Ramo's Theorem Research Articles

The charge collection in single side silicon microstrip detectors

The transient current technique has been used to investigate signal formation in unirradiated silicon microstrip detectors, which are similar in geometry to those developed for the ATLAS experiment at LHC. Nanosecond pulsed infrared and red lasers were used to induce the signals under study. Two peculiarities in the detector performance were observed: an unexpectedly slow rise to the signal induced in a given strip when signals are injected opposite to the strip, and a long duration of the induced signal in comparison with the calculated drift time of charge carriers through the detector thickness—with a significant fraction of the charge being induced after charge carrier arrival. These major effects and details of the detector response for different positions of charge injection are discussed in the context of Ramo's theorem and compared with predictions arising from the more commonly studied phenomenon of signal formation in planar pad detectors.

Direct conversion X-ray sensors: sensitivity, DQE and MTF

Imaging characteristics of photoconductive flat panel X-ray image detectors, such as X-ray sensitivity, detective quantum efficiency (DQE), and resolution in terms of the modulation transfer function (MTF) are examined with applications to amorphous selenium (a-Se), polycrystalline HgI2 and CdZnTe detectors. A theoretical model has been developed for the calculation of X-ray sensitivity of a pixellated detector by using the Shockley–Ramo theorem, the weighting potential of the individual pixel and the final trapped charge distribution across the photoconductor. The X-ray sensitivity of pixellated X-ray detectors mostly depends on the mobility and lifetime product of charges that move towards the pixel electrodes, and the extent of dependence increases with decreasing pixel per unit detector thickness. A cascaded linear system model that includes incomplete charge collection and interaction depth dependent conversion gain and charge collection stages is considered for the calculation of the zero spatial frequency detective quantum efficiency DQE(0) of a direct conversion X-ray image detector. The DQE(0) performance of a-Se, HgI2 and CdZnTe detectors is examined for fluoroscopic applications. It is shown that, in addition to high quantum efficiency, both high conversion gain and high charge collection efficiency are required to improve the DQE performance of an X-ray image detector. The factors affecting the MTF of a-Se flat-panel detectors for digital mammography are investigated. Both theoretical and experimental methods are examined. A theoretical model has been developed based on cascaded linear system analysis with parallel processes to take into account the effect of K-fluorescence. This model has been used to understand the performance of a small-area prototype detector with 85 μm pixel size. The presampling MTF of the prototype has been measured and compared to a theoretical calculation based on the model. The calculation shows that K-fluorescence accounts for a 15% reduction in the MTF at the Nyquist frequency of the prototype detector. The measurement of presampling MTF of the prototype detector reveals an additional source of blurring, which is probably due to charge carrier trapping in the blocking layer at the interface between a-Se and the active matrix. This introduces a drop in presampling MTF at high spatial frequencies.

Timing in thick silicon detectors for a Compton camera

In the scope of construction of a PET apparatus based on detection of Compton scattering in silicon (Compton camera), the timing properties of 1-mm-thick silicon pad and double-sided microstrip, detectors are studied. Timing in pad detectors is also investigated for 140.5 keV /sup 99m/Tc and 364.5 keV /sup 131/I gamma rays in view of a SPECT application. Compton scattering and energy loss of the Compton electron in the silicon detector are simulated using the GEANT package. The electric field in the detector is calculated numerically for a fully depleted detector in the abrupt junction approximation, taking into account the geometry and varying the reverse voltage. Signal formation is studied using Ramo's theorem and pulse shaping properties of the trigger circuit. A time-walk cut is seen to be directly corresponding to a deposited-energy cut. At 10 keV threshold in 1-mm-thick detectors, 10 ns timing windows are shown to reject a significant portion of events, degrading efficiency or limiting the angular range in a prohibitive way. More involved techniques are therefore suggested, either in the electronics circuit or in later stages of the trigger.

Measurements of charge collection profiles in virgin and strongly irradiated silicon diodes by means of the micro-IBICC technique

Ion beam-induced charge collection (IBICC) method is a very sensitive technique to investigate the electronic features of semiconductor materials and devices. This technique consists in measuring the charge induced at the electrode by the motion of free carriers generated by a spatially scanned focused energetic ion beam. The measurement of the charge collection efficiency as a function of the ion impact position allows the electronic features of semiconductor materials and devices to be mapped. We used the micro-beam facility of the Ruder Boskovic Institute in Zagreb (HR) to perform lateral IBICC measurements of virgin, Au doped and strongly irradiated (frontal irradiation with 6.5 MeV He 2+ ions for a total dose of 2E12 ions/cm 2) p +/n/n + silicon diodes in order to evaluate charge collection profiles (CCPs) under different applied bias conditions. Basic transport parameters (minority carrier diffusion length, depletion region width) have been extracted from the experimental profiles by using a mathematical procedure based on the extended Ramo's theorem. The irradiated sample shows CCPs strongly influenced by the radiation damage as evidenced by a drop of charge collection efficiency which occurs at the end of the He 2+ ion penetration range.

