Shockley Model Research Articles

I-V and impedance characterization of a solution processed perovskite based heterojunction photodetector

Herein, we report the investigation of electrical and photo response properties of hybrid organic-inorganic perovskite based photodetector. Surface morphology of perovskite thin film examined through Field emission scanning electron Microscope (FESEM) revealed smooth surface with complete coverage of the glass substrate. Different elements present in perovskite thin films, as well as their quantitative ratio are identified by Energy-dispersive X-ray spectroscopy (EDX) analysis. The current-voltage (I-V) measurements are carried out in dark and under illumination conditions to investigate the charge transport process and photodetector performance. The important diode electrical parameters (ideality factor, saturation current, and barrier height) are extracted by the analysis of I-V characteristics using Shockley model. The observed rectification and photo response behavior is analyzed using energy level diagram. The photodetector demonstrates good photoresponse and exhibits responsivity and detectivity value of 0.4 A/W and detectivity of 3 × 1011 Jones, respectively at -5 V. The charge transport mechanisms are examined under the framework of space charge limited conduction (SCLC) model, which helps to extract the mobility, trap density and trap energy. Moreover, electrochemical impedance spectroscopy (EIS) analysis is performed at different applied voltage and employed to determine the different device parameters such as recombination and charge transport resistance, chemical capacitance, mobility and life time. The mobility estimated from impedance analysis show good agreement with the mobility calculated from SCLC model.

Model for large magnetoresistance effect in p–n junctions

We present a simple model based on the classic Shockley model to explain the magnetotransport in nonmagnetic p–n junctions. Under a magnetic field, the evaluation of the carrier to compensate Lorentz force establishes the necessary space-charge region distribution. The calculated current–voltage (I–V) characteristics under various magnetic fields demonstrate that the conventional nonmagnetic p–n junction can exhibit an extremely large magnetoresistance effect, which is even larger than that in magnetic materials. Because the large magnetoresistance effect that we discussed is based on the conventional p–n junction device, our model provides new insight into the development of semiconductor magnetoelectronics.

On a rapidly converging iterative algorithm for diode parameter extraction from a single IV curve

We measure the I–V curve of a p–n diode on a range of currents where the well known Shockley model exhibits its shortcoming. We show that the behavior of the I–V curve, on the whole range of currents, can be captured by a modified four parameter Shockley model; the parameters of the model are obtained by a numerical procedure, consisting of an iterative rapidly converging nonlinear fitting algorithm. The fit returns sound estimates of the parameters for I–V curves, as function of temperature. The method is validated on a well known 1N4148 diode, but it can also be applied to any other devices described by single-diode models such as solar cells. We also show that the knowledge of the temperature dependence of the parameters can be used to obtain a quantitative estimate of other physical properties of the system, such as the energy gap of junction materials.

A Shockley‐Type Polymer: Fullerene Solar Cell

AbstractCharge extraction rate in solar cells made of blends of electron donating/accepting organic semiconductors is typically slow due to their low charge carrier mobility. This sets a limit on the active layer thickness and has hindered the industrialization of organic solar cells (OSCs). Herein, charge transport and recombination properties of an efficient polymer (NT812):fullerene blend are investigated. This system delivers power conversion efficiency of >9% even when the junction thickness is as large as 800 nm. Experimental results indicate that this material system exhibits exceptionally low bimolecular recombination constant, 800 times smaller than the diffusion‐controlled electron and hole encounter rate. Comparing theoretical results based on a recently introduced modified Shockley model for fill factor, and experiments, clarifies that charge collection is nearly ideal in these solar cells even when the thickness is several hundreds of nanometer. This is the first realization of high‐efficiency Shockley‐type organic solar cells with junction thicknesses suitable for scaling up.

