Depletion Approximation Research Articles

Identification of the free boundary of a space-charge region with the aid of a Laplacian potential

The possibility of identifying the free boundary of the space-charge region around a conductor, without going through the solution of the related Poisson’s problem, is discussed in the depletion approximation. The use of Green’s second theorem leads to an integral identity that involves this boundary and a Laplacian potential for the same configuration. The identity is applied to a spherical and a right-angle concave conductor. For the sphere, the method yields the exact solution; in the wedge case, with some additional conditions, an approximate boundary identification is obtained, which is in good agreement with published numerical results.

An Improved Interpretation of Depletion Approximation in p–n-Junctions

The conventional treatment of depletion approximation in p-n-junctions often leaves a student with an erroneous impression that the approximation essentially involves complete neglect of the fraction of the space charge region (SCR) where charge density makes a transition to zero. This work describes a simple analytical model for clarifying the relationship of depletion approximation to the SCR.

Modelling of the dynamic threshold MOSFET

Despite the inappropriateness of the depletion approximation when the source-substrate junction is slightly forward biased, conventional SPICE models are being used to simulate digital integrated circuits implemented with dynamic threshold voltage MOSFETs (DTMOS). Based on PISCES simulations and experimental data, the correct modelling of DTMOS is discussed in this work. It is found that the dependence of the threshold voltage on the source-substrate forward bias can differ importantly from the conventional model at voltages above ∼0.4 V for long channel technologies, but it could agree with conventional models for voltages as high as ∼0.5 V for deep submicron technologies. This is explained in terms of the difference in substrate doping levels between such technologies, which is at least one order of magnitude. Mobility and transverse electric field always improve in the dynamic operation. However, the conventional models used by SPICE to calculate these two parameters, which are based on the depletion approximation, lead to important miscalculations. This is proven using the PISCES numerical solution of the Poisson equation considering the presence of mobile charge in the depletion region. With these results, some guidelines for the correct modelling of this device are generated.

AC measurements and modeling of WO 3 thick film gas sensors

Polycrystalline WO 3 films were fabricated through thick film technology. Stimulated temperature measurements showed that the intergranular Schottky barrier is not high enough to apply the depletion approximation to evaluate the fundamental parameters related to the sensing properties. Therefore, a semi-analytical approach to the Poisson's equation with the complete charge density was chosen. As a result, we could estimate the dependence on frequency of both apparent permittivity and depletion region width by impedance spectroscopy.

Average Drift Mobility and Apparent Sheet-Electron Density Profiles in Strained-Si–SiGe Buried-Channel Depletion-Mode n-MOSFETs

In this paper, we describe a simple method to extract the average drift mobility and the apparent sheet electron density versus the applied gate voltage and the vertical effective electric field in strained-Si-SiGe buried-channel depletion-mode metal-oxide semiconductor field-effect transistors (n-MOSFETs). For this, we adapted an established technique used in evaluating mobility profiles in Schottky-gate MESFETs, by taking into account the existence in our devices of the gate-oxide capacitance C/sub ox/ and by introducing an effective junction capacitance C/sub D/, which follows the ideal Schottky's depletion approximation. By applying our method on fabricated transistors we were able to obtain the average drift mobility profile versus the applied vertical effective field and monitor values as high as 618 cm/sup 2//Vs. We also extracted the apparent sheet electron density profile with values reaching as high as 3.4 /spl times/ 10/sup 12/ cm/sup -2/. Although the layer design had not been optimized, the results show mobility enhancement in the strained silicon channel and, to our view, point to a unique regime of operation for these devices, which should benefit the low-power and low-voltage applications. The proposed method could be used as a nondestructive tool for monitoring the transport properties in Si-SiGe modulation-doped MOSFETs. It could also serve as a useful platform for determining explicit modeling links between the layer design and the device performance.

