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  Review of the Physics of Semiconductor Detectors

Pulse Formation Process

fig5.jpg (10279 bytes)

Fig. 5. Equivalent Circuit of Semiconductor
Detector. Where l(t) is the current generator;
CDis the capacitance of the depletion region;
RD is the resistance of the depletion region;
and Z is the series impedance.
The equivalent circuit of a semiconductor detector operated as a spectrometer is shown in Fig. 5. In most cases, effects of high resistance of the reverse-biased junction are negligible. If a zero-electric-field radiation-insensitive region is present in the detector, its impedance (a parallel RC combination) appears in series with the circuit and is indicated in Fig. 5 by the impedance Z. The impedance also accounts for any resistance (or resistance-capacitance combination) appearing in series with the contacts.

When semiconductor detectors are used as spectrometers, they are invariably connected to a charge-sensitive (integrating) preamplifier with a high dynamic input capacitance. The charge-sensitive preamplifier integrates on its feedback capacitance the current signal delivered by the detector and feeds the resulting voltage signal to the filter amplifier (main amplifier). The time behavior of the current signal at the input of the charge-sensitive preamplifier is determined by the current signal's shape and by the effect of the equivalent circuit shown in Fig. 5. The effect of the equivalent circuit is usually either negligible or easily calculated, whereas detailed considerations on the charge collection process in the detector are needed to calculate the induced current signal l(t).

Charge Collection Process and the Resulting Induced Current Signal

The current delivered by the signal generator l(t) is induced on the contacts of the detector by the motion of the charge carriers created by the ionizing radiation. Therefore, the first problem in determining l(t) is calculating the motion of the charge carriers in the detector's electric field. When this problem is solved, the induced charge can be calculated by electrostatic considerations.

The charge carriers created by the ionizing radiation drift to the contacts of opposite polarity, following the lines of force of the electric field established by the applied voltage. In the case of heavily ionizing particles such as fission fragments, the drift process does not begin immediately due to the creation of the charge cloud (see subsection "Plasma Effects").

The electric field E(r) in the detector can be calculated from known quantities: applied bias voltage, detector geometry, and resistivity of the bulk material. Once the electric field is known, the motion of a charge carrier created at a given point r0 of the detector volume can be calculated by using the values for the drift velocity Vd as a function of the electric field E given in the referenced literature. Thus the differential equation

eq5.jpg (7479 bytes)

 (5)

can be written for every charge carrier and can be solved if the initial positions r0 are known only when the charge carriers are created along a well-defined track (heavy charged particles). In the case of beta, x, and gamma radiation, the only information on r0 values is of statistical nature. The integration of Eq. (5) leads to r(t) for every created charge carrier. The charge induced by every carrier can then be calculated by electrostatic considerations. For instance, in the case of a detector with plane parallel contacts and a field E(x) across a distance W, the charge induced by a carrier q moving along a length Dx in the direction of the field is given by
 

eq6a.jpg (6792 bytes)

 (6)

independently of the shape of E(x). Equations (5) and (6) (or the appropriate induction equation) yield the contribution to l(t) of every single charge carrier and, by integration over all the created charge carriers, the total l(t) function.

Rise Time

The rise time Tt of the pulse generated by a semiconductor detector can be measured at the output of a charge-sensitive preamplifier. If the preamplifier is sufficiently fast, Tt is determined by the following factors:

1. The charge collection time TR ,
2. The rise time of the detector equivalent circuit, in most cases a negligible quantity, and
3. The plasma time (see "Plasma Effects").

In most cases TR is the dominant factor. Although a precise calculation of TR can be quite complex, the order of magnitude of TR can be easily obtained by the following formulas: 

TR @ W x 10–7s

(7)

for silicon detectors at room temperature, and
 

TR @ W x 10–8s

(8)

for germanium detectors at LN2 temperature.

In these formulas, W is the thickness of the depletion region measured in mm. For silicon detectors and for planar HPGe detectors, the value of W is provided with each detector. For coaxial Ge detectors, W is the radius of the cylinder (specified in the detector instruction manual).

The formulas given above are indicative only of orders of magnitude and do not give exact values.

The previous discussion did not consider trapping effects, which result in a loss of charge to the collection process and consequent distortion of the shape of the peak as observed with a multichannel analyzer.

Trapping Effects

Trapping of a charge carrier in a semiconductor occurs when the carrier is captured by an impurity or imperfection center and is temporarily lost to any charge transport process. In semiconductor detectors, it is useful to introduce the quantity t+ (mean free drift time):

t+ = (Nt σ Vth )–1

(9)

where

Nt = density of trapping centers,
σ = trapping cross section,
Vth = thermal velocity.

Note that t+ does not ordinarily coincide with the classical lifetime in photoconductivity theories. This is because in photoconductivity the traps are generally filled, while in a depleted detector, the traps are generally empty.

The trapped charge carrier can be reemitted in the relevant band and take part again in the charge transport process. The average time spent by a carrier in a trap is called the mean detrapping time tD and is strongly temperature dependent:

eq10.jpg (8838 bytes)

 (10)

where

C = a constant,
Et = activation energy of the trap,
K = Boltzmann's constant,
T = absolute temperature.

If the mean detrapping time is of the same order of magnitude as, or larger than, the electronic shaping constants, the charge carrier is lost to the charge collection process or is collected with significantly reduced efficiency. The result is poor energy resolution and peak tailing. On the other hand, if the mean detrapping time is orders of magnitude shorter than the charge collection time due to drift of the carriers, then the trap has no effect on the charge collection process. For this reason, normally used dopants such as Li, P, B, and Ga, which are shallow donors or acceptors, do not act as traps.

It can be shown that to first-order approximation, the efficiency of collection of a charge carrier subjected to trapping with a mean free drift time t+ is given by

eq11.jpg (8911 bytes)

 (11)

where is the collected fraction of the created charge.

In a modern germanium gamma-ray spectrometer the charge collection efficiency is of the order of 0.999, and as TR is of the order of 10–7 s, then t+, according to Eq. (11), is of the order of 10–4 s.

As typical values of Vth and σ are 107 cm · s–1 and 10–13 cm2 respectively, the maximum concentration of trapping centers permissible in the detector is of the order of 1010 cm–3 , corresponding to approximately 1 for every 1012 atoms of germanium.

Plasma Effects

Of particular interest in heavy-ion spectroscopy are plasma effects. In silicon charged-particle detectors heavy charged particles produce a dense cloud of electron-hole pairs into which the electric field, created by the applied bias voltage, cannot penetrate at the onset. Only when the cloud has been sufficiently dispersed by bipolar diffusion will the charge carriers begin to drift under the influence of the electric field. This phenomenon has the following effects:

1. A delay is generated between the creation of the electron-hole pairs (which can be considered instantaneous) and the appearance of the rising edge of the charge pulse in the detector. This delay results in an additional component to the time jitter of the signal delivered by the detector.

2. The rise time of the charge signal from the detector is slowed down; this also increases the value of the time jitter.

3. Because of the existence of a dense cloud of charge in an initially zero-electric-field region, charge carriers can recombine, with consequent loss of pulse amplitude. This phenomenon is unimportant in the detection of light particles, gamma, or x rays because the probability of carrier recombination in a semiconductor region with a high electric field is negligible.

For further information on this subject see ORTEC's application note AN-40, "Heavy-Ion Spectroscopy with Surface Barrier Detectors."