Electrical and optical performances with extracted minority carrier lifetimes of InAs/GaSb SL photodetector operating in the mid wavelength infrared range

We report a study on the temperature dependence of electrical and optical performance of InAs/GaSb based type-II superlattice (T2SL) pin photodetectors in the mid wavelength infrared range (MWIR). The SL structure exhibits an optical response of 50% cut-off wavelength at 4.9 μm at 79 K. Deduced from current density–voltage (J–V) measurements, dark current density under 0.1 V reverse bias is measured as 7.6 × 10−6 A/cm2 with a corresponding differential–resistance–area product (R0A) of 3.3 × 104 Ωcm2 at 100 K. Minority carrier lifetimes of the T2SL detectors are analysed by Shockley's Model where experimental data for dark current densities are fitted by diffusion and generation-recombination (GR) components at different temperatures.

A modified Shockley equation taking into account the multi-element nature of light emitting diodes based on nanowire ensembles

In this work we study how the multi-element nature of light emitting diodes (LEDs) based on nanowire (NW) ensembles influences their current voltage (I–V) characteristics. We systematically address critical issues of the fabrication process that can result in significant fluctuations of the electrical properties among the individual NWs in such LEDs, paying particular attention to the planarization step. Electroluminescence (EL) maps acquired for two nominally identical NW-LEDs reveal that small processing variations can result in a large difference in the number of individual nano-devices emitting EL. The lower number of EL spots in one of the LEDs is caused by its inhomogeneous electrical properties. The I–V characteristics of this LED cannot be described well by the classical Shockley model. We are able to take into account the multi-element nature of such LEDs and fit the I–V characteristics in the forward bias regime by employing an ad hoc adjusted version of the Shockley equation. More specifically, we introduce a bias dependence of the ideality factor. The basic considerations of our model should remain valid also for other types of devices based on ensembles of interconnected p–n junctions with inhomogeneous electrical properties, regardless of the employed material system.

Systematic investigation of the misorientation- and temperature-dependent Kapitza resistance in CeO2

The misorientation- and temperature-dependent grain boundary thermal (Kapitza) resistance in CeO2 is investigated using molecular dynamics simulations. A few empirical potentials for molecular dynamics simulations are evaluated for their predicted properties such as the phonon dispersion curves, bulk thermal conductivity, and grain boundary structures. Through the comparison of these properties with experimental results, the most reasonable potential (Gotte2007) is selected. The Kapitza resistances of tilt and twist grain boundaries with misorientation angles ranging from 3° to 87° are calculated and a clear transition angle at about 16° is observed. The Kapitza resistance is found to increase almost linearly with misorientation angle in the low-angle regime but remain nearly constant at the high-angle regime, a behavior very similar to the grain boundary energy. A nearly linear correlation between Kapitza resistance and grain boundary energy is thus obtained. Similar to the grain boundary energy, the Read–Shockley model can well describe the misorientation-dependent Kapitza resistance at low-angle regime. The Kapitza conductance (the inverse of Kapitza resistance) is found to increase almost linearly with temperature in our simulations.

A generalized Read–Shockley model and large scale simulations for the energy and structure of graphene grain boundaries

The grain boundary (GB) energy is a quantity of fundamental importance for understanding several key properties of graphene.

Growth rate distributions for regular two-dimensional grains with Read–Shockley grain boundary energy

Understanding the effects of an anisotropic grain boundary energy model is difficult analytically, so simulations provide a valuable insight into the dynamics and characteristics of grain structure evolution. A brute-force analysis of the common Read–Shockley model in two dimensions for grains of the topological class of n-sided grains with 3 ⩽ n ⩽ 8 is performed. In this approach, we rotate all cubic two-dimensional crystal lattices of n-sided grain neighbors stepwise with high resolution in two dimensions, and compute the growth rates of each configuration to obtain detailed growth rate distributions. To perform the computationally intense task of computing the growth rates for billions of configurations, a vertex type model and a supercomputer are utilized. Modern computation facilities deliver the required computation power to carry out such studies in a reasonable amount of time. The vertex model is validated by comparison with 700 phase-field simulations. Our findings support the isotropic behavior of grain structures with randomly oriented lattices and Read–Shockley grain boundary energy, where the average values of the growth rate distributions follow the isotropic Neumann–Mullins relation. Further, all computed growth rate distributions are not symmetric and adapt a normal distribution for n → 8. Abnormal grain growth does not occur for the Read–Shockley model with neighbors in the range 3 ⩽ n ⩽ 8. The brute-force method of scanning the parameter space in high resolution reaches its limit fast, but contributes a new perspective.

