Detectors Based on Ionisation in Semiconductor Materials

Abstract

Semiconductor detectors are based on the detection of electron–hole pairs created in a semiconductor material by ionising radiation. Compared to gas-based detectors, an advantage of semiconductors is that the amount of energy needed to produce one pair of free charge carriers is about a factor 10 less. Another important difference is that the density of a semiconductor is typically 1000 larger than the density of a gas. The primary charge signal is therefore larger than in gas-based detectors. In semiconductors, however, it is much more difficult to obtain charge multiplication. All semiconductor detectors in use today are without any built-in charge amplification mechanism. As a result, the signals in semiconductor detectors are very small and extremely good low-noise electronics is essential.

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Semiconductor detectors are based on the detection of electron–hole pairs created in a semiconductor material by ionising radiation. Compared to gas-based detectors, an advantage of semiconductors is that the amount of energy needed to produce one pair of free charge carriers is about a factor 10 less. Another important difference is that the density of a semiconductor is typically 1000 larger than the density of a gas. The primary charge signal is therefore larger than in gas-based detectors. In semiconductors, however, it is much more difficult to obtain charge multiplication. All semiconductor detectors in use today are without any built-in charge amplification mechanism. As a result, the signals in semiconductor detectors are very small and extremely good low-noise electronics is essential.

Semiconductor detectors are expensive, because extremely pure starting material is required and only small detectors can be made. The most commonly used detector materials are germanium and silicon. Due to recent technological advances, cadmium–telluride and cadmium–zinc–telluride have now also been improved to the point where these materials have become useful as particle detectors.

References [1, 2, 3, 4] provide a more in-depth discussion of semiconductor radiation detectors, semiconductor devices and the fundamentals of semiconductors.

5.1 Introduction to Semiconductors

In this section, we briefly review some properties of semiconductors that are relevant to their use as detectors for ionising radiation. We always mention silicon, but similar considerations apply to germanium and other semiconductors.

The electronic configuration of silicon is the configuration of neon, plus two electrons in the 3s level and two electrons in the 3p level. These four valence electrons are involved in the chemical bonding of silicon. In a silicon crystal, all these electrons are shared between the atoms, i.e. the electrons are not bound to one particular atom and travel freely in the crystal. Each silicon atom forms four covalent bonds with four neighbouring silicon atoms. These four neighbours are located at the corners of a regular tetrahedron surrounding the atom. This gives rise to a lattice with the same structure as diamond. A silicon crystal therefore is not isotropic and properties such as electron drift velocity will depend on the orientation of the movement relative to the lattice structure. These non-equivalent directions are designated by the Miller indices, e.g. (1,1,1), (1,0,0) etc. We refer the reader to the literature on solid state physics, for example [3], for the meaning of this notation.

In a crystal, the energy levels of individual atoms form so-called ‘bands’ with closely spaced energy levels. In between the allowed energy bands there are forbidden energy regions called `band gaps’. An electron can never be in a stationary state with an energy corresponding to the band gap. The different energy levels inside a band are distinguished by the wave number k. The quantity p ^* = kћ is sometimes referred to as the ‘quasi momentum’, because it shares some properties with the momentum of a free particle. This analogy, however, should be used with caution. The relation between the electron energy and the wave number in the different bands is illustrated in Fig. 5.1. This diagram is called the electronic band structure, and it is obtained by solving the Schrödinger equation for the lattice. Let us assume that all the levels in the first and the second band are filled with electrons and that nearly all the levels in the third band are empty. In this case, the bands will be labelled core band, valence band and conduction band as indicated in Fig. 5.1.