Effect of non-abrupt doping profiles on the carrier sheet density in one-side modulation-doped GaAs/AlGaAs quantum wells

A theoretical study on the influence of non-abrupt doping profiles on the electronic subband structure in one-side modulation-doped GaAs/AlGaAs quantum wells with graded interfaces is presented. Emphasis is given on the dependence of the carriers sheet density on both the spacer thickness and the donors diffusion length. A proper discretization method for solving the coupled Schrodinger and Poisson equations self-consistently over a nonuniform grid is proposed. This furnishes numerically stable results for the electron sheet density even when non-abrupt doping and heterointerface profiles are included in the model calculation. The depletion approximation is surpassed and a significant increase in the electron sheet density is obtained when non-abrupt doping profiles are taken into account. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Approximate analytical solution to the space charge problem in nanosized Schottky diodes

An analysis is given of the space charge region that is induced in a semiconductor by a circular Schottky contact. Using the depletion approximation, the resulting free-boundary problem for Poisson’s equation is formulated and approximately solved in oblate spheroidal coordinates. Expressions are derived for the boundary of the space charge region and the related depletion potential. Calculations of the thickness of the Schottky barrier as a function of the diode size, down to the nanometer range, show good agreement with published results obtained numerically.

Out-of-well carrier screening in a strainedInxGa1−xN/GaNmultiple quantum well structure

Steady state and time-resolved differential transmission spectroscopy in a strained In xGa12xN/GaN multiple quantum well structure embedded within a p-i-n structure has been studied to identify the dominant screening mechanisms and measure the carrier sweep-out time. The results are consistent with a long-range out-of-well carrier screening where the photoexcited carriers escape the wells and separate, creating a space-charge field that induces an increase of the in-well field and shifts the excitonic transition to lower energies. A redshift of excitonic resonance under photoexcitation suggests that the direction of in-well field is opposite to that in the barriers due to the piezoelectric polarization in this strained In xGa12xN well. Compared with GaAs multiple quantum wells, a long carrier sweep-out time of 650 ps was observed in this structure. DOI: 10.1103/PhysRevB.68.233304 PACS number~s!: 78.67.De, 71.35.2y, 78.47.1p Time-resolved differential transmission measurements on GaAs multiple quantum wells ~MQW’s! embedded within a p-i-n structure have shown that the photoexcited electronhole pairs in the quantum well ~QW! under the influence of the electric field escape the well and travel in opposite directions to produce a space-charge field which opposes the built-in field. 1‐ 4 This out-of-well carrier screening mechanism, through the quantum-confined Stark effect ~QCSE!, shifts the excitonic resonance and modifies the absorption of the QW. Measurements of the change in the absorption as a function of time delay and wavelength provide a unique way to determine the dynamics of perpendicular carrier transport and carrier sweep-out times in p-i(MQW)-n structures. It is well known that hexagonal wurtzite III-nitride semiconductors have pronounced piezoelectric constant. 5,6 For InxGa12xN/GaN QW’s, the strength of the in-well electric field generated by spontaneous polarization and piezoelectric polarization can be as large as several MV/cm. Several groups have attributed the blueshift of the photoluminescence peak energy in undoped InxGa12xN/GaN QW’s with increasing carrier injection to the reduction of the QCSE due to in-well carrier screening. 7,8 Furthermore, built-in electric fields due to the p-n junction are present in these p-i(MQW)-n structures. Therefore, it is of interest to determine if the dominant carrier screening mechanism is in-well or long range out-of-well in such structures. A clear understanding of these carrier dynamics can provide great value for the design and optimization of InxGa12xN based emitters as well as for the optimization of material growth conditions. More importantly, such a study is expected to determine the carrier sweep-out times and provide insight into the potential profile in the strained InxGa12xN/GaN MQW’s. In particular, the sweep-out rate, a key property that limits the response time of nitride based p-i-n photodetectors, can be determined. In this Brief Report, we report experimental results on steady state and time-resolved differential transmission measurements of an InxGa12xN/GaN p-i(MQW)-n structure at room temperature. The differential transmission spectra are used to identify the dominant screening mechanisms and simultaneously to estimate the spatial band potential profile. A redshift of excitonic resonance immediately following pulsed excitation and a carrier sweep-out time of 650 ps were observed, indicating that photoexcited carriers sweep out of the well and screen the entire MQW structure. The increase of the in-well field under photoexcitation indicates that the direction of the in-well field is opposite to that in the barriers due to the piezoelectric polarization in the strained InxGa12xN wells. The sample described in this work was grown using metalorganic vapor phase epitaxy. Initially, a 4 mm thick GaN:Si layer was deposited on a c-face sapphire substrate. This layer was followed by the growth of ten periods of 7 nm thick GaN barriers and 2.5 nm thick In0.2Ga0.8N wells. The last QW was capped with a GaN barrier region followed by a 0.2 mm thick GaN:Mg p-contact layer. In order to determine the magnitude of the field distribution, we solved the Poisson equation for a p-i(MQW)-n structure within the depletion approximation. Neglecting the screening effects of background carriers, the net electric field within the wells ~barriers! Ew (Eb )i s

