The discrete Einstein relation

In this paper the connection between massM, resistance and commute time for random walks on graphs is further explored, and the relation=2M · is proved. An extension of the result is made to multigraphs, which are an extension of the graph concept where a black box is treated like an edge.     

A generalized Einstein relation for semiconductors

The Einstein relation relates the diffusion coefficient to the mobility and is frequently used in semiconductor device analysis and design. A flux equation governing the behavior of mobile particles in semiconductor material is derived from the Boltzmann transport equation. The particles are assumed to obey quantum statistics. The flux equation provides the formal means for defining the transport parameters D and μ.A generalized Einstein relation valid for any particle density under equilibrium as well as non-equilibrium conditions is derived. The relation is given in terms of Fermi integrals and a scattering parameter. Once the scattering parameter is specified, the Einstein relation can be easily evaluated. The asymptotic limit for degenerate material is also given.     

The Einstein relation for degenerate semiconductors with nonuniformband structures

Modification of the Einstein equation for semiconductors with nonparabolic energy bands and doped nonuniformly with impurity atoms is suggested. The suggestion is based on a new approximation of the Fermi-Dirac integral of order 1/2, namely, F_1/2(η_n), where η_n is the reduced Fermi level for electrons. The relation reduces to that for semiconductors with parabolic energy bands and doped uniformly with impurity atoms under appropriate boundary conditions. A comparison of the calculated and exact results for F_1/2(η) is found to be very encouraging     

Generalization of Einstein relation for organic semiconductor thin films

Under the basis of Gaussian energy distributions of the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO), analytical expressions of generalized Einstein relation for electron and hole transport in organic semiconductor thin films are developed. Numerical calculations show that, although traditional Einstein relation is still valid for low carrier concentrations, when the carrier concentration is high, the diffusion coefficient–mobility ratio increases rapidly with the carrier concentration. A relative turning carrier concentration, characterizing the upper limit of the validity of traditional Einstein relation is defined. The dependences of the relative turning concentration on the variance of LUMO or HOMO energy distributions as well as the sample temperature, and the applications of the generalized Einstein relation in the analysis of organic light-emitting device are discussed.     

The quantum correction of the Einstein relation for high frequencies

For nondegenerate semiconductors the low-frequency Einstein relation qD₀ = kTμ₀ must be replaced by qD′(ω) = kTμ′(ω)p(ω) at high frequencies, where D′ and μ′ are the real parts of the complex diffusion coefficient D and the mobility μ, respectively, and p(ω) is the quantum correction factor for thermal noise. This is shown, first by treating thermal noise as a case of velocity-fluctuation noise, and secondly by using a simple statistical model. The results are extended to degenerate semiconductors and metals.     

On the validity of the Einstein relation for nonequilibrium conditions

The flux equation governing the behavior of mobile particles in semiconductor material is derived from the Boltzmann transport equation. It is then shown that under suitable assumptions the Einstein relation frequently used in device analysis is valid for nonequilibrium conditions.     

Dependence of general Einstein relation on density of state for organic semiconductors

The Einstein relation (ER) about the diffusion coefficient D and mobility μ of charge carriers has been suspected for disordered organic semiconductors. The general Einstein relation (GER) is popular in recent years, and usually been calculated based on the Gaussian DOS. A clearly cutting inverse-exponential (CCIE) DOS [Org. Elect. 30 (2016) 60–66] is proposed. The mobility is obtained by solving variable range hopping (VRH) equations. The results show that the experimental mobility-density data can be well fitted by using single CCIE DOS in the wide ranges of density, but cannot be fitted by using single Gaussian or un-cutting exponential-type DOS. In this work, the coefficient ζ in the GER (D/μ = ζkT/q) is calculated based on the Gaussian and CCIE DOSs. The variations of coefficient ζ with temperature and density are analyzed. It is shown that the ζ are a gradually decreasing function with temperature and similar for both DOSs. But variations of ζ with density are very different for both DOSs. The ζ is a gradually increasing function of density for the Gaussian DOS, but a non-monotonously increasing function of density for the CCIE DOS with a platform located in the typical range of density. The ζ is assumed as a constant to analyze the data of ideality factor for two organic diodes based on rr-P3HT and OC₁C₁₀-PPV in literature, the theoretical results are in agreement with experimental data.     

Relaxing avalanche mode and its relation to TRAPATT, a.r.p. and IMPATT oscillators

A large-signal computer study has been used to examine the initiation of TRAPATT oscillations and the multifrequency nature of TRAPATT amplifiers. A relaxing avalanche mode is observed to lead to trapped-plasma generation below the critical current density qNd vs, merging smoothly to relaxation oscillations without a trapped plasma. The analysis shows that certain diode-circuit-impedance relationships should be met for self-initiation of TRAPATT oscillations and provides an explanation of the TRAPATT amplification mechanism of Prager, Chang and Weisbrod.     

Noise in Gunn oscillators

We estimate the ultimate noise of Gunn oscillators in the absence of 1/fnoise. The basic noise source considered is thermal or Johnson noise augmented by intervalley noise of carriers hopping between the high and low mobility bands. For example, for Qext= 10²and Pout=10^-1W we estimate δfrms≈ 1-2 Hz, and AM noise relative to the carrier of -156 dD, both measured in 1-kHz bandwidths.     