Fig. 5.1

Electronic band structure for a one-dimensional lattice. If all the levels in the first and the second band are filled with electrons, these are called the core and the valence band. The first empty band is called the conduction band

In the conduction band there is one value of p ₀ ^* = kћ, where the energy is minimum and around this minimum the energy can be parametrised as

$$E=E_c +(p^* -{p_0}^*)^2/{2m_e}^*.$$

Unlike the situation shown in Fig. 5.1, in silicon the minimum of the conduction band does not occur at the same k-value as the maximum of the valence band. If there are any electrons present in the conduction band, these electrons will sink to the bottom of the band. These electrons will behave as almost free negative particles with effective mass m _e ^*. If any electrons are present in this band, they can move around freely, and for this reason this band is called the conduction band. If there is an electric field, these electrons will give rise to a net electric current.

Nearly all the levels in the valence band are occupied by an electron. If all the levels are occupied by an electron, there is no net electric current. If any electrons are missing, these vacancies are called holes. The holes rise to the top of the band. Near the top of the band the energy of the levels around the maximum can be parametrised as E = E _v − (p ^* − p ₀ ^*)²/2m _h ^*. It can be shown that the vacancies near the top of the band behave as almost free positive particles with effective mass m _h ^*. If there is an electric field present, these holes will also give rise to a net electric current.

In silicon, the four valence electrons form the valence band. The next higher electron energy level in silicon forms the conduction band. In between these two bands there is a band gap. The width of the energy gap between the valence band and the conduction band is the most important fact in determining the electrical properties of the material. If this energy gap is significantly larger than 1 eV, the material is an insulator. In a metal there is no energy gap between the two bands and the material is a conductor. If this energy gap is of the order of 1 eV, the material is a semiconductor. These different possibilities are illustrated in Fig. 5.2.

Fig. 5.2

Energy band structure of conductors, insulators and semiconductors. The vertical axis represents the electron energy, the horizontal axis the position in the lattice

At absolute zero temperature all levels below a critical value E _F, called the Fermi energy, are occupied by electrons and all levels above the Fermi energy are empty. At any temperature different from zero, and in the condition of thermal equilibrium, the probability f(E) that a particular energy level is occupied by an electron is given by the Fermi–Dirac distribution

$$f(E) = \frac{1}{{e^{\frac{{E - E_F \\ \\}}{{kT}}} + 1}}$$

((5.1))

In this equation, T represents the absolute temperature and k the Boltzmann constant. The electron state where the energy of the electron is equal to E _F has 50% chance of being occupied by an electron. All the states with energy larger than the E _F have a probability of being occupied less than 50%, and all the states with energy smaller than E _F have a probability of being occupied larger than 50%. In insulators or semiconductors the Fermi energy is situated in the band gap. Except at absolute zero temperature there are always some electrons present in the conduction band and some unoccupied levels, or holes, in the valence band. In an insulator, the band gap is large and the probability of having electrons in the valence band or holes in the conduction band is extremely small. The conductivity caused by such electrons or holes is negligible. In semiconductor materials, the band gap energy is much smaller, and this probability is no longer negligible. This conductivity is much smaller, however, than the typical conductivity of true conductors such as metals, hence the name semiconductors.

The density of the electrons in the conduction band of a semiconductor is given by

$$\frac{{dn(E)}}{{dE}} = \rho (E) \,\,f(E)$$

((5.2))

where ρ(E) represents the density of electron states. Close to the bottom of the conduction band, this density of states is similar to the density of states of free particles enclosed in a cubic potential well. This last density is derived in Exercise 6 and is given by

$$\rho (E)\,\, dE = 4\pi \left( {\frac{{2m_e }}{{h^2 }}} \right)^{3/2} \sqrt E \,\,dE$$

The density of states for electrons near the bottom of the conduction band is given by the same equation but with the energy replaced by (E – E _c), where E _c is the energy of the bottom of the conduction band and the electron mass replaced by the effective electron mass ‘m _e ^*’ in the lattice.

