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Estimation of important properties of p-n junctions such as
reverse leakage current and capacitance is greatly facilitated by the
depletion approximation. Computation of the electrostatic potential and
electron and hole concentrations within this approximation are
commonplace in the microelectronics industry. The depletion
approximation requires appropriate and accurate boundary conditions.
Thus far, such reasonable boundary conditions have been widely applied
only for the case in which the p-type and n-type impurity concentrations
are spatially uniform. The author derives the solution of the depletion
approximation with appropriate boundary conditions for the case in which
the impurity concentration on one side of the diode decays exponentially
with distance. He plots the potential and charge density of this
exponential depletion approximation and compares these results to full
numerical solution of the semiconductor equations. The agreement between
the proposed approximation and the numerical solution validates this
approximation scheme

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... The magnitude of built-in (diffusion) electric field in the emitter E pr can reach the values compared with the electric field strength in the DL of ð n junction. The occurrence of the field which distribution is not restricted by an immediate vicinity of metallurgical boundary of the junction, as it takes place for a stepwise impurity profile, as well as high concentration of free (majority) carriers result in characteristic features SQO, 5(2), 2002 of structure of the space-charge region [22]. Indeed, since the Debye screening length in the emitter l Dp is rather small (l Dp << x 0 ), for its main part (≈ x 0 ) the quasineutrality condition is valid: ρ(x) = 0, E pr (x) = const, where ρ(x) is the charge density. ...

... As distinct from a stepwise (quasi-uniform) impurity distribution, in the considered case besides the main DL in the vicinity of metallurgical boundary of pn junction, there is an additional DL [22] at the boundary with uniform p + region of the emitter at x e ≤ x ≤ x 0 , with the extension of W e = x e x 0 (Fig. 1). Setting the field E(x e ) = 0, and the potential ϕ(x e ) = U bi , we find for the considered DL the field distribution E(x) = E pr (x+x e )/W e , the potential ϕ(x) = U bi + [(x+x e ) 2 /(2W e )], as well as the potential difference ∆ϕ e ≡ ϕ(x e ) ϕ(x 0 ) = W e /2, where W e = λ Dp 2 [here the Debye screening length is determined by maximum value of the acceptor concentration in the emitter λ Dp = (ε 0 k B T/4πq 2 N Am ) 1/2 ]. ...

A theoretical investigation of the influence of a nonuniform doping concentration to the temperature response curve of diode temperature sensors is presented, which is the first effort in this field important for diffused diodes used as the temperature sensors. The current-voltage characteristic, from which the temperature response curve can be obtained, has been calculated using the model of a one-dimensional exponentially graded pn junction with uniformly doped base region and the diffusion current of the minority carriers through the pn junction. We show that depending on the doping gradient both contributions to the current coming from the electron and hole current components appear to be of the same order of magnitude. That is in contrast to the prediction of the widely used asymmetrical step junction model. It follows from numerical calculations that an effective shift of the tempera-ture response curve due to nonuniformly doped emitter region in the temperature equivalent can reach the value of about 20 K. The limiting temperature T m in the temperature response curve that restricts its extent into the high temperature range has been analyzed depending on the excitation current, the doping concentration of the base, and the pn junction depth.

... Hence, k = D b /(ln(N b / N epi )) 1/2 . The depletion approach of an exponential graded PN junction [12] is used to calculate the potential and electric field distributions from the one-dimensional Poisson equation. This has been solved by taking into account the continuity of the electric field in each junction, by setting x = 0 at the collector N + P + junction and by assuming the V = 0 to be located in the P-base region between the two depletion regions. ...

... Hence, k = D b /(ln(N b / N epi )) 1/2 . The depletion approach of an exponential graded PN junction [12] is used to calculate the potential and electric field distributions from the one-dimensional Poisson equation. This has been solved by taking into account the continuity of the electric field in each junction, by setting x = 0 at the collector N + P + junction and by assuming the V = 0 to be located in the P-base region between the two depletion regions. ...

A quasi-analytical model addressed to predict the breakdown voltage in four-layer transient voltage suppressor (TVS) diodes based on the punch-through effect is reported in this paper. For breakdown voltage in excess of 1V, a closed form expression is derived. In addition, the three-layer TVS diode can also be described with the developed model. Finally, results obtained from the model are in good agreement with simulation and experimental data.