Jitter in ring oscillators

Jitter in ring oscillators is theoretically described, and predictions are experimentally verified. A design procedure is developed in the context of time domain measures of oscillator jitter in a phase-locked loop (PLL). A major contribution is the identification of a design figure of merit κ, which is independent of the number of stages in the ring. This figure of merit is used to relate fundamental circuit-level noise sources (such as thermal and shot noise) to system-level jitter performance. The procedure is applied to a ring oscillator composed of bipolar differential pair delay stages. The theoretical predictions are tested on 155 and 622 MHz clock-recovery PLL's which have been fabricated in a dielectrically isolated, complementary bipolar process. The measured closed-loop jitter is within 10% of the design procedure prediction     

Describing functions and oscillators

In this article the "Describing Function" technique is applied to the analysis and design of oscillators. It is shown that, with some simplifications of the expressions involved in the analysis, this technique allows one not only to quantify the amplitude but also to determine the degree of distortion of the generated sinusoidal signal. The advantage of the describing function is that it allows the inclusion of the nonlinear behavior of a system while maintaining the simplicity often associated with the linear system analysis. This will be demonstrated through an example presenting the analysis and design of a sinusoidal oscillator in the frequency domain. The method as such allows formulation of closed-form expressions for the amplitude and distortion levels in much the same way classical linear techniques are used     

Jitter and phase noise in ring oscillators

A companion analysis of clock jitter and phase noise of single-ended and differential ring oscillators is presented. The impulse sensitivity functions are used to derive expressions for the jitter and phase noise of ring oscillators. The effect of the number of stages, power dissipation, frequency of oscillation, and short-channel effects on the jitter and phase noise of ring oscillators is analyzed. Jitter and phase noise due to substrate and supply noise is discussed, and the effect of symmetry on the upconversion of 1/f noise is demonstrated. Several new design insights are given for low jitter/phase-noise design. Good agreement between theory and measurements is observed     

Pole movement in electronic and optoelectronic oscillators

An RLC circuit with poles on the left half of the complex frequency plane is capable of executing transient oscillations. During this period, energy conversion from potential to kinetic and from kinetic to potential continuously goes on, until the stored energy is lost in dissipation through the resistance. On the other hand, in an electronic or opto-electronic oscillator with an embedded RLC circuit, the poles are forcibly placed on the right-half plane (RHP) and as far as practicable away from the imaginary axis in order to help the growth of oscillation as quickly as possible. And ultimately, it is imagined that, like the case of an ideal linear harmonic oscillator, the poles are frozen on the imaginary axis so that the oscillation neither grows nor decays. The authors feel that this act of holding the poles right on the imaginary axis is a theoretical conjecture in a soft or hard self-excited oscillator. In this article, a detailed discussion on pole movement in an electronic and opto-electronic oscillator is carried out from the basic concept. A new analytical method for estimating the time-dependent part of the pole is introduced here.     

Spurious oscillations in IMPATT oscillators

Incorrect circuit design of cavity-controlled IMPATT oscillators may cause low-frequency instabilities and frequency jumping, which are simply avoidable. A simple technique for identifying the avalanche resonance frequency of packaged IMPATT oscillators is also described.     

Phase noise in LC oscillators

Analytical methods for the phase-noise analysis of LC-tuned oscillators are presented. The fundamental assumption used in the theoretical model is that an oscillator acts as a large-signal LC-tuned amplifier for purposes of noise analysis. This approach allows us to derive closed-form expressions for the close-to-carrier spectral density of the output noise, and to estimate the phase-noise performance of an oscillator from circuit parameters using hand analysis. The emphasis is on an engineering approach intended to facilitate rapid estimation of oscillator phase noise. Theoretical predictions are compared with results of circuit simulations using a nonlinear phase-noise simulator. The analytical results are in good agreement with simulations for weakly nonlinear oscillators. Complete nonlinear simulations are necessary to accurately predict phase noise in oscillators operating in a strongly nonlinear regime. To confirm the validity of the nonlinear phase-noise models implemented in the simulator, simulation results are compared with measurements of phase noise in a practical Colpitts oscillator, where we find good agreement between simulations and measurements     

EMI-induced failures in crystal oscillators

An electromagnetic interference (EMI) induced failure mode pertaining to crystal-based voltage-controlled oscillators (VCO) has been studied. The failure consists of a transition to a frequency of oscillation that differs from the crystal's fundamental resonant frequency, when the circuit is temporarily exposed to continuous or pulsed radio-frequency electromagnetic fields. The new state persists even after the EMI source is removed and leads to hang-up in digital systems. This mode transition has been observed experimentally. Its essential properties have been predicted theoretically and simulated numerically, using simplified oscillator models. The likelihood of observing such a failure in a noisy electromagnetic environment is assessed with respect to the radiated susceptibility levels given in MIL-STD-461B     

InP Gunn oscillators in V-band

Output powers of 200 mW and efficiencies up to 6.5% have been achieved with CW InP Gunn oscillators in V-band (50?75 GHz). Fixed frequency as well as frequency tunable oscillators have been developed. Tuning bandwidths of 19% and 6.1% have been achieved with mechanical and varactor tuning, respectively.     

Domain configurations in transferred-electron oscillators

An analysis is given of the field configuration of the drifting high-field domain in transferred-electron oscillators. It is shown that the field configuration is triangular in most samples used for microwave generation.