Space charge limited and emitter current limited injections in space charge region of semiconductors

If, in a space charge region of a semiconductor device, a unipolar drift current is a main component of the total current, an analogy with motion of electrons in vacuum exists. In the space charge limited emission, injection of minority carriers is limited by space charges, as in a punched-through p-n-p diode. On the other hand, in a transistor, minority carriers are injected into the depletion layer between the base and the collector, after diffusion in the neutral base region. This injection is limited by emitter current, and this mode of injection corresponds to a temperature limited emission in a vacuum tube. In this paper, equations of minority carrier impedances will be calculated in both these cases, and negative resistances can be expected at some transit angles. Then a generalized equation, which includes these two modes of injections as special cases, will be obtained. This generalized equation corresponds to the Llewellyn-Peterson equation of vacuum tubes, which include the space charge limited emission and the temperature limited emission as special cases.     

On the additivity of ohmic and space charge limited currents

The current density is calculated for the case where both ohmic and space charge limited (SCL) contributions are important. It is shown that, although the actual current is less than the sum of the currents which would be obtained individually from the ohmic or SCL conduction mechanisms, the error involved in assuming their additivity is not large.     

Modeling anomalous charge carrier transport in disordered organic semiconductors using the fractional drift-diffusion equation

Anomalous transport is ever-present in many disordered organic semiconductor materials. The long-tail behavior observed in the transient photocurrent is a manifestation of anomalous transport. Owing to the fact that anomalous transport has dispersive and non-Gaussian transport dynamics, thus anomalous transport cannot be adequately described by the standard drift-diffusion equation which is a framework commonly used to model normal diffusive transport. In this work, we generalized the standard drift-diffusion equation to time fractional drift-diffusion equation (TFDDE) using the fractional calculus approach to model the anomalous transport in the regio-random poly(3-hexylthiophene) (RRa-P3HT) and regio-regular poly(3-hexylthiophene) (RR-P3HT). Physical elucidation of TFDDE is given by stressing how the influence of the multiple-trapping mechanisms and energy disorder lead to the long-tail behavior in the transient photocurrent curves. TFDDE is solved numerically using finite difference scheme to obtain the profiles of charge carriers density evolution and hence to reproduce the corresponding transient photocurrents of RRa-P3HT and RR-P3HT. Poisson solver is also included in the model to account for the fluctuation of localized electric field due to the evolution of charge carriers. It is found that charge carriers acquire additional energy from high electric field that helps them to escape from the trap centers more easily and then propagating at higher velocity, which yields higher transient current. Higher concentration of charge carriers can be generated at higher light intensity and they can occupy energy levels close to the mobility edge, where charge carriers will encounter smaller capturing rate and hop at a longer length in each hopping event. Thus, the transport dynamic of charge carriers at high light intensity is less dispersive than that of the low light intensity. Besides, the transport dynamic of charge carriers in RR-P3HT is relatively less dispersive and has higher mobility than that of the RRa-P3HT since RR-P3HT has lower capturing rate and is less energy disordered.     

Noise in space charge limited field emission devices

The noise suppression in space charge limited field emitter cathodes is calculated, and it is shown that the noise can be more strongly space charge suppressed than in thermionic cathodes. The velocity fluctuation noise of the field emitter cathode is also calculated under arbitrary space charge conditions. The results are used to evaluate the noise figure of traveling wave tubes. Low noise figures are possible if the noise suppression of the space charge limited field emitter remains intact at microwave frequencies.     

Thermal equilibrium noise of space charge limited current in silicon for holes with field-dependent mobility

Measurements of the limiting white noise of space charge limited hole current in silicon were made at d.c. voltages where the hole mobility becomes field-dependent. The surprising result is established that the hole temperature is equal to the lattice temperature at 110°K within 10 per cent at voltages where the deviations from constant mobility reach 50 per cent. This result contradicts accepted theories relating carrier temperature to field-dependent mobility.     