The number of electrons per unit volume in the conduction band n _e is hence given by

$$\begin{array}{l} n_e = \int {\rho (E)\,\, f(E) \,\,dE} \\ =\,\, 4\pi \left( {\frac{{2m_e^* }}{{h^2 }}} \right)^{3/2} \int\limits_{E_c }^\infty {\sqrt {(E - E_c )} } \frac{1}{{e^{\frac{{E - E_F }}{{kT}}} + 1}}\,\, dE \\ =\,\, 4\pi \left( {\frac{{2m_e^* }}{{h^2 }}} \right)^{3/2} e^{ - \frac{{E_C - E_F }}{{kT }}} (kT)^{3/2} \int\limits_0^\infty {\sqrt x }\,\, e^{ - x}\,\, dx \\ =\,\, 4\pi \left( {\frac{{2m_e^* }}{{h^2 }}} \right)^{3/2} (kT)^{3/2}\,\, \frac{{\sqrt \pi }}{2} e^{{ - \frac{{E_C - E_F }}{{kT }}}} \\ \end{array}$$

Similarly, the density of hole states near the top of the valence band is given by the same equation but with the energy replaced by (E _v – E), where E _v is the energy of the top of the valence band and the electron mass replaced by the effective hole mass in the lattice. In silicon, the effective electron mass and the effective hole mass are not very different from the true electron mass. This is not always the case for other semiconductors. The probability to have a hole in the valence band is given by 1 – f(E), where f(E) is the Fermi–Dirac distribution given by Eq. (5.1). The number of holes per unit volume in the valence band n _h is therefore given by

$$ n_h = 4\pi \left( {\frac{{2m_h^* }}{{h^2 }}} \right)^{3/2} (kT)^{3/2}\,\, \frac{{\sqrt \pi }}{2} \,\,e^{ - \frac{{E_F - E_V }}{{kT }}}$$

In very pure semiconductor materials the Fermi energy is halfway between the top of the valence band and the bottom of the conduction band. The density of electrons in the conduction band n _e and the density of holes in the valence band n _h are equal. This carrier density is called the intrinsic carrier density n _i and is proportional to

$$n_i \propto T^{3/2}\ e^{\left( - \dfrac{{E_g }}{{2kT}}\right)}$$

((5.3))

where E _g is the width of the band gap between the conduction and the valence band. Equation (5.3) shows that the conductivity in semiconductors will strongly increase with the temperature. In metals the conductivity is less dependent on the temperature and tends to decrease with temperature.

It is important to realise that the relation n _i = n _e = n _h only holds for extremely pure and defect free materials. Let us now assume that the silicon contains trace amounts of phosphorus. Phosphorus is the next element after silicon in the periodic table and hence it has three 3p electrons. The additional 3p electron occupies an energy level that corresponds to the conduction band in silicon. Because of the higher nuclear charge Z of the phosphorus atoms, the energy of this level is a little lower than the corresponding level of silicon. If a silicon atom in the crystal is replaced by a phosphorus atom, this will create a localised level just below the conduction band and the 3p electron will occupy this level. The same holds for a number of other elements with the same electron structure as phosphorus, such as arsenic or antimony. The energy difference between these levels and the conduction band is very small and due to thermal agitation these electrons will jump to the conduction band and move freely around in the lattice, leaving the 3p phosphorus level empty most of the time. In a state of thermal equilibrium, the Fermi–Dirac Eq. (5.1) still describes the electron distribution, but the Fermi energy is now no longer in the middle of the band gap but much closer to the conduction band.

Similarly, if the silicon contains trace amounts of trivalent materials such as gallium, boron or indium, there will be empty acceptor levels just above the valence band, and these will give rise to holes in the valence band. In this case the Fermi level is close to the valence band. We see that trace amounts of impurities will strongly influence the concentration of electrons and holes in the crystal and hence the electrical properties of the material.

