Charge Collection

The ionization of particles traversing the detector leads to the creation of free electron-hole pairs as discussed in the previous section. Fig.  shows the principal layout of a solid state detector with opposite electrodes.

Figure: A charged particle traversing the detector generates free electron-hole pairs along its track, which are moved by the electric field.

An electric field between the electrodes is required to move these carriers according to the relation

$$v=\mu E \quad,$$ (2.4)

where is $v$ is the mean carrier drift velocity, $\mu$ the mobility and $E$ the electric field. The mobilities and thus drift velocities are different for electrons and holes. The linear relation of eq.  is only valid for weak electric fields; at high electric field strength, the increasing number of collisions of the carriers with the crystal lattice finally leads to saturation of the average velocity. For silicon, empirical functions have been found for electrons and holes [14,15],

$$v_{e} = \frac{\mu_{e} E}{\sqrt{1+\left(\frac{\mu_{e} E}{v_{e,\rm\,sat}}\right)^2}}\quad {\rm and}$$ (2.5)

$$v_{h} = \frac{\mu_{h} E}{1+\frac{\mu_{h} E}{v_{h,\rm\,sat}}} \quad ,$$

where $v_{h,\rm\,sat}$ and $v_{h,\rm\,sat}$ denote the saturation velocities, which are in the order of $10^7\,\rm cm/s$. These general relations for electron and hole velocities are graphically shown in fig. .

Figure: Electron and hole velocities vs. the electric field strength in silicon.

While the charges move inside the detector bulk, a current is induced in the electrodes, no matter whether the carriers finally reach the electrodes or not. This current $i$ is proportional to the sum of all carrier velocities [16],

$$i=\frac{e}{D} \left( \sum{v_{e}} + \sum{v_{h}} \right) \quad ,$$ (2.6)

with the elementary charge $e$ and the detector thickness $D$. The integrated current gives the total collected charge $Q_c$, which is usually measured with integrating and thus charge-sensitive amplifiers,

$$Q_c=\frac{e}{D} \int \left( \sum{v_{e}} + \sum{v_{h}} \right) dt \quad .$$ (2.7)

After the generation of a free electron-hole pair, the electron moves to the positive electrode while the hole moves to the negative. If no charges are trapped, the sum of the distance they travel equals the detector thickness, regardless of their initial position. Thus, the integral term in eq.  equals the detector thickness multiplied by the number of pairs $n$, and the total collected charge is

$$Q_c=n\,e \quad .$$ (2.8)

The collected charge is stated in terms of electrons, which might mislead to the false conclusion that only the electrons contribute to the charge measured at the electrode. In fact, both electrons and holes are responsible for the charge collection in equal parts, since on average both carriers travel through half of the detector if no charge trapping occurs.

In usual silicon detectors, virtually all charges finally reach the electrodes. After heavy irradiation however, charge traps emerge such that the mean travel distance shrinks. In other solid state detector materials, such as diamond [17] or gallium arsenide [18], charge traps are always present, resulting in a mean pair travel distance less than the detector thickness. The charge collection efficiency $\eta_{c}$ can thus be expressed by the mean travel distance, which is also called ``charge collection distance'' $d_{c}$, divided by the detector thickness,

$$\eta_{c}=\frac{d_{c}}{D} \quad .$$ (2.9)

Depending on the sensor material, the electric field can be homogeneous or not. Diamond detectors merely consist of a thin film with ohmic contacts on both sides. An applied voltage results in a homogeneous field between the electrodes. With silicon however, it is impossible to operate a detector this way, because the intrinsic number of free carriers due to thermal excitation would be orders of magnitude higher than the expected detector signal. Thus, one either has to cool down the sensor to very low temperature or introduce a reverse-biased pn-junction, which is obviously more practical.

A junction is introduced by doping with acceptors and donors resulting in zones of p and n types. Starting with a homogeneously doped material, a thin layer with a high doping density of the other type is applied onto the surface, resulting in a pn-junction. On the opposite surface, which is known as backplane, the bulk type doping is enhanced to get a good ohmic contact. Fig.  shows the schematic layout of a silicon detector based on n-type bulk material, which is mostly used. Once the junction is under reverse bias, all free carriers in the bulk are drained by the electric field. The thickness of the implants with high doping concentration ($\rm p^+$ and $\rm n^+$) is in the order of a micrometer, so that the difference between bulk and total detector thickness is negligible.

Figure: Schematic cross-section of a silicon detector with implant thicknesses not to scale. The electric field results from a bias voltage above the depletion voltage.

The bulk donor (or acceptor) density $N$ can be obtained from its resistivity $r$. In the case of an n-type bulk the relation is

$$N=\frac{1}{r\, e\, \mu_{e}} \quad .$$ (2.10)

The electric properties in a one-dimensional detector model are illustrated by fig. . Backplane and implant thicknesses are neglected and an abrupt junction is assumed.

Figure: Charge density, electric field and potential in a one-dimensional model of a silicon detector at full depletion.