Recently, we developed a general laser control scheme, i.e., the Stark control of electrons at interfaces (SCELI), based on Stark shifts, that is able to manipulate the electron dynamics at material interfaces [A. J. Garzón-Ramírez and I. Franco, Phys. Rev. B 98, 121305 (2018)]. Here, we investigate how SCELI is influenced by the band bending effects introduced by interfacial dipoles and by the laser screening due to the polarization response of the material. For this, we follow the quantum dynamics of a model one-dimensional tight-binding semiconductor-semiconductor heterojunction driven by nonresonant few-cycle laser pulses of intermediate intensity. Band bending effects are introduced through an interfacial electrostatic potential term dictated by the depletion approximation of a neutral p−n junction. In turn, screening effects are captured through the general boundary conditions of the field vectors at interfaces dictated by Maxwell's equations. For field amplitudes where SCELI dominates (E0≤0.55 V/Å in the model), laser screening leads to a 46% reduction of the effect that can be partially compensated by increasing the intensity of the incident field. Surprisingly, band bending mildly affects SCELI. When both band bending and screening are considered simultaneously, the charge transfer is reduced, on average, 40% for E0≤0.46 V/Å. Overall, we observe that screening and band bending change the magnitude of SCELI but leave the underlying mechanism for electron transfer intact.

The hydrogenation effects on HgCdTe diode performance are presented and the mechanism of hydrogenation is revealed. By the
hydrogenation, R0A is increased by 30 times and photo-response is also improved. It is supposed that these are explained by the increased minority
carrier lifetime by the hydrogenation. However, it is found from LBIC measurements that the minority carrier lifetime doesn’t
increase by the hydrogenation. An important clue that explains the hydrogenation effects is found from Hall measurements.
It is found that, after the hydrogenation, the doping concentration of Hg-vacancy doped substrate decreases and the mobility
increases. For the heavily hydrogenated bulk substrate, it is also found that the hydrogen passivates the whole Hg-vacancy
and reveals the residual impurity and p-type doping concentration is exponentially graded. From these measurements, the diffusion
current model of gradually doped diode is proposed. This model shows that the diffusion current of the graded junction diode
is 2 orders of magnitude smaller than that of the abrupt junction diode, which clearly explains the R0A increase by the hydrogenation. Medicisimulation to investigate the change of LBIC signal by the doping grading also coincides
with the measurements. From these measurements and model, the hydrogenation effects are attributed to the grading of Hg-vacancy
doped p-type substrate by the diffused hydrogen.

It is important to study an exponential-constant p-n junction because it gives a realistic approximation for many diffused p-n junction profiles. To calculate the space-charge layer capacitance for this junction we use an abrupt space-charge edge approximation with a correction which includes the effect of the mobile carriers at the edges of the space-charge region. In this approach the offset voltage voff is used in place of the built-in potential as obtained from the depletion approximation. An analytical model for the space-charge region capacitance for an exponential-constant junction is developed. This model holds well for zero bias, for small forward voltages, and for reverse voltages. It shows good agreement when compared with the Chawla-Gummel model. It is simple and gives a direct relationship between the depletion capacitance and the applied voltage.

An analytical model for the spatial distribution of potential, electric field and carrier densities is presented, assuming uniform doping density and constant quasi-Fermi potentials in the direction of modeling. Starting from the current relations a differential equation respecting two-dimensional effects is developed and solved approximately by decomposition in three regions with different preconditions: (a) flatband region, (b) constant spacecharge density and (c) strong inversion or accumulation. Inaccuracies arise mainly from violated preconditions at the interfaces between the different regions. Two-dimensional effects are respected in the analytical model of (b) only but they influence region (c) by means of boundary conditions at the interface of (b) and (c).The investigations of this paper are focused to region (c). This part of our model is identical to the model of Hauser and Littlejohn[1]. They integrated a simplified form of the semiconductor-Poisson equation twice but were restricted to one-dimensional applications and thermal equilibrium. Our derivation allows for large electric fields and current densities perpendicular to the direction of modeling. From analytical considerations confirmed by numerical experiments we suggest preference to the gradual channel condition , with x and y defined according to Fig. 1. This definition allows the potentials to vary significantly with respect to y. We found that this holds for the quasi-Fermi potentials also. Furthermore an empirical model for the maximum width of strong-inversion layers in uniformly doped silicon films is presented in eqn (46).