Influence of mosaicity on the fracture behavior of sapphire

The resistance to breakage of sapphire samples of different crystallographic orientations and produced by the Kyropoulos and Verneuil processes was investigated. The fracture strength was determined by using four-point bending tests, whereas the single-edge V-notch beam method was used to measure the fracture toughness. For both mechanical properties, it was found that (i) Verneuil sapphire has values up to two times higher than Kyropoulos sapphire and (ii) the data scatter is also generally higher for Verneuil sapphire. The main factor responsible for this behavior is believed to be the presence of mosaicity in Verneuil and its absence in Kyropoulos sapphire. By using the Read–Shockley model to estimate the energy of low-angle grain boundaries, together with the Griffith energy criterion for intergranular crack propagation, it is demonstrated that intergranular fracture along the mosaic block boundaries is energetically more favorable than transgranular fracture for crack-to-boundary orientations smaller than a critical deflection angle. Nevertheless, both the location of the crack in respect to the boundary, as well as the 3-D nature of these features, must be taken into consideration for a complete description of the intergranular deflection toughening mechanism.

Electron dynamics in semiconductors, Markovian and beyond

In earlier calculations of the dynamics of a single electron interacting with lattice vibrations an attempt was made to find a first principles' quantum mechanical description matching the original Haynes–Shockley model for an electron in a semiconductor, a model taken from Einstein's theory of Brownian motion. The equation of motion for the electron's propagator was derived from the exact double path-integral representation of a polaron by Feynman et al. (FHIP). The equation was then converted to the Wigner form of quantum mechanics, and solved in Wigner variables. The purpose of the present calculation is to remove a singularity that arose earlier from scattering by phonons with wave lengths greater than the Einstein length. We find here that the singularity can only be removed by including non-Markovian terms, terms left out in the earlier calculations. The complementary addition of the new terms enhances the plausibility of the modified solution, now no longer Markovian. In this calculation, as a first step, we omit the effect of the electron's recoil energy originating with Markovian scattering by the troublesome long phonon waves. The effect of the neglected recoil energy would be helpful but insufficient in removing the singularities. Of particular interest is the non-Markovian correction to the Einstein diffusion relation, a relationship at the heart of the Haynes–Shockley model. The present calculation, then, may be viewed as yielding an upper limit to the correction, at about 10%.

Shockley equation parameters of P3HT:PCBM solar cells determined by transient techniques

The Shockley equation (SE), originally derived to describe a p--n junction, was frequently used in the past to simulate current--voltage (j/V) characteristics of organic solar cells (OSC). In order to gain a more detailed understanding of recombination losses, we determined the SE parameters, i.e. the ideality factor and the dark saturation current, from temperature dependent static j/V-measurements on poly(3-hexylthiophene-2,5-diyl)(P3HT):[6,6]-phenyl-C$_{61}$ butyric acid methyl ester (PCBM) bulk heterojunction solar cells. As we show here, these parameters are directly related to charge carrier recombination and become also accessible by transient photovoltage and photocurrent methods in the case of field-independent charge carrier generation. Although determined in very different ways, both SE parameters were found to be identical. The good agreement of static and transient approaches over a wide temperature range demonstrates the validity of the Shockley model for OSC based on material systems satisfying the requirement of field-independent polaron-pair dissociation. In particular, we were able to reproduce the photocurrent at various light intensities and temperatures from the respective nongeminate recombination rates. Furthermore, the temperature dependence of the dark saturation current $j_0$ allowed determining the effective band gap of the photoactive blend perfectly agreeing with the literature values of the energy onset of the photocurrent due to charge transfer absorption. We also present a consistent model directly relating the ideality factor to recombination of free with trapped charge carriers in an exponential density of tail states. We verify this finding by data from thermally stimulated current measurements.