A model of ohmic contacts to semiconductors

A unified and detailed model of both thermionic and tunneling ohmic contacts to semiconductors is described. The model takes into account the actual profile of the energy barrier as it is determined by the difference between the metal work function and the semiconductor electron affinity, by the ionized impurities in the semiconductor, by the interfacial quantum electric dipole and by the classical and quantum penetration of the charge carriers into the depletion layer. The current, above and below the energy barrier peak, is computed by using the Kemble generalized transmission coefficient, by taking into account the anisotropy of the effective masses as well as their dependence on impurity-concentration, and by employing the Fermi-Dirac statistics. Good agreement between theoretical and experimental results is observed. The model enables the design of ohmic contacts to semiconductors, i.e. the determination of the semiconductor impurity concentration required to achieve pre-assigned features.     

Generation of the second harmonic in the space charge limited diode

The generation of second harmonics of voltage and current in a space charge limited diode is considered. An equation for the second harmonic of the field is derived by an iteration procedure and the efficiency of the yield of the second harmonic is calculated.     

Grain boundary space charge region potential barrier in polycrystalline semiconductors

The grain boundary space charge region potential barrier is derived in an analytical form under equilibrium as well as non-equilibrium conditions, using various energy distributions to represent the grain boundary surface states, and the Shockley- Read-Hall theory for the carrier generation-recombination processes. The energy value of the grain boundary states is taken as a single level in the energy band gap of the semiconductor and located at the intrinsic Fermi level or displaced from it by a given amount. The restriction of the single level is later removed by permitting a certain width to the energy distribution of these states. It is shown that the results derived for non-equilibrium case reduce to the equilibrium case results derived separately, under the appropriate limit of zero excitation potential. The numerical results obtained from this formulation for the case of polycrystalline silicon are presented graphically for different values of the bulk doping concentration, grain boundary surface state density, grain boundary state energy distributions and the excitation potential, under equilibrium and non-equilibrium conditions. It is found that the grain boundary energy states located higher than the intrinsic Fermi level give greater values of grain boundary potential barrier height for a p-type semiconductor and the reverse applies to an n-type semiconductor. Further a wider energy distribution of these states gives rise to a towering of the potential barrier height at the grain boundary. Under non-equilibrium conditions, it is seen that the grain boundary barrier height remains unaltered for values of the excitation potential below a certain threshold which in turn depends upon the bulk dopant concentration in the semiconductor and the grain boundary surface state density and its energy distribution. Above this threshold value the barrier height reduces nearly linearly with the increase of the excitation potential with a slope lying-between ?0.5 and ? 1.0. For comparison with the available experimental results of Seager and Castner for n-type polycrystalline silicon we have obtained the grain boundary surface density and energy distribution parameters that provide excellent matching with the experimentally determined variation of the grain boundary potential barrier height with the dopant concentration in the range of 10¹³ to 10¹⁷/cm³, and based upon these parameters we have calculated the non-equilibrium barrier heights at different excitation potentials for this experimental case. A comparison is also made with Seager's experimental observations of non-equilibrium grain boundary barrier height for n-type silicon bicrystals in the illumination range of 3 × 10^?6 to 3 × 10^?2 SUN. For particular values of the density of grain boundary surface states and their energy distribution parameters, we could match our theoretical results with his experimental observations of barrier height.     

Transient space charge limited currents in light-pulse excited silicon

Space charge limited current (SCLC) transients have been studied by exciting over-depleted $\text{gold-on}\text{-n-}\text{type}$ silicon surface barrier diodes. The excitation was done by short (1 nsec) light pulses from a supperradiant laser and currents due to transport of electrons through the diodes were studied. Deviations from expected pulse shapes according to the theory of Many et al. for SCLC in an insulator are caused by (a) the influence of doping atoms on the field, (b) by field-dependent mobility and (c) by lateral spreading of the discharge. Effect (a) is pronounced at low fields (voltages only slightly exceeding the depletion voltage). It causes a shortening of the cusp time with up to $\text{? 20\%}$, a general increase in current, and a decrease in the ratio of cusp to the starting current values. Effect (b) which occurs at high fields, causes a lengthening of cusp time and changes in current magnitude. The latter is shown to give deviations from the expected quadratic dependence of SCLC on voltage. Effect (c) gives an increase in apparent space charge limited current when the light excitation is increased above the level which is sufficient for SCLC from the irradiated area. An increase in light with a factor $\text{? 7}$ approximately doubles the transient current.     

