{e}_{ef}_{e}_{TOF,e}(t)_{ef}_{e1}_{Mq,e}/τ_{TOF,e}_{e2}_{Mq,e}/τ_{M,Ndef}_{ef,e1}_{Mq,e}/τ_{TOF,e})(U_{ef,e}/(U_{FD}))_{ef,e2}_{Mq,e}/τ_{M,Ndef})(U_{ef,e}/(U_{FD}))_{ef,e}_{e}d(Ψ_{e}^{0})^{1/2}_{0}_{e}^{0}_{ef,h1}_{Mq,h}_{TOF,h}_{ef,h}/(U − U_{FD}))_{ef,h2}_{Mq,h}_{M,Ndef}_{ef,h}/(U − U_{FD}))_{C,ef,h}_{h}d(Ψ_{h}^{0})^{1/3}_{0}_{e,h}_{ef}

As mentioned above, a vast variety of possible pulsed current transients, composed of drift, diffusion and displacement current components exists depending on the detector design and different external factors, such as the injected charge quantity, applied voltages, presence of traps,

The simulated specific transient shapes associated with different regimes of the injected charge drift are illustrated in

Variations of current transients due to the injected charge drift, observed in experiments, are illustrated in ^{+}nn^{+} structures. The reverse bias voltage was kept fixed with the rather moderate values _{FD}

The transients, —characteristic to the pulse durations controlled by the ambipolar diffusion lifetime, are illustrated in _{e}^{2}/Vs value of the electron mobility. The enhancement of the injected charge density, proportional to the _{ex}_{0} injected carrier concentration, leads to the increasing delay (of the rear kink in current transient) and to an increase of the current (proportional to _{e}_{D}^{2}^{2}_{a}_{a}^{2}/s using the measured _{D}

Variation of current transients measured at extremely small excitation densities (close to those possible to detect at a threshold sensitivity of the measurement system equipped with proper current amplifier) is illustrated in _{r}_{ex0}^{14} n/cm^{2}. The transient waveform inherent to the drift dominated current (curve 1 in

The models of the formation of the injected charge pulsed currents have been developed concerning the junction-type detectors. The partial and full depletion regimes have been analyzed. It has been shown, that, in junction detector, the drift time for the rather small density of the injected charge is shortened relatively to that of the capacitor-like detectors when a proper frame of reference (for comparison) is accepted and the characteristic relaxation times are matched. The description of the current pulse shape for the large injected charge drift in a finite area detector is coincident with that derived for the correlated drift (Ramo's-type) expressions. However, the induced currents obtained for the regimes of the small injected charge and of partial depletion lead to deviations from the Ramo's expressions. The analysis of the drift velocity field revealed the current increase within a vertex of the current pulse, for the monopolar drift regime. It has been shown, that presence of carrier traps considerably modifies the shape of the current transients. For the extremely large density of the injected charge _{g}U

This study was funded by the European Social Fund under the Global Grant measure project VP1-3.1-ŠMM-07-K-03-010.

The authors declare no conflict of interest.

_{eff}) in the space charge region of p-n junction detectors

(_{e}_{h}_{e}_{h}_{e}_{h}_{1} = E_{qh}_{σ}_{NDX}_{h}_{σ}_{eNDX}_{2} = E_{qe}_{σ}_{NDX}_{e}_{3} = E_{σ}_{NDX}_{e,h}_{e}_{h}_{0}_{0} is the unit ortho-vector in the uni-directional coordinate system; _{h}_{e}_{σ}_{L}_{0}_{q}(t)

(_{TOF,e}/q_{e}_{L}

Transients of the normalized current density _{TOF,e}/q_{e}_{C}

Transients of the normalized current density _{TOF,e}/q_{e}

(_{FD}