Simulated and experimental spectroscopic performance of GaAs X-ray pixel detectors

In pixel detectors, the electrode geometry affects the signal shape and therefore the spectroscopic performance of the device. This effect is enhanced in semiconductors where carrier trapping is relevant. In particular, semi insulating (SI) GaAs crystals present an incomplete charge collection due to a high concentration of deep traps in the bulk. In the last few years, SI GaAs pixel detectors have been developed as soft X-ray detectors for medical imaging applications. In this paper, we present a numerical method to evaluate the local charge collection properties of pixel detectors. A bi-dimensional description has been used to represent the detector geometry. According to recent models, the active region of a reverse biased SI GaAs detector is almost neutral. Therefore, the electrostatic potential inside a full active detector has been evaluated using the Laplace equation. A finite difference method with a fixed step orthogonal mesh has been adopted. The photon interaction point has been generated with a Monte Carlo method according to the attenuation length of a monochromatic X-ray beam in GaAs. The number of photogenerated carriers for each interaction has been extracted using a gaussian distribution. The induced signal on the collecting electrode has been calculated according to the Ramo's theorem and the trapping effect has been modeled introducing electron and hole lifetimes. The noise of the charge preamplifier have been also taken into account. A comparison between simulated and experimental X-ray spectra from a 241Am source acquired with different GaAs pixel detectors has been carried out.

Review of the Shockley–Ramo theorem and its application in semiconductor gamma-ray detectors

The Shockley–Ramo theorem is reviewed based on the conservation of energy. This review shows how the energy is transferred from the bias supplies to the moving charge within a device. In addition, the discussion extends the original theorem to include cases in which a constant magnetic field is present, as well as when the device medium is heterogeneous. The rapid development of single polarity charge sensing techniques implemented in recent years on semiconductor γ-ray detectors are summarized, and a fundamental interpretation of these techniques based on the Shockley–Ramo theorem is presented.

Signals in non-irradiated and irradiated single-sided silicon detectors

A simulation of signals in silicon microstrip detectors (p +–n–n +) has been written. Electron–hole pairs are created by electrons from 90Sr beta source with Landau fluctuations considered. Simulated induced currents calculated according to Ramo's theorem are integrated and shaped. For irradiated sensors, trapping is included in the drift simulation. Using many Monte Carlo generated events, the charge collection efficiency (CCE) is calculated as a function of shaping time, applied voltage, and temperature. Results are compared with CCE measurements of unirradiated and irradiated strip detectors using readout chip (SCT 32A) with 25 ns peaking time.

X-ray sensitivity of photoconductors: application to stabilizeda-Se

The x-ray sensitivity of a high-resistivity photoconductor sandwichedbetween two parallel plate electrodes and operating under a constant field isanalysed by considering charge carrier generation that follows the x-ray photonabsorption profile and taking into account both electron and hole trappingphenomena but neglecting recombination, bulk space charge and diffusion effects.The amount of collected charge in the external circuit due to distributedgeneration of electrons and holes through the detector is calculated byintegrating the Hecht collection efficiency with Ramo's theorem across thesample thickness. The results of the model allow the x-ray sensitivity to becalculated as a function of the applied field, detector thickness and electronand hole ranges (µτ), given the field and energy dependence of theelectron and hole pair creation energy, W±, and the energy spectrum ofincident x-ray radiation. The sensitivity model was applied to stabilized a-Sethat is currently used as a successful x-ray photoconductor in the recentlydeveloped flat panel x-ray image detectors. Recent free electron-hole paircreation energy versus electric field data at room temperature and appropriateelectron and hole drift mobilities were used to calculate the sensitivity formonoenergetic x-rays at 20 and at 60 keV. For the 20 keV radiation, it was shownthat a typical detector thickness of 200 µm (4 × attenuation depth at20 keV) with currently attainable electron and hole trapping parameters in a-Sewas operating optimally, the sensitivity of which can only be increased byfurther increasing the applied field. With the receiving electrode positively biased,the sensitivity was much more dependent onthe hole lifetime than electron lifetime. The absence of hole transport resultsin a reduction in sensitivity by a factor of about 4.4, whereas the absence ofelectron transport results in a sensitivity degradation of only 22%. The ratioof hole trapping limited sensitivity to electron trapping limited sensitivity isabout 0.3. For a detector of thickness 200 µm operating at 10 V µm-1,the maximum sensitivity is about 220 pC cm-2 mR-1, and thissensitivity degrades by more than 10% when either the electron lifetime fallsbelow ~20 µs or the hole lifetime falls below ~5 µs. When thehole lifetime is very short so that the sensitivity is substantially reduced,the sensitivity versus thickness dependence at a given field exhibits a maximum(an optimal thickness) that is less than that for full absorption. In the caseof 60 keV x-ray photons, it is more useful to examine the sensitivity as afunction of detector thickness given the practical bias voltage limit. Thesensitivity versus thickness behaviour for a given bias voltage exhibits amaximum, that is an optimal thickness, that is less than that for nearly fullabsorption. Electron lifetimes longer than ~200 µs and hole lifetimeslonger than ~10 µs do not significantly affect the sensitivity.