Modeling the performance of solar cells with and without the depletion approximation : Abenante, L. Solar Energy Materials and Solar Cells, 2002, 73, (2), 109–116

Modeling the performance of solar cells with and without the depletion approximation : Abenante, L. Solar Energy Materials and Solar Cells, 2002, 73, (2), 109–116

Effect of residual acceptors on electron mobility in single asymmetric quantum wells

We present the results of the calculation of the low-temperature electron mobility in one-side modulation-doped single asymmetric quantum wells, taking into account the presence of residual impurities. The effects of ionized acceptors were included in the variational calculation within the depletion approximation. All relevant scattering rates were taken into account, namely acoustic phonons, ionized impurities (remote and background), alloy disorder and interface roughness. The mobility dependencies on the electron gas density and residual acceptor concentration were found and we show that the background impurity scattering rate is the main scattering mechanism for acceptor concentrations above ∼3×10 15 cm −3 , in good agreement to recent experimental works involving high-quality GaAs/AlGaAs modulation-doped quantum wells.

Implant dose monitoring by MOS C– V measurement

The most critical parameter for deep sub-micron MOS field effect transistors is the threshold voltage, which is highly dependent on processing specifically, the ion implanted channel dose. Monitoring the channel doping on product wafers is highly desirable and is a major issue for process engineers. MOS C– V methods are widely used for process ramp up and monitoring and MOS C– V doping profiling is an introduced method for monitoring of low dose implants. However, the failure of the depletion approximation in the near surface region implies that conventional MOS C– V measurements yield erroneous doping profiles in that region. Integrating MOS C– V doping profiles yields only a partial implant dose excluding the important near surface dose portion. Here, we report a new approach, which enables the determination of the entire implant dose, taking into account the crucial surface region. Moreover, the MOS threshold voltage can be obtained self-consistently. The method is also applicable to MOS structures with ultra thin gate oxides.

Effects of Doping Profile on Characteristics of InAs Quantum Dots

Capacitance-voltage measurements were carried out to investigate the effects of the doping profile on characteristics of self-assembled InAs quantum dots. Using this technique, we observed features of zero-dimensional electron confinement indicating the presence of quantum dots. However, the number of confined states differed depending on the doping profile, and this fact was confirmed by photoluminescence measurements. The equation in the depletion approximation led us to calculate the distribution of carriers as a function of depth from the sample surface, and the results are in agreement with the depth of the quantum dot layer.

Modeling the performance of solar cells with and without the depletion approximation

The measured internal spectral response and open-circuit voltage of two silicon solar cells were modeled with the simulation program PC1D version 4.4 by both applying and not applying the depletion approximation (DA). A comparison between the results from the two modeling approaches shows that dividing the device under exam into regions according to the DA is preferable in order to get both modeling reliability and physical insight into the device operation. Not using the DA leads in fact to both overestimate the bulk lifetime and neglect the optical properties of highly doped regions.

On the high-temperature subthreshold slope of thin-film SOI MOSFETs

This paper addresses the validity of the classical expression for the subthreshold swing (S) in SOI metal-oxide semiconductor field effect transistors (MOSFETs) at high temperature. Using numerical simulation, it is shown that two effects invalidate the classical expression of S at high temperature. Firstly, the depletion approximation becomes invalid and intrinsic free carriers must be taken into account to determine the effective body capacitance. Secondly, the charge-sheet model for the inversion layer becomes inaccurate due to a lowering of the electric field at the surface and a broadening of the inversion layer thickness in weak inversion. These effects must be taken into account to predict accurately the high-temperature subthreshold characteristics of both partially depleted and fully depleted SOI MOSFETs.