Allowing for current spreading in semiconductors during measurements of the contact resistivity of ohmic contacts

A modification of the contact-area pattern with radial geometry, which has certain advantages in determining the contact resistivity of ohmic contacts (ρ _c) fabricated on substrates and low-resistance semiconductor layers, is proposed. Different variants of its application for both the transmission line method (TLM) and methods based on a numerical calculation of the resistance of the semiconductor with allowance for current spreading are considered. It is shown that the transmission line method makes it possible to obtain an upper estimate of the contact resistivity on substrates. The errors of such estimates are also calculated as a function of the parameters of the semiconductor and the contact. The TLM estimate is a good first approximation for determining the exact value of ρ _c by numerically calculating the resistance of the semiconductor. The results obtained are used to study the contact resistivity of Ni-based ohmic contacts on n-6H-SiC substrates. Fiz. Tekh. Poluprovodn. 32, 832–837 (July 1998)     

Space charge limited current in an insulator at high fields

It is shown that the correct high field current/voltage characteristic in space charge limited single-injection solid state diodes is obtained at sufficient high fields.     

Theoretical insights into the charge transport in perylene diimides based n-type organic semiconductors

We employed a tunneling enabled hopping model to investigate the charge transport properties for four n-type organic semiconductors perylene diimides compounds. The molecular parameters are calculated by density functional theory and the transport is modeled by kinetic Monte Carlo simulation. It is found that the substitutions at the bay positions of the perylene core have large influences on the charge transport properties through modifications in molecular conformation, the charge reorganization energy as well as the stacking networks in the crystals. The temperature dependence of the mobility shows typical “band-like”, in agreement with the recent experiment, but we ascribe it to be the characteristic of nuclear tunneling effect for a localized charge, not by a delocalized band. The largest charge mobility is calculated to be 16.96 cm²/V s for the cyano substitution, in good comparison with the experimental value of 6 cm²/V s.     

Analysis of charge transfer complex at the interface between organic and inorganic semiconductors

Improving the electrical performance of organic semiconductors is critical to use them for optoelectronic applications. In this study, we analyze the mechanism of charge transfer complex (CTC) formation at the interface between organic and inorganic semiconductors through extensive optical and electrical measurements. N,N′-Bis(naphthalen-1-yl)-N,N′-bis(phenyl)benzidine (NPB) and molybdenum oxide (MoO₃) were sequentially deposited to form a donor/accepter heterojunction structure. The CTC formation and conductivity of the films were determined using UV–visible spectroscopy and transmission line method, respectively. Compared with the single layer devices, the donor/accepter heterojunction exhibits significantly enhanced conductivity. In addition, the conductivity and CTC generation efficiency of the heterojunction display strong dependence on NPB layer thickness, which originates from the variation of dipole interactions at the heterojunction interface. These results provide useful insights on interfacial doping properties, which is potentially beneficial for enhancing the understanding of organic/inorganic interfaces.     

Modeling of charge transport across disordered organic heterojunctions

Organic electronic devices often consist of a sandwich structure containing several layers of disordered organic semiconductors. In the modeling of such devices it is essential that the charge transport across the organic heterojunctions is properly described. The presence of energetic disorder and of strong gradients in both the charge density and the electric field at the heterojunction complicates the use of continuum drift-diffusion approaches to calculate the electrical current, because of the discrete positions of the sites involved in the hopping transport of charges. We use the results of three-dimensional Monte Carlo simulations to construct boundary conditions in a one-dimensional continuum drift-diffusion approach that accurately describe the charge transport across the junction. The important effects of both short- and long-range Coulomb interactions at the junction are fully accounted for. The developed approach is expected to have a general validity.     