In the presence of impurities, the densities of electrons and holes in thermodynamical equilibrium are given by

$$\begin{array}{l} n_e \propto T^{3/2}\ e^{ - \frac{{E_c - E_F }}{{kT}}} \\ n_h \propto T^{3/2}\ e^{ - \frac{{E_F - E_v }}{{kT}}} \\ \sqrt {n_e n_h } = n_i \propto T^{3/2}\ e^{ - \frac{{E_g }}{{2kT}}} \\ \end{array}$$

If we call E _i the intrinsic energy level, i.e. the energy of the middle of the band gap, these equations can be written as

$$n_e = n_i\ e^{ + \frac{{E_F - E_i }}{{kT}}} ; \,\,\,\, n_h = n_i \,\,e^{ + \frac{{E_i - E_F }}{{kT}}}$$

If a pure sample of silicon is doped with a small concentration of electron donor atoms, the material is called n-type silicon. Similarly, if a silicon crystal is doped with an electron acceptor atoms, the material is called p-type silicon.

Consider a piece of n-type silicon. The position of the Fermi level is determined by the condition that the total charge in the crystal is zero. Therefore, the total number of electrons in the conduction band equals the number of holes in the valence band plus the number of empty donor levels. Usually, the number of holes in the valence band and the number of electrons left in the donor levels is negligible compared to the number of electrons in the valence band and we can write

$$\begin{array}{l} n_e \approx N_D \approx n_i\ e^{+ \frac{{E_F - E_i }}{{kT}}} \\ E_F - E_i = kT\ln \frac{{N_D }}{{n_i }}\\ \end{array}$$

The position of the Fermi level moves closer to the conduction band as the density of donors gets larger. Similarly, for p-type silicon, the position of the Fermi level moves closer to the valence band as the density of acceptor levels is larger. If both types of dopant are present, the dopant with the largest density determines the position of the Fermi level.

The electrons and holes in the crystal will move under the influence of an electric field. In the same way as charges in gases, the electrons or holes will constantly collide with ‘obstacles’ in the lattice and in each collision completely change direction. The time between collisions is of the order of 10⁻¹² s. The thermal velocity of the electrons and holes is much larger than the drift velocity. At the macroscopic level, the drift of the electrons and holes under the influence of an electric field will look like a smooth process, but at the microscopic level the motion of the charge carriers will be completely chaotic. In analogy with the motion of charges in gases, the dependence of the electron velocity on the electric field is written as v _e =$\mu$ _e E, where E is the electric field and v _e and μ _e are the electron velocity and the electron mobility, respectively. The corresponding quantities for holes are defined in the same way.

The electron and hole velocities are illustrated in Fig. 5.3. Notice that the velocities of electrons and holes are similar and that these are comparable to the electron velocities in gases. Notice also that, unlike the situation in gases, the drift velocity of electrons is only somewhat larger than the drift velocity of holes. This reflects the fact that the collision cross sections for electrons and holes are only slightly different. Because of this, it is much more difficult to achieve stable charge multiplication by the avalanche formation mechanism in semiconductors than in gases.

Fig. 5.3

Electron and hole drift velocities in silicon as a function of the electric field. The velocity shown here is the velocity in a plane parallel to the crystallographic <111> direction. The different curves correspond to different values of the temperature in degrees Kelvin. Figure from [5], © 1975 IEEE

Table 5.1 lists some relevant physical properties of silicon and germanium. Notice the difference in radiation length and the difference in band gap energy.

Table 5.1 Some physical properties of silicon and germanium. Unless otherwise noted, data for silicon are at room temperature and data for germanium at 77 K

To illustrate how trace impurities will strongly influence the electrical properties of a semiconductor, let us consider a cuboid of semiconductor material with a metal contact on two opposing surfaces (see Fig. 5.4).