Theory of ion beam induced charge collection in detectors based on the extended Shockley–Ramo theorem

An analysis of the charge collection process induced by focused MeV ion beams in semiconductor devices is presented. It is based on the extended Shockley–Ramo theorem that provides a rigorous mathematical tool for the calculation of the induced charge and current under the assumption of a quasi-steady-state operation of the semiconductor device. A complete description of the theory and underlying assumption is given as well as a simple application of the method aimed to evaluate the main transport properties of fully depleted semiconductors from the analysis of frontal and lateral ion beam induced charge collection (IBICC) measurements.

Statistical models for charge collection efficiency and variance in semiconductor spectrometers

Charge collection efficiency and the variance in the collected charge of semiconductor spectrometers are modeled. The model is based on a statistical approach and the extended Ramo theorem. The model yields an expression for variance in charge collection efficiency as a function of photon energy, bias voltage, and semiconductor parameters. These calculations as a function of absorption depth are particularly important in semiconductors with high atomic numbers, such as CdZnTe, since in these materials a uniform absorption cannot be assumed for a wide range of energies. Three different spectrometer configurations were considered: resistive, partially depleted Schottky barrier, and fully depleted Schottky barrier. An analytical model for the resistive configuration is presented and the results are compared to numerically obtained results of the Schottky configuration.

Evaluation of active layer properties and charge collection efficiency of GaAs particle detectors

According to Ramo's theorem the charge collection efficiency of a particle detector is mainly influenced by the field distribution between the contacts of a Schottky diode. In semi-insulating GaAs material a space charge layer is formed due to deep levels needed for the compensation of acceptors. In this paper the deep levels and their influence on the distribution of the electric field is studied experimentally by different methods of electrical characterization. It is found that the electrical active concentration of the midgap donor of ∼10 15 cm −3 at an energy of 0.67 eV below the conduction band is only about one tenth of its total concentration of ∼16 16 cm −3 as measured by infrared absorption. The Schottky barrier leakage current is found to be responsible for the variation of the electrically active deep centers and it therefore influences the charge collection efficiency (c.c.e.). The c.c.e. turns out to be inversely proportional to the active concentration of deep centers. These results are supported by our modelling of the c.c.e.: Using the transport and the Poisson equation the electrical field distribution can be calculated through the coupling of the quasi-Fermi levels and the compensation mechanism. The model calculations of charge collection efficiencies for both alpha particles and protons are confirmed by the experimental results. The work is performed within the framework of the RD8 project.

Generalization of Ramo's Theorem and Its Application to Semiconducting Materials

Ramo's theorem is generalized for the current statically induced in neighboring conductors by a specified movement of a charge carrier to include the higher frequency domain. As an example of application of the generalized theory, the current associated with carriers generated in a rectangular semiconducting material is computed. When taking the velocity difference between electrons and holes into account, the current normalized at low frequencies due to spatially uniform generation of carrier pairs in an intrinsic semiconductor is formulated as follows: I(ω)=∑n=1,k(n/ωτ)2[(1+ jωτ/n)exp (- jωτ/n)-1], where k(>1) is the ratio of the electron velocity to the hole velocity, ω is the angular frequency, and τ is the transit time for a hole traversing through the device.

The effect of velocity saturation on the shape of current signals in germanium cylindrical detectors

The current signals delivered by High Purity Germanium (HPGe) coaxial detectors are calculated analytically using Ramo's theorem, taking into account the dependence of the mobility on the electric field strength. The integrated signal corresponding to the rise time of a charge sensitive preamplifier can be derived staightforwardly from the same expressions.

