Lucky Drift Research Articles

Impact Ionization Rate of Electrons in Bilayer Graphene Nanoribbons

An analytical model is developed for calculating the electric field-dependent electron ionization rate in bilayer graphene nanoribbons. All probable arrangements of optical phonon and inter-electron scattering phenomena before the occurrence of an ionizing collision event have been considered in this model, which is viewed as the multistage scattering model of impact ionization. This model considers both lucky ballistic and lucky drift electrons moving under an applied electric field. The numerical results calculated by using this model are compared with the results obtained from an earlier developed analytical model.

One‐dimensional lucky‐drift model with scattering and movement asymmetries for impact ionization in amorphous semiconductors

AbstractA lucky‐drift (LD) model for impact ionization has been recently successfully used to account for avalanche phenomenon in amorphous semiconductors. This model however leads to the dependence of the impact ionization coefficient on the sample thickness. However such dependence has not been confirmed experimentally. Recently LD model has been improved taking into account the scattering and movement asymmetry of charge carriers in the applied electric field. As a result, the impact ionization coefficient was obtained independent on the sample thickness. We apply the improved LD model to study the field dependence of the impact ionization coefficient in a‐Se, a‐Si:H and Ge2Sb2Te5. We show that even in one‐dimensional formulation of the improved LD model the agreement between theoretical results and experimental data evidenced in a‐Se is better than that in the formulation with scattering and movement symmetry. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Avalanche multiplication in amorphous selenium and its utilization in imaging

Amorphous selenium (a-Se) is a well known photoconductor that is currently used in X-ray image detectors and HARP video tubes. Recent advances have made it practical to operate a-Se at extremely high electric fields F, where avalanche multiplication occurs. At sufficiently high fields, the effective quantum efficiency (η∗) (or the overall yield) of the photoconductor can be increased several orders of magnitude above unity and renders avalanche a-Se photoconductors a prospective alternative to vacuum photomultiplier tubes (PMTs) and silicon avalanche photodiodes (APDs). In this work we report our study of η∗ and charge transport for a-Se avalanche photoconductors with different photoconductive layer thicknesses in wide range of F. Our study shows that a-Se is able to produce a gain of ∼1000 with a rise time of ∼1ns, both of which clearly point to the potential (or realized) use of this photoconductor in a variety of imaging applications. Furthermore, our work supports the validity of the so-called modified Lucky drift model to explain the nature of impact ionization and avalanche in this material.

Lucky drift impact ionization in amorphous semiconductors

The review of avalanche multiplication experiments clearly confirms the existence of the impact ionization effect in this class of semiconductors. The semilogarithmic plot of the impact ionization coefficient (α) versus the reciprocal field (1∕F) for holes in a-Se and electrons in a-Se and a-Si:H places the avalanche multiplication phenomena in amorphous semiconductors at much higher fields than those typically reported for crystalline semiconductors with comparable bandgaps. Furthermore, in contrast to well established concepts for crystalline semiconductors, the impact ionization coefficient in a-Se increases with increasing temperature. The McKenzie and Burt [S. McKenzie and M. G. Burt, J. Phys. C 19, 1959 (1986)] version of Ridley’s lucky drift (LD) model [B. K. Ridley, J. Phys. C 16, 3373 (1988)] has been applied to impact ionization coefficient versus field data for holes and electrons in a-Se and electrons in a-Si:H. We have extracted the electron impact ionization coefficient versus field (αe vs F) data for a-Si:H from the multiplication versus F and photocurrent versus F data recently reported by M. Akiyama, M. Hanada, H. Takao, K. Sawada, and M. Ishida, Jpn. J. Appl. Phys.41, 2552 (2002). Provided that one accepts the basic assumption of the Ridley LD model that the momentum relaxation rate is faster than the energy relaxation rate, the model can satisfactorily account for impact ionization in amorphous semiconductors even with ionizing excitation across the bandgap, EI=Eg. If λ is the mean free path associated with momentum relaxing collisions and λE is the energy relaxation length associated with energy relaxing collisions, than the LD model requires λE>λ. The application of the LD model with energy and field independent λE to a-Se leads to ionization threshold energies EI that are quite small, less than Eg∕2, and requires the possible but improbable ionization of localized states. By making λE=λE(E,F) energy and field dependent, we were able to obtain excellent fits to α vs 1∕F data for both holes and electrons in a-Se for both EI=Eg∕2 and EI=Eg. In the former case, one expects occupied localized states at EF(=Eg∕2) to be ionized and in the second case, one expects excitation across the bandgap. We propose that ionization excitation to localized tail states very close to the transport band can explain the thermally activated α since the release time for the observed activation energies is much shorter than the typical transit times at avalanche fields. For the a-Se case, EI=Eg≈2eV leads to the following conclusions: (a) For holes, λE has negligibly little field dependence but increases with energy. At the ionization threshold energy λE∼4nm. (b) For electrons, λE increases with energy and the field with λE∼2nm at the ionization threshold and at impact ionization fields. For electron impact ionization in a-Si:H, the data can be readily interpreted in terms of near bandgap ionization EI=Eg and a λE that decreases with increasing field, and having very little energy dependence. The energy relaxation length has opposite tendencies in a-Se and a-Si:H, which probably reflects the distinctly different types of behavior of hot carriers in the transport band in these two amorphous semiconductors.