Shockley model description of surface states in topological insulators

Surface states in topological insulators can be understood based on the well-known Shockley model, a one-dimensional tight-binding model with two atoms per elementary cell, connected via alternating tunneling amplitudes. We generalize the one-dimensional model to the three-dimensional case representing a sequence of layers connected via tunneling amplitudes $t$, which depend on the in-plane momentum $\mathbit{p}=({p}_{x},{p}_{y})$. The Hamiltonian of the model is a $2\ifmmode\times\else\texttimes\fi{}2$ matrix with the off-diagonal element $t(k,\mathbit{p})$ depending also on the out-of-plane momentum $k$. We show that the existence of the surface states depends on the complex function $t(k,\mathbit{p})$. The surface states exist for those in-plane momenta $\mathbit{p}$ where the winding number of the function $t(k,\mathbit{p})$ is nonzero when $k$ is changed from 0 to $2\ensuremath{\pi}$. The sign of the winding number determines the sublattice on which the surface states are localized. The equation $t(k,\mathbit{p})=0$ defines a vortex line in the three-dimensional momentum space. Projection of the vortex line onto the space of the two-dimensional momentum $\mathbit{p}$ encircles the domain where the surface states exist. We illustrate how this approach works for a well-known model of a topological insulator on the diamond lattice. We find that different configurations of the vortex lines are responsible for the ``weak'' and ``strong'' topological insulator phases. A topological transition occurs when the vortex lines reconnect from spiral to circular form. We apply the Shockley model to Bi${}_{2}$Se${}_{3}$ and discuss applicability of a continuous approximation for the description of the surface states. We conclude that the tight-binding model gives a better description of the surface states.

Valuing an Early‐Stage Biotechnology Investment as a Rainbow Option

In an article published in this journal in 2003, Richard Shockley and three of his students presented a detailed valuation of an early‐stage biotechnology investment using a binomial lattice option pricing model. The article demonstrates how investments with multiple stages can be treated as “compound sequential options”—that is, as series of options in which investments in one option provide the opportunity to invest in the next in the series.In this article, the author uses the same business case analyzed by Shockley et al. to demonstrate how to value this early‐stage biotechnology investment by separately modeling the two types of risks: technology and product market. An option that has two distinct kinds of risk that develop differently over time is known as a “rainbow option.” The key adjustment to the option pricing model required to value such an option is that, instead of the standard binomial option pricing model with two outcomes at each point in time, the author uses a “quadranomial” option pricing model with four outcomes at each point in time. By distinguishing technology risks from product market risks and allowing them to develop differently over time, the author's analysis leads to a very different valuation and, indeed, a different decision about the initial investment than the one produced by Shockley's model.

Verification of p-n junctions in polymer light-emitting electrochemical cells via electrical characterization

We report micrometer thick sandwich light-emitting electrochemical cells (LECs) based on a blend of poly[5-(2′-ethylhexyloxy)-2-methoxy-1, 4-phenylene vinylene] and poly(ethylene oxide) complexed with lithium trifluoromethanesulfonate. These LECs exhibit very similar properties as those of thin LECs including bipolar current-voltage characteristics and light emission. Mixing of aluminum nanoparticles into polymer layers improves electroluminescence because of smaller series resistance and larger light-emitting area. Taking series resistance into account, we confirm the operating mechanism of an LEC is the formation of a p-n junction by in situ electrochemical doping via fitting the steady state current-voltage characteristics to the expression for the Shockley model of a p-n diode.