Model for Schottky barrier and surface states in nanostructured n-type semiconductors

A semiclassical model for Schottky contacts to be applied to nanosized polycrystalline n-type semiconductors was developed. To this purpose we determined the density of surface states as a function of the mean grain radius, which establishes the Schottky barrier height. The intergranular potential shape was investigated in depletion approximation under spherical geometry and a critical revision of this method was proposed. The model was then extended to also include nanostructured materials, which could not be considered in the previous approach. Thus we were able to explain the flattening of the band bending and the decrease in the surface state density, which are experimentally observed when the granulometry is very fine.

Apparent band-gap narrowing doping functions for silicon in the Dhariwal and Ojha's form facilitating solar cell modeling

It is shown that reported doping functions for apparent band-gap narrowing (ABGN) in silicon that are written in the Slotboom and De Graaff's form (Slotboom, De Graaff, Solid-State Electron. 19 (1976) 857) can be set in the Dhariwal and Ojha's form (Dhariwal, Ojha, Solid-State Electron. 25 (1982) 909) at all dopings without any significant deviation in their capability of fitting the experimental data. The use of the Dhariwal and Ojha's form is preferable because it allows for a straightforward application of the depletion approximation in the non-uniformly doped regions of silicon devices like ion-implanted and diffused solar cells, facilitating the device performance modeling and analysis. This is demonstrated by an application to the internal spectral responses of two ion-implanted silicon solar cells and a comparison with a modeling method which does not use the depletion approximation.

Validation of the small-signal model of a forward-biased p–n junction diode

Small-signal parameters of an abrupt p–n junction diode under forward bias are computed numerically by solving Poisson's equation and continuity equations together. The parameter values are compared with those obtained with analytical expressions. It is found that the depletion capacitance is underestimated when the depletion approximation is used. The diffusion conductance and capacitance values agree with theory only in a limited range of forward bias voltages, and this range becomes smaller as the minority carrier lifetime decreases. The reasons behind the discrepancy are discussed.

On the precise measurement of the wavevector mismatch in quadratic nonlinear crystals

The exact determination of wavevector mismatch is a crucial issue for many modern nonlinear optical applications such as all-optical signal processing, beam shaping and spatial quadratic solitons. We extend an approach to measure the wavevector mismatch in quadratic nonlinear crystals towards moderate conversion efficiencies where the low pump depletion approximation is invalid. An approximate analytical formula derived for the mismatch is experimentally verified in a KTiOPO4 crystal cut for type-II phase matching.

Adatom lifetime in film growth at solid surfaces in the framework of the Johnson–Mehl–Avrami–Kolmogorov model

On the basis of the quasi-static approximation and for simultaneous nucleation the adatom lifetime, τ, during film growth at solid surfaces has been computed by Monte Carlo (MC) simulation. The quantity DN 0 τ, N 0 and D being respectively the cluster density and the adatom diffusion coefficient, is found to depend upon the portion of surface covered by clusters and, very weakly, on N 0. Moreover, a stochastic approach based on the Johnson–Mehl–Avrami–Kolmogorov (JMAK) theory has been developed to obtain the analytical expression of the MC curve. The collision factor of the mean island has been calculated and compared with those previously obtained from the uniform depletion approximation and the lattice approximation.

Electric-field distribution in Au–semi-insulating GaAs contact investigated by positron-lifetime technique

Positron-lifetime spectroscopy has been used to investigate the electric-field distribution occurring at the Au--semi-insulating GaAs interface. Positrons implanted from a ${}^{22}\mathrm{Na}$ source and drifted back to the interface are detected through their characteristic lifetime at interface traps. The relative intensity of this fraction of interface-trapped positrons reveals that the field strength in the depletion region saturates at applied biases above 50 V, an observation that cannot be reconciled with a simple depletion approximation model. The data, are, however, shown to be fully consistent with recent direct electric-field measurements and the theoretical model proposed by McGregor et al. [J. Appl. Phys. 75, 7910 (1994)] of an enhanced ${\mathrm{EL}2}^{+}$ electron-capture cross section above a critical electric field that causes a dramatic reduction of the depletion region's net charge density. Two theoretically derived electric field profiles, together with an experimentally based profile, are used to estimate a positron mobility of $\ensuremath{\sim}95\ifmmode\pm\else\textpm\fi{}35{\mathrm{cm}}^{2}{\mathrm{}\mathrm{V}}^{\mathrm{\ensuremath{-}}1}{\mathrm{}\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$ under the saturation field. This value is higher than previous experiments would suggest, and reasons for this effect are discussed.