Lucky drift model for non-local impact ionisation incorporating a soft threshold energy

The lucky drift model of impact ionisation is extended to consider non-local effects. Analytical expressions are developed for the ionisation pathlength probability distribution function (PDF) and good agreement is found with a numerical method based on lucky drift. The lucky drift expressions agree reasonably well with ionisation pathlength PDFs calculated using the Monte Carlo technique at moderately high electric fields (3×107 V m−1) but not at very high fields (108 V m−1) and a reason for this is proposed. Expressions for the non-local ionisation coefficient are found based on the lucky drift model and the expressions are evaluated numerically. Lucky drift is also used to find the probability of injection across a potential barrier and comparison with measurements shows good agreement except at very high electric field values. Reasons for the discrepancy at high fields are proposed.

Lucky-drift model for nonlocal impact ionisation

The probability-distribution function (PDF) for impact-ionisation path length is a crucial quantity for understanding and modelling the low-noise behaviour of avalanche photodiodes with short multiplication regions. In these devices the high electric fields needed to produce avalanche narrow the PDF, reducing the randomness in ionisation position and hence the noise in the multiplication. A simple method is presented for calculating PDFs using the `lucky-drift' model. The results are compared with those predicted by corresponding Monte Carlo calculations employing a parabolic energy band, deformation-potential optical-phonon scattering and hard-threshold impact ionisation. The overall behaviour expected for the PDF is reproduced by the simple model. However the unphysical, catastrophic assumption made for energy-relaxing collisions in lucky drift results in an unphysical delta function in the PDF which carries a significant weight at high electric fields.

Analysis of SiC IMPATT device in millimeter-wave frequencies

SiC IMPATT devices have an efficiency and power advantage over Si and GaAs IMPATT devices at millimeter-wave frequencies. A lucky drift model describing ionization rates as a function of electric fields at high temperatures has been incorporated into a Read-type large-signal model. The simulation demonstrates that the efficiency (and dc power density) at operating temperature (800 K) for SiC p+n single-drift flat-profile IMPATT structures is 12.4% (6.7 MW/cm2), 15% (4.5 MW/cm2), and 15.8% (3.3 MW/cm2) for frequencies of 200, 100, and 50 GHz, respectively. © 1998 John Wiley & Sons, Inc. Microwave Opt Technol Lett 18: 167–168, 1998.

Analysis of SiC IMPATT device in millimeter‐wave frequencies

SiC IMPATT devices have an efficiency and power advantage over Si and GaAs IMPATT devices at millimeter-wave frequencies. A lucky drift model describing ionization rates as a function of electric fields at high temperatures has been incorporated into a Read-type large-signal model. The simulation demonstrates that the efficiency (and dc power density) at operating temperature (800 K) for SiC p+n single-drift flat-profile IMPATT structures is 12.4% (6.7 MW/cm2), 15% (4.5 MW/cm2), and 15.8% (3.3 MW/cm2) for frequencies of 200, 100, and 50 GHz, respectively. © 1998 John Wiley & Sons, Inc. Microwave Opt Technol Lett 18: 167–168, 1998.

Spatial limitations to the application of the lucky-drift theory of impact ionization

Multiplication characteristics predicted by the lucky-drift (LD) theory of impact ionization are compared to experimental results on a range of thin GaAs PIN diodes with i-region thicknesses, w, from 1 /spl mu/m down to 0.025 /spl mu/m. Whereas lucky-drift and experimental results are in agreement for w/spl ges/0.1 /spl mu/m, significant differences are observed for thinner structures where nonlocal effects are important. Multiplication characteristics predicted by Monte-Carlo (MC) and lucky-drift simulations which use the same material parameters and produce the same bulk ionization rates are also compared and differences are again found in the multiplication characteristics of thinner structures. These differences are attributed to the lucky-drift description of carrier transport and establish a lower spatial limit to the application of this theory.

Spatio-temporal impact ionisation transients: A Lucky drift model study in GaAs

The impact ionisation occurring under a high field in a semiconductor can be described in terms of the ionisation coefficient, the number of ionisation events directly produced by a carrier per unit length it travels. The position dependence of this impact ionisation coefficient is at present poorly understood, and is usually assumed to be constant, possibly with the inclusion of a deadspace length (where the coefficient is zero). It is more complicated in reality however as spatial oscillations can occur. In this paper we employ a Lucky drift model to give an explanation of the origin of spatial oscillations in the coefficients, and the way their features vary with a range of model parameters is investigated. In addition to constant fields, a piecewise linear/constant field is modelled, as is a quantum barrier placed in a region of constant fields. The results are specifically for GaAs, but have qualitative validity for the other wide bandgap III–V's. A realistic value for effective electron deadspace in GaAs is obtained, which is approx. 1.3 times the minimum deadspace.