A model of the collector-base depletion layer is worked out. Doping profiles of base and epitaxial layer to substrate transition are approximated by exponential functions. The depletion approximation is used. The model provides relations between transistor doping profiles and collector-base capacitance as a function of bias voltage and collector current density. Higher order derivatives of the capacitance with respect to bias voltage and collector current are calculated and their influence on third order harmonic distortion is shown. Relations between transistor doping profiles and third order distortion are evaluated. Since the model is essentially one-dimensional it is only valid at low and intermediate collector current densities.

The contribution of the graded region of implanted p-n junctions is analysed using an exponential profile. Though previously neglected, the authors have recently shown that this contribution to the saturation current of HgCdTe diodes is significant. Assuming a dominant Auger recombination, an analytical solution to the continuity equation is obtained. An expression for the current generated by the graded region is presented for both ohmic and reflecting boundary conditions. A revised condition for a 'wide' region is derived. When the region is 'narrow', the current differs drastically from that of the zero-gradient case. The effects of the junction depth and the substrate and surface concentrations on the current are investigated. It is shown that the reverse current does not saturate.

Excess carrier distribution and saturation current generated by a graded p‐n junction are investigated, approximating the dopant profile by an exponential function. Analytical solutions of the steady‐state continuity equation are presented for lifetime dominated by Auger, radiative, or Shockley–Read recombination mechanisms, for both ohmic and electrically reflecting boundary conditions. The saturation current generated by the graded region is derived for each of the dominant mechanisms and boundary conditions. Unlike previous published works the result of this analysis is a set of concise analytical expressions that make them a useful tool for simulation and investigation of graded structures such as silicon solar cells. The results are calculated for HgCdTe diodes, demonstrating the effect of the profile slope as determined by surface and bulk concentrations and junction depth. It is shown that the contribution of the graded region to the saturation current, as compared to that of the substrate, is significant, and that by proper selection of the diode parameters it is possible to substantially reduce this contribution.

In a graded junction, the formalism for handling reflecting boundary conditions must be modified. Since a significant drift term is present, zero recombination velocity at the surface does not imply a zero excess carrier gradient but rather zero overall flux. A model for analyzing p‐n junctions fabricated by implantation or diffusion is presented, assuming the dominant recombination mechanism in the graded region is Auger. The model enables optimization of diode design. By proper selection of parameters, mainly by reducing surface concentration or by increasing the steepness of the dopant profile, it is possible to drastically reduce the saturation current generated by the graded region.

A model is presented for the stored collector charge in a bipolar transistor. Exponential functions are used to describe base and substrate doping. The model takes account of space charge regions as well as injection regions. Furthermore, it includes dissipation losses in the neutral collector, the Kirk effect and built-in electric fields in the neutral regions. Smooth transistions are made between the various space charge and injection modes. The model provides a direct relation between doping profiles, bias conditions and stored collector charge. The model is applied to the problem of signal distortion, to which stored collector charge is a major contributor, especially at low voltages and high currents. Distortion measurements are performed on a transistor in a test circuit. To describe the linear parts of the transistor crude estimates of relevant transistor parameters like e.g. Early voltage and saturation current are made. Therefore a simple neutral base model is used. Finally, circuit analysis has to be performed to calculate distortion. This is done using an almost periodic Fourier transform in order to calculate intermodulation products arising from two input signals to the circuit. Good agreement between model and experiment is shown. The large influence of the stored charge on distortion is demonstrated. The model is suited for predicting distortion properties of new designs as well as for assessing the effects of process spreading.

Experimental results show that the contribution of the graded region to the current of Hg(1-x)Cd(x)Te diodes is not negligible, as compared to that of the p type bulk. The theoretical analysis reveals the influence of the electric field present outside the depletion region on the current generated by the graded region. The analysis shows the importance of the lifetime profile in the graded region, which is a function of the specific recombination mechanism and its dependence on the local dopant concentration. The effect of parameters such as substrate concentration, surface concentration, and junction depth on this current is discussed.

A quasi-analytical model addressed to predict the breakdown voltage in four-layer TVs diodes with Gaussian epitaxial profile is developed for the first time in this work. The model yields the breakdown voltage value in terms of technological and/or geometrical device parameters, being suitable for cases where the punch-through takes place before the avalanche breakdown. For breakdown voltages in excess of 3 V, a closed form expression can be inferred, simplifying the quasianalytical model. In addition, the existent three-layer structure model is obtained when proper boundaries are included in the proposed model. Analytical results are in satisfactory agreement with the simulation and experimental data.