When the pn-junction is fully depleted (i.e. there are no free carriers in the bulk), only the core excess charges of donors and acceptors remain, such that the charge densities of bulk $\rho_{\rm bulk}$ and $\rm p^+$ implant $\rho_{\rm p}$ are

$$\rho_{\rm bulk}=e\,N_{\rm bulk} \quad {\rm and} \quad \rho_{\rm p}=-e\,N_{\rm p} \quad.$$ (2.11)

The bulk charge density is constant over the full width of the bulk. Since the global charge must be balanced in a steady state, positive and negative charges have to match. This condition is leads to the width $d_p$ of constant non-zero implant charge density which is equal to the bulk width but scaled down by the ratio of doping concentrations,

$$d_p=D\frac{N_{\rm bulk}}{N_{\rm p}} \quad.$$ (2.12)

The charge density $\rho$ can be obtained by the Poisson equation

$$\frac{dE}{dx}=\frac{\rho}{\epsilon}$$ (2.13)

with the dielectric constant $\epsilon$. In the case of full depletion, the electric field is of triangular shape, rising from zero at the backplane ($x=0$) to its maximum at the junction ($x=D$). In the implant, the field quickly drops to zero again. The maximum electric field $E_{\rm max}$ is given by

$$E_{\rm max}=\frac{e N_{\rm bulk} D}{\epsilon} \quad.$$ (2.14)

The relation

$$\frac{d\varphi}{dx}=-E$$ (2.15)

defines the electric potential $\varphi$. The potential difference between backplane and implant electrodes is the voltage $V$ applied to the detector. At full depletion, $V_{\rm depl}$ is given by

$$V_{\rm depl}=\frac{e N_{\rm bulk} D^2}{2 \epsilon}=\frac{D^2}{2 r \mu_{e} \epsilon} \quad.$$ (2.16)

If the applied voltage is higher than the depletion voltage, a constant offset adds to the electric field as shown in fig. . With $V \gg V_{\rm depl}$, the electric field can be approximately considered constant. When the applied voltage is below full depletion, the field does not extend over the whole bulk. With its maximum still at the junction, only a fraction of the sensor is depleted. Obviously, the charge collection is inefficient in that case, since the carriers do not move outside the electric field. The width of the depletion zone, or collection distance $d_c$, is given by

$$d_c=D\sqrt{\frac{V}{V_{\rm depl}}} \quad.$$ (2.17)

Thus, the efficiency of a silicon detector is given by

$$\begin{array}{ll} \eta_{c}=\sqrt{\frac{V}{V_{\rm depl}}} & \... ...\eta_{c}=1 & \mbox{for $V \geq V_{\rm depl}$}\quad. \end{array}$$ (2.18)

Fig.  compares the theoretical efficiency (eq. ) to measurement [12].

Figure: Theoretical and measured efficiency of a silicon detector. While theory suits well at depletion and above, it does not perfectly describe the low voltage regime.

In a homogeneous electric field, the carrier velocities are constant. Initially all carriers move towards the electrodes and gradually they are drained at the electrodes until all charges are gone. As the mobilities of electrons and holes differ, the current waveform is a superposition of two triangles of the same area, but different slope.

In silicon detectors however, the behavior is different because the electric field depends on the position and moreover, the carrier velocities are on the edge of saturation. For a better understanding of the influence of the various parameters, the current waveform has been simulated [19,20] based on the above model including the nonlinear velocities (eq. ). The movement and resulting currents of both electrons and holes are calculated in finite time steps.

Figure: Electron and hole currents induced by a MIP in a standard silicon detector ( $D=300\rm\, \mu m$, $r=4\rm\, k\Omega cm$) at the depletion voltage and at $V=2V_{\rm depl}$.

Fig.  shows the current contributions of electrons and holes in a standard silicon detector at $V_{\rm depl}$ and at $2V_{\rm depl}$. Neglecting the nonlinear velocities and the quantization of the carriers, it can be shown that the electron component follows an exponential decay at full depletion. A single electron starting at the junction ($x=D$) moves according to

$$D-x=\int v\,dt \quad .$$ (2.19)

Substituting the velocity with eq.  and using a linear dependence of the electric field on the position, one gets the differential equation

$$-\frac{d}{dt}x=\frac{\mu_e E_{\rm max}}{D} x$$ (2.20)

with the solution

$$x(t)=D e^{-\frac{\mu_e E_{\rm max}}{D}t}\quad,$$ (2.21)

using the initial condition $x=D$. In the case of twice the depletion voltage as shown in the insert of fig. , the current shapes are approximately linear since the electric field is almost constant.

Figure: Currents induced by a MIP in silicon detectors ( $r=4\rm\, k\Omega cm$) of various thicknesses ( $D=300, 400, 500\rm\, \mu m$) at their depletion voltages ( $V_{\rm depl}=79,141,220\rm\, V$).

Fig.  shows the sum current of detectors with various thicknesses at their respective depletion voltages, which scale with the square of the thickness (eq. ). The current waveforms approximately scale with the thickness. Although the distances, which the carriers have to travel, are farther, the higher electric field results in faster movement. If the linear relationship between electric field and carrier velocities (eq. ) held, the collected charge would exactly scale with the detector thickness at the respective depletion voltage. The overall drift time $t_{\rm drift}$ is given by

$$t_{\rm drift}=\frac{D}{v}\quad.$$ (2.22)

The velocity will be substituted with the integral average of eq.  over $x$,

$$v=\frac{1}{D}\int_0^D \mu E(x) dx\quad.$$ (2.23)

With the linear dependence of the electric field on the position, we get

$$t_{\rm drift}=\frac{D^2}{\int_0^D \frac{\mu N_{\rm bulk}}{\epsilon}x\,dx}=\frac{2\epsilon}{\mu e N_{\rm bulk}},$$ (2.24)

which is independent of the thickness. Although the maximum electric field in silicon detectors of $300$ to $500\rm\, \mu m$ touches the saturation region of the carrier velocities, the nonlinear influence is not yet dominant, as seen in fig. . Thus, the benefit of silicon detectors thicker than the canonical $300\,\rm\mu m$ is an approximately proportionally higher signal output.