Modification of the lucky drift theory of impact ionisation for investigation of soft thresholds

The lucky drift theory for a soft impact ionisation threshold due to Burt and Marsland is generalised and improved. New, accurate values for ionisation thresholds are used to fit the model to experimental data for GaAs and AlGaAs at room temperature. The best fit parameters give information on the impact ionisation scattering rate, in particular what its dependence on energy is, and its softness of threshold factor. Low temperature ionisation coefficients in GaAs are modelled for the first time. A Monte Carlo implementation of the soft threshold lucky drift model is described, which is used for verification of our model and is also suitable for use in device simulations.

Substrate hot electron injection modelling based on lucky drift theory

Substrate hot electron injection modelling based on lucky drift theory

Bandstruwre dependence of impact ionisation in semiconductors from high pressure studies

Abstract Hydrostatic pressure has been used to study the bandstructure dependence of impact ionisation in Si, GaAs and Ge photodiodes. The results were modeled using lucky drift and Monte Carlo calculations, with ionisation threshold energies calculated from model pseudopotential bandstructures. The results for Si, GaAs and Ge were significantly different, and were interpreted in terms of the different band structures. For Si, multiplication is dominated by electron ionisation in the A valleys due to the large intervalley separations. However for GaAs and Ge ionisation gives rise to final electron states which are not only in the lowest conduction-band valley as previously assumed, but throughout the Brillouin zone. This has important implications for our understanding of impact ionisation and for modelling of avalanche multiplication in semiconductor alloys and multilayers.

Lucky drift estimation of excess noise factor for conventional avalanche photodiodes including the dead space effect

A technique for estimating the excess noise factor in conventional avalanche photodiodes has been developed. It is based upon a computer simulation of carrier motion using the lucky drift concept. The importance of the impact ionization dead space is demonstrated, and an established theory is shown to overestimate the excess noise factor due to the neglect of the dead space phenomenon in conventional avalanche photodiodes. >

Temperature dependence of ionisation coefficients in silicon derived from physical model

Electron and hole ionisation coefficients from 200 to 500 K have been calculated using the lucky drift model of impact ionisation calibrated to measurements made at 300 K. The results are fitted to two- and three-parameter empirical relationships for the ionisation coefficient against field, which can be used in device modelling.

Power series approximation used in soft threshold lucky drift model of impact ionisation

The soft threshold lucky drift model of impact ionisation in semiconductors due to Marsland (1987) has been developed by replacing an approximation with a more accurate power series approximation. The model has been fitted to measure ionisation coefficients and the new results give better agreement to the soft threshold lucky drift model due to Ridley (1987), especially for InP and GaAs. Further agreement for Si is obtained when the same experimental data is used. The variation of mean free paths and the 'softness of threshold' factor due to the choice of different experimental data and different ionisation threshold energies is estimated in silicon. The lucky drift model with the power series approximation is also fitted to ionisation coefficients measured in AlxGa1-xAs and Ge and a chemical trend in the mean free path is observed for all the materials reported.

A lucky drift model, including a soft threshold energy, for the relation between gate and substrate currents in MOSFETs

Substrate and gate currents in MOSFETs are formulated by interpreting the lucky drift model with a soft threshold energy from the distribution function point of view. A modified Keldysh formula is introduced to the ionization relaxation time. Unknown parameters in the formulation are determined by fitting the calculation to the measured relations between gate and substrate currents in MOSFETs. Impact ionization coefficient reported by Lee et al. [16], is also subsidiarily used for determining the parameters. The model can eliminate the nonphysical fitting parameter on the Schottky barrier lowering effect, which was necessary in past models. Fitting parameters concerning the ionization relaxation time, thus obtained, are compared with various theoretical calculations. It is recognized that the relation between gate and substrate currents in MOSFETs reflects the energy dependent ionization rate which originates from electron-electron interaction in the complicated energy band structure of silicon.

Comments, with reply, on 'comparison of multiquantum well, graded barrier, and doped quantum well GaInAs/AlInAs avalanche photodiodes: a theoretical approach by K. Brennan

It is pointed out that the lucky drift model complements the Monte Carlo simulation used in a previous paper (ibid., vol.QE-23, no.8, p.1273-82, 1987) to predict the performance of avalanche photodiodes containing multilayer structures. In his reply, the author agrees with the basic promise raised by the commenter, that models such as luck drift serve to complement detailed, first principles models like the Monte Carlo approach, aiding in their interpretation. >

Erratum: Lucky drift models of multilayered structures and approximate forms of lucky drift expressions

Erratum: Lucky drift models of multilayered structures and approximate forms of lucky drift expressions

Non-localised impact ionisation using a modified lucky drift theory

The lucky drift theory developed by B.K. Ridley (1983) is modified to include the effect of field variation over the 'dead space'. The new theory is applicable to small-geometry logic devices for which the total energy available from the supply voltage is comparable to the ionisation threshold energy. It is shown that the new theory predicts a substantially lower level of multiplication than do conventional theories based on an ionisation coefficient.