Fig. 5.4

To measure the resistivity of bulk silicon, a parallelepipedal block of silicon is equipped on two opposing surfaces with ohmic metal contacts

From the definition of the resistivity ρ we have, where R is the resistance of this block

$$\begin{array}{l} R = \frac{t}{A}\rho \\ \\ i = \frac{V}{R} = \frac{{A.\,V}}{{t.\,\rho }} \\ \end{array}$$

On the other hand, the current in this block is related to the velocity of the electrons and holes by

$$i = e\ A (n_e\,\, v_e + n_h\,\, v_h )$$

Using

$$v_e = \mu _e E = \mu _e \frac{V}{t}$$

We therefore have

$$\rho = \frac{1}{{e(n_e \mu _e + n_h \mu _h )}}$$

((5.4))

Often the number of charge carriers of one type is much larger than the number of charge carriers of the other type. In that case the resistivity of the material is entirely determined by charge carrier density of the majority charge carriers. The intrinsic carrier density and the mobilities of the charges in silicon are given in Table 5.1. From these data, one can readily calculate that the resistivity of intrinsic silicon is 230,000 Ω cm. A donor concentration as low as 0.2 ppb (2 donor atoms in 10¹⁰ silicon atoms) will reduce this bulk resistivity to 463 Ωcm.

This example shows that, when trying to collect the ionisation current in silicon, we will always draw a fairly large current. As is shown in Chap. 8, this gives rise to a large noise. To avoid this current we take advantage of the property of semiconductors to form diodes. If the diode is polarised in the reverse bias mode, a strong reduction of the dark current is obtained.

5.2 The Semiconductor Junction as a Detector

The operation of nearly all present-day electronic devices is based on the formation of n–p semiconductor junctions. The same principle is the basis of the use of semiconductors as detectors for ionising radiation.

An n–p junction is schematically represented in Fig. 5.5. In this junction, a region of n-type silicon is in contact with a region of p-type silicon. This should not be realised by pressing together two pieces of silicon, since the crystalline lattice should be continuous over the junction region. Both the n-type region and the p-type region are electrically neutral. However, in the n-type region there are many free electrons in the conduction band and almost no holes in the valence band, while in the p-type region there are many free holes in the valence band and almost no electrons in the conduction band. Owing to the thermal agitation, these charges will diffuse to the adjacent region of the other type, creating an excess of negative charge in the p-region and an excess of positive charge in the n-region. These charges will build up a potential difference and therefore an electric field over the junction region. This diffusion process will stop when the electric field generated is sufficiently large to prevent any further build-up of a charge difference. At this point the Fermi levels in the n-type silicon and in the p-type silicon are at the same level. In this contact area, the number of free charge carriers will be strongly reduced because free charges are removed by the electric field. This region with a reduced number of charge carriers is called the depletion region. This field will push any charges created in this region to the n-type side or the p-type side. Any charges created outside the depletion region will not be collected, but will simply recombine until thermal equilibrium is again reached.

Fig. 5.5

(a) Schematic diagram of a n–p junction, (b) diagram of electron energy levels showing the creation of a contact potential V ₀, (c) charge density, (d) electric field intensity

It should be mentioned that this potential difference between the two sides of the silicon will not be observed as a potential difference between the two metal contacts applied to the two sides of the device. This is because at each metal–silicon contact there is also a contact potential and this will compensate the potential difference caused by the diffusion. Under condition of thermal equilibrium the Fermi levels in the silicon and in both metal contacts must be the same and therefore both metal parts must be at the same potential.

The structure represented in Fig. 5.5 will function as a diode. Let us assume we have made, so-called, ohmic metal contacts on the n-type and p-type silicon side. Such contacts only add or remove the majority charge carriers of each side and have a negligible resistance. Ohmic contacts will be discussed further at the end of this section. If, using these external metal contacts, we reduce the potential difference between the two sides, the majority charge carriers will again be able to flow to the other side and a current related to the concentration of majority charge carriers will flow. If, using these external metal contacts, we increase the potential difference between the two sides, the majority charges will not be able to flow to the other side, and the current will be related to the concentration of minority charge carriers. That is, we have a large current if the junction is forward biased and a small current if the junction is reverse biased. As a result, the junction behaves as a diode. A diode is an electrical device that allows a large current to flow in one direction and only a small current in the other direction.