A new approach to analyze the current-voltage (I-V) and 1 MHz
capacitance-voltage (C-V) characteristics of a non-abrupt p/n-InP
epitaxial junction was developed. The theoretical model took into
account the diffusion of shallow-dopant impurities during the
fabrication process and also provide a non-destructive way to determine
shallow-dopant profiles in p/n-semiconductor epitaxial junctions. Our
results suggest that the I-V and C-V characteristics are independent of
the diffusion of shallow-dopant impurities in the n-side for a n/p-InP
epitaxial junction with a donor impurity concentration N<sub>A</sub>=10
<sup>18</sup> cm<sup>-3</sup>. Moreover, for an acceptor impurity
concentration is N<sub>D</sub>⩾10<sup>-3</sup> N<sub>A</sub>, the
C-V characteristics can not be analyzed in terms of the Mott-Schottky
equation. The verification for our theory was done by comparing the
calculated values of the capacitance at 1 MHz and electrical current for
a n/p-InP epitaxial junction with those reported in the literature and
extracted from the I-V data. Excellent agreement between these values
strongly support the validity of our theoretical expression for the I-V
and 1 MHz C-V characteristics

A detailed analysis of the depletion layer of exponentially graded p-n junctions is presented, which improves previous efforts in this field in different aspects. (1) We obtain analytical expressions for the depletion layer widths which are nearly as simple as those given by the widely used step and linearly graded junction models, but by far more accurate. In addition, we derive simple closed-form expressions for the punch-through and reach-through voltages. (2) The analysis is extended to the ease of substrates with nonuniform doping profiles, a case rarely treated in the literature, but of great importance for the design of many devices.

For a graded p-n junction under sufficient forward bias, the usual space-charge approximation to the potential breaks down, and a numerical solution of the differential equation satisfied by the potential is required. A procedure is described which avoids the difficulties associated with direct numerical integration of the stiff differential equation, and which yields a pair of very close upper and lower bounds to the potential at all points. For a linearly graded junction, tables of bounds are given which nowhere differ by as much as 1 per cent, and which effectively bridge the gap between the space-charge case and the neutral case.
The computer solutions are used to calculate the voltage dependence of the stored charge (low-frequency ac capacitance) of a graded junction numerically, as a function of the bias voltage across the junction. The expression for the capacitance is split into two parts, one of which dominates in the neutral case and the other in the space-charge case. With properly normalized variables, it is possible to give a universal plot of small-signal ac capacitance against applied voltage. The results differ from the usual approximate formulas by amounts ranging up to nearly 10 per cent.

The exponential p-n junction shows some peculiarities, which make its theoretical study interesting. In this paper some characteristics of the transition layer of such a junction are analysed, taking into account the mobile carriers in this region. Following a method introduced by Sab, the Poisson-Boltzmann equation is linearized using a parameter α, which is a measure of the relative importance of the fixed, ionized impurity space charge compared with the mobile carrier charge in the transition layer of the p-n junction. The width of the transition layer, on the left and on the right of the junction, the built-in voltage, the total potential difference across the transition region and the total capacitance are derived. The d.c. theory of the junction capacitance is compared with experimental data obtained on implanted silicon junctions : ft satisfactory agreement is found.

In a single crystal of semiconductor the impurity concentration may vary from p-type to n-type producing a mechanically continuous rectifying junction. The theory of potential distribution and rectification for p-n junctions is developed with emphasis on germanium. The currents across the junction are carried by the diffusion of holes in n-type material and electrons in p-type material, resulting in an admittance for a simple case varying as (1 + iωτp)1/2 where τp is the lifetime of a hole in the n-region. Contact potentials across p-n junctions, carrying no current, may develop when hole or electron injection occurs. The principles and theory of a p-n-p transistor are described.

Space‐charge, field, and potential distributions in inhomogeneously doped crystals are determined by a nonlinear differential equation which requires numerical computation in order to be solved. By means of numerical computation an attempt is made to obtain the potential, field, and space charge throughout the crystal. For exponential impurity distributions bounding curves can be obtained which allow one to obtain the complete distributions. A few examples are computed which demonstrate the difficulties encountered in obtaining solutions.