Use of the BFCPIC and BFCRAY codes to describe space charge limited emission in electron guns for gyrotrons

Numerical modelling of an electron gun in the space charge limited regime requires determining the current density distribution as well as the electric fields and electron trajectories. This is a rather complicated self-consistent problem, since the space charge influences the electric field, which in turn influences the electron trajectories. Previous simulations of magnetron electron guns using the BFCPIC and BFCRAY codes used a simple emission model (constant current density) that is approximately valid for thermionic emission. The code has been modified to include space charge limited emission. Several different ways of doing this are considered. One of the models considered uses Gauss’s law to force the electric field on the emitter to vanish; it was used in the original version of BFCPIC for the simulation of ion diodes. A second is based on the use of Child’s law (locally), which may be more appropriate for extension to fully electromagnetic particle-in-cell (PIC) codes. Calculations were performed with both models, and the results compared with each other and with experiments performed at FZK.     

Recombination current measurements in the space charge region of MOS field-induced pn junctions

Electrically active defects in the device region are routinely monitored by CV measurements of reverse-biased field-induced (FI) pn junctions in MOS structures. While useful, this approach is sensitive only to near mid gap defects. Here, we demonstrate a method for interrogation of forward-biased FI pn junctions, which can reveal defect levels over a significantly wider region of the band gap. The method proposed is based on a simultaneous measurement of the gate current and the high frequency gate capacitance in non-equilibrium non-steady state in response to a linear gate voltage ramp which drives the MOS capacitor from inversion equilibrium towards accumulation. This recombination-sensitive technique enables a self-consistent determination of the forward current–voltage characteristic of the FI pn junction. It makes a wider range of important impurities, especially metallic contaminants, accessible to detection by MOS CV approaches. Since the approach satisfies the low-injection condition, the results can be directly related to the properties of the defect centres, thus facilitating defect identification and control.     

An improved model to explain ohmic contact resistance of f-GaAs and other semiconductors

A model is proposed which explains the inverse proportionality between specific contact resistance ϱ_c and the carrier concentration (N_D) of n-GaAs ohmic contact obtained experimentally very well and can be extended to the ohmic contact of other semiconductors. This model assumes that ϱ_c is composed of two parts ϱ_c1 and ϱ_c2. The contact resistivity ϱ_c1 is due to the contact between the alloy and the underlying heavily doped contact region (N_DC for n-type contact, N_AC for p-type contact) formed by doping from the contact alloys during annealing. The contact resistivity ϱ_c2 is caused by the barrier height (Φ₂) of the high-low junction between the heavily doped contact region and the bulk material. There are two aspects: (1) If bulk material is degenerated, i.e. N_D >N_C (N_C: effective density of states in conduction band) or N_A >N_V (N_V: effective density of states in valence band), the barrier height Φ₂ vanishes and ifϱ_(c) is mainly determined by ifϱ_c1 which depends solely on concentrations N_DC and N_AC. It is calculated theoretically. (2) When the bulk material is lightly doped, i.e. N_D < N_Cor N_A < N_V, then Φ₂ appears and increases with the decreasing of N_D(N_A). If N_DC or N_AC are high enough, the field emission is the main mechanism to control the carrier transport between the contact alloy and the underlying layer. In this case ϱ_c is predominantly determined by ϱ_c2 and an inverse proportionality between ϱ_c and N_D or N_A can be found.     

Space-charge and injection limited current in organic diodes: A unified model

Space-charge and injection limited currents in organic diodes have been analyzed using a unified model. Both currents have been modeled using the transport equations combined with a proper value of the free charge density at the metal–organic interface. The method has been applied to diodes with different organic materials, metal contacts and lengths of the organic materials. This unified model accurately reproduces published current–voltage curves for a variety of diode structures operating at different temperatures and voltages. The results of the method, in the injection regime, have been compared with the results of pure injection models developed by other authors. The reduction of the computational time and the number of parameters in our model are important advantages of our procedure. Moreover, it is an alternative where the injection models start to fail: at low applied voltages, close to the Ohmic regime, and for low heights of the energy barrier at the interface. The treatment is complemented with a compact model that relates the current density j with the free charge-carrier density at the interface p_f(0): p_f(0) = K₁j^m + K₂, where the parameter K₁ depends on the barrier height at the interface, m depends on the organic material and K₂ controls the flat zone at low currents to include the dependence with thermal carriers and impurities.