Calculation of beam loading using the induced-current method in passive cavities

A method for calculating the loaded Q and beam detuning of a passive (gain or idler) cavity in an electron device is presented. It is based on the determination of induced current by a technique often referred to as Ramo's theorem. The induced current is used in conjunction with a lumped equivalent circuit representing the cavity. This leads to a solution for the self-consistent cavity fields. Although the induced-current expression is usually developed from low-frequency models, it is shown that Ramo's theorem is valid for high-frequency, steady-state analysis when the unloaded passive cavity Q is high. The method is used to calculate the loaded Q of the passive cavity of a new type of gyro-resonant electron device, the magnicon. >

The effect of velocity saturation on the shape of the current signals in silicon detectors

Current signals delivered by a silicon p +−n planar junction are computed using Ramo's theorem, taking into account the dependence of the mobility on the electric field strength.

Ultrafast semiinsulating InP:Fe-InGaAs:Fe-InP:Fe MSM photodetectors: modeling and performance

The response of InGaAs metal-semiconductor-metal (MSM) detectors as a function of all relevant detector parameters is investigated experimentally and theoretically. The calculations of the intrinsic detector response are carried out by taking fully into account the two-dimensional character of the problem. The generalized Ramo theorem is applied to obtain the induced current, which is the only measurable quantity. The calculations clearly demonstrate the distinct contributions of electrons and holes to the response. The extrinsic response is derived taking parasitics, which are obtained in different ways, into account. A full-wave analysis of the detector circuit, solving Maxwell equations, is performed. The resulting S/sub 11/ scattering parameter is measured by means of a network analyzer. Fitting of the calculated and measured S/sub 11/ parameters to an equivalent circuit diagram yields the values of the parasitic elements. Perfect agreement between calculated and measured data is found. >

Polarization effects in a Si(Li) detector

AbstractIn previous work time‐ and position‐dependent artifacts and a time‐dependent increase in the background radiation of x‐ray spectra was observed in planar Sl(Li)detectors for x‐rays in the keV region. In the present experiments the time‐dependent background was studied by use of a thin x‐ray beam. The time‐dependent background is considered to be due to incomplete charge collection. The amount of background is greater when the detector is left unexposed to x‐rays during the bias time than when the detector is constantly irradiated. The time‐dependent background was observed to decay with half‐lives of the order of 30 s, 10 min and 1–4 h. The results are interpreted such that polarization of the detector material occurs by traps that are formed under the influence of the bias voltage. During irradiation some of these traps are filled by charges from the hole‐electron pairs from the x‐ray beam. By using Ramo's theorem and converting the energy scale in the x‐ray spectra to detector depth, the polarization effect in the detector can be estimated to proceed with a speed of the order of 1 μm min−1 in an electric field of about 3000 V Cm−1.

The validity of Ramo's theorem

It is shown that the original formulation of Ramo's theorem is still valid in the presence of a space charge. The generalized version of this theorem which has been around for a very long time, is proved to be wrong. The consequences for the theories of semiconductor detectors and generation-recombination noise in p- n junctions are described.

Transport equations and Ramo's theorem: Applications to the impulse response of a semiconductor detector and to the generation-recombination noise in a semiconductor junction

Two methods, the transport equations method and the Ramo's theorem method, for describing the charge carrier transport phenomena in the space-charge region of a semiconductor junction are discussed. Remarks on the current induction effect and the validity of Ramo's theorem in semiconductor junctions are given and impulse response of a semiconductor detector is treated using both methods. These are then applied to the calculations of the impulse response of a photodiode. The applications of the two methods to the generation-recombination noise in a p-n junction, known as the collective approach and the corpuscular approach respectively, are discussed between these two Both approaches are considered with special regard to the coupling problem. The discrepancies between these two methods as applied to the impulse response and the g-r noise problems are explained. The importance of the Ramo's theorem method for the proper description of phenomena in semiconductor detectors is pointed out.

Comments on the collective and corpuscular approach of generation-recombination noise in a p- n junction

Earlier results by van Vliet (collective approach) and by Lauritzen (corpuscular approach) for the generation-recombination current noise in the space-charge region of a p- n junction are discussed. It is shown that the noise caused by generation-recombination processes in the space-charge region can be described well by the corpuscular approach, including the properly formulated Ramo's theorem. These results are more general than those obtained in the collective method with the adiabatic approximation.