The standard physical model by which p‐n junctions in semiconductors are generally analyzed is shown to need significant revisions when applied to strongly asymmetrical junctions, such as commonly used in diffused silicon transistor emitters and solar cells. The potential, field, and space‐charge density have much wider spatial distributions on the heavily doped side than generally thought, thus, in effect, widening the ’’junction region’’ there. In addition, Auger processes on that side can reduce the minority‐carrier lifetime sufficiently to cause recombination without defects in the widened space‐charge region and a consequent excess junction saturation current not previously recognized.

This paper presents a numerical analysis of the mechanisms of operation within a linearly-graded p-n junction. Considered in this analysis are three important modes of junction operation: equilibrium, forward bias, and reverse bias in the collector junction. In addition, calculations of electrical space-charge layer capacitance are presented for the forward-biased linearly-graded junction. The conclusions derived are compared, in graphical form, with the results of previous investigations of the linearly-graded junction.

A study of the depletion-layer characteristics of double-diffused p-n junctions, formed by successive diffusion of opposite type impurities into a semiconductor, is presented. The general form of the impurity profile is taken to be N(r) = NSA e−ar(ar+2k) where NS, A, a and k are constants and r is the distance. This general form reduces to Gaussian, erfc and two-step diffusion profiles as particular cases. Results are shown graphically for typical values of these constants.It is shown that the double-diffused junctions can be well approximated by equivalent double-exponential profiles in the impurity range of practical interest and C-V relations are obtained analytically. The results obtained are convenient for ready engineering calculations. The accuracy and the range of validity of the approximation are discussed.A simple and accurate analytical method is outlined for the calculation of reverse-biased sidewall capacitance of double-diffused structures, with special reference to the emitter junction of a planar transistor. It is shown that by approximating the impurity profiles by Gaussian distributions, the metallurgical junction formed is given by an ellipse and this leads to a simple evaluation of the sidewall capacitance. An expression obtained for the impurity gradient along the sidewall enables the computation of forward-biased depletion-layer capacitance when the mobile carriers in the depletion region are to be considered.

The fundamental one-dimensional transport equations, applied to a single-junction device in steady state, are solved `exactlyÂ¿ on a digital computer (IBM 7094/7040). The relevant internal distributions and terminal properties are obtained, without approximations, to the desired generality and accuracy. An efficient numerical alogrithm limits to approximately 1min the computation time for one set of accurate solutions.

The classical capacitance-voltage relations based on abrupt space-charge edge approximations, while adequate at large reverse bias, do not adequately describe the capacitance near zero bias. This paper presents explicit capacitance-voltage relations valid near zero bias for linearly graded and exponential-constant profiles. For linearly graded junctions, the intercept in a 1/C<sup>3</sup>versus voltage plot is shown to be well approximated by the "gradient voltage" defined by V g = 2/3 kT/q ln a<sup>2</sup>εkT/q/8qn i <sup>3</sup>. Also presented is an accurate numerical technique for machine computation of the transition region capacitance for any doping profile. Explicit relations obtained by dimensional considerations and curve fitting on numerical solutions are free of singularities, hence useful in computer-aided device design and doping profile determination.

A numerical iterative method of solution of the one-dimensional basic two-carrier transport equations describing the behavior of semiconductor junctions under both steady-state and transient conditions is presented. The method is of a very general character: none of the conventional assumptions and restrictions are introduced and freedom is available in the choice of the doping profile, recombination-generation law, mobility dependencies, injection level, and boundary conditions applied solely at the external contacts. For a specified arbitrary input signal of either current or voltage as a function of time, the solution yields terminal properties and all the quantities of interest in the interior of the device (such as carrier densities, electric field, electrostatic potential, particle and displacement currents) as functions of both position and time.

A self-consistent iterative scheme for the numerical calculation of dc potentials and currents in a one-dimensional transistor model is presented. Boundary conditions are applied only at points representing contacts. Input data are: doping profile, parameters governing excess carrier recombination, parameters describing the dependence of mobility on doping and on electric field, applied emitter and collector voltages, and a trial solution for the electrostatic potential. The major limitation of the present approach results from use of Boltzmann rather than Fermi statistics. Convergence of the iteration scheme is good for low and moderate injection levels.

Theory and Practice of Microelectronics

- S K Ghandhi

S. K. Ghandhi, Theory and Practice of Microelectronics. New York:

Theory of p-n junctions in semiconductors and p-n junction transistors

- W Schockley

Handbook of Mathematical Functions

- M Abramowitz
- I Stegun