Modelling of the evolution of micro-grain misorientations during creep of tempered martensite ferritic steels

Tempered martensite ferritic steels are prone to low-angle boundary (LAB) vanishing and micro-grain size increase during creep deformation. A physically based model of LAB vanishing during creep deformation is presented. The LABs are modelled by simple dislocation arrays following the Read and Shockley model. Depending on the activated slip systems, mobile edge/screw dislocations annihilate with LAB parallel dislocations of opposite sign. The LAB misorientation frequency evolution versus creep strain can be analytically computed. The material parameters are the fractions of edge/screw dislocations in the LABs and the critical edge/screw dislocation annihilation distances. Different LAB misorientation frequencies obtained by electron backscatter diffraction (EBSD) and available in the literature are used as input data of the simulations. The computed LAB misorientation frequencies are then compared with the experimental frequencies obtained at different creep strains. If the material parameters belong to physical ranges, their influence on the predictions remain rather weak. The model generally permits a reasonable prediction of misorientation and micro-grain size evolutions during creep. Finally, the effects of precipitate evolution, climb and internal stresses are discussed.

Elucidation of the Excited-State Dynamics in CuInS2 Thin Films

Transient absorption (TA) and photoluminescence spectroscopy have been performed on spray-deposited CuInS2 thin films. Sulfur and indium vacancies introduce electronic states in the bandgap located at 1.5 and 0.15 eV above the valence band, respectively. Deep donor and deep acceptor doublet states at 1.1 and 0.2 eV are assigned to copper/indium antisite defects. The excited-state dynamics, which are derived from TA experiments, show electronic coupling between indium antisite defects and indium vacancies. The Shockley, Read, and Hall recombination model has been modified to account for this coupling and to simulate the TA results. Furthermore, the lifetime of the 1.1 eV state is found to be 20−50 μs, which is related to the low photovoltage of CuInS2 based solar cells.

High-temperature PbTe diodes

We describe the preparation of high-temperature PbTe diodes. Satisfactory rectification was observed up to 180–200 K. Two types of diodes, based on a p-PbTe single crystal, were prepared: (1) by In ion-implantation, and (2) by thermodiffusion of In. Measurements were carried-out from ~ 10 K to ~ 200 K. The ion-implanted diodes exhibit a satisfactorily low saturation current up to a reverse bias of ~ 400 mV, and the thermally diffused junctions up to ~ 1 V. The junctions are linearly graded. The current–voltage characteristics have been fitted using the Shockley model. Photosensor parameters: zero-bias-resistance × area product, the R 0 C 0-time constant and the detectivity D ⁎ are presented.

Natural ageing effects of Schottky-gate GaAs FETs in current-voltage characteristics and in 1/f noise spectrum

We study the current-voltage (I–V ) characteristics and 1/f (flicker) noise spectra of experimental samples (prototypes) of submicron GaAs field-effect transistors (FETs) with Schottky planar gate in the linear regime. The model of I–V characteristics, in which the FET channel is described by the Shockley model and the crowding of current lines in passive regions is taken into account, is proposed. Analysis of the measured set of I–V characteristics allowed us to define geometric and electric parameters of the devices more precisely. The natural ageing effects were revealed by analyzing the measurements performed at a 3-year interval. We make the conjecture that excess defects are present in the structure and they are probably represented by dipoles (donor-acceptor pairs). Concentration of excess defects is estimated and its significant decrease with time is detected. For analysis of 1/f noise, we used the hypothesis of the existence of bistable point defects, assuming that the latter result in resistivity fluctuations of the conducting regions. It is shown that defects in the channel give the main contribution to 1/f noise and their concentration decreases with time. Increased content of defects near the buffer layer was detected in some devices. The experimental data obtained allow us to suppose that the observed 1/f noise is caused by the excess defects revealed in the analysis of I–V characteristics of FETs with Schottky planar gate.

Grain boundary stiffness based on a dislocation model

The classical Read–Shockley model for grain boundary energy is employed to derive an expression for grain boundary stiffness. The stiffness increases linearly with boundary misorientation and is maximum for asymmetric tilts with equal densities of the constituent dislocation sets. We also show that the intrinsic waviness in the boundary morphology is coupled to the stiffness.