In practice, an n–p junction is usually made by starting from a homogeneous block of, say, n-type silicon. A dopant is diffused in the material from one side by exposing the silicon to a vapour of the dopant material, creating a region of p-type silicon with a finite depth. The resulting dopant concentrations in the junction are represented in Fig. 5.6(a).

Fig. 5.6

(a) Profile of dopant concentration in an n–p junction. (b) Idealised profile of the dopant concentration in an n–p junction diode, as used in the calculation

Below we want to derive some essential properties of p–n junctions when used as detectors for ionising radiation. To derive these results we use a simplified model of such a p–n junction. We model the dopant concentration by assuming a concentration that is constant both in the n-type silicon and in the p-type silicon. This situation is illustrated by Fig. 5.6(b). For the calculation, we will furthermore assume the depletion layer is completely devoid of free charge carriers. The charge density in the depletion layer is eN _d and −eN _a in the n-type and p-type region, respectively, and N _a >> N _d. Outside the depletion layer, the number of free electrons and holes exactly compensates the positive or negative space charge caused by the dopant atoms and, because of the presence of free charges, the electric field must be zero. The widths of the two parts of the depletion region are written as x _n and x _p, with x = 0 taken to be at the border between the two types of silicon. Applying the Maxwell equation, $\int {D\,\,dv = \int {q\,\,dv} }$, over a pillbox-shaped volume containing the junction, one can see that the total charge in the junction should be zero, therefore we have N _d .x _n = N _a .x _p, and hence x _n >> x _d.

The electric potential in a piece of silicon with a junction can be described by a Poisson equation. In this equation, ρ(x) represents the charge density and e is the dielectric constant of the medium.

$$\frac{{d^2\,V}}{{dx^2 }} = - \frac{{\rho (x)}}{\varepsilon }$$

The diode can be seen as made up of four regions with different charge densities. In each region, the charge density is constant and it is therefore quite easy to solve the Poisson equation in each region separately. At the boundary between the regions, the potential and the electric field should be continuous. The four regions are

$$\begin{array}{*{20}c} {{\rm{Region1}},\quad{V}_{\rm{1}} (x):} \hfill & { - \infty< x <- x_p } \hfill & {\rho= 0} \hfill\\ {{\rm{Region2}},\quad{V}_{\rm{2}} (x):} \hfill & { - x_p< x < 0} \hfill & {\rho=- eN_a } \hfill\\ {{\rm{Region3}},\quad{V}3(x):} \hfill & {0 < x < x_n } \hfill & {\rho= eN_d } \hfill\\ {{\rm{Region4}},\quad{V}_{\rm{4}} (x):} \hfill & {x_n< x < \infty } \hfill & {\rho= 0} \hfill\\\end{array}$$

In the equations above e represents a positive number equal in magnitude to one electron charge. In region 1 and 4, there are free charge carriers and there is no electric field. We therefore have

$$\begin{array}{l} \dfrac{{dV_1 }}{{dx}} = 0, \Rightarrow V_1 (x) = {\rm{constant}} \\ \\ \dfrac{{dV_4 }}{{dx}} = 0, \Rightarrow V_4 (x) = {\rm{constant}} \\ \end{array}$$

The potential V ₃(x) is found by solving the Poisson equation in region 3, i.e. for 0<x<x _n.

$$\begin{array}{l} \displaystyle\frac{{d^2 V_3 }}{{dx^2 }} = - \frac{{eN_d }}{\varepsilon } \\ \displaystyle\int {\frac{{d^2 V_3 }}{{dx^2 }} dx = \frac{{dV_3 }}{{dx}}} = - \frac{{eN_d }}{\varepsilon }.\, x + C_1 \\ \displaystyle\int {\frac{{dV_3 }}{{dx}} dx = V_3 (x) } = - \frac{{eN_d }}{\varepsilon } \frac{{x^2 }}{2} + x.\, C_1 + C_2 \\ \end{array}$$

Using the continuity condition at the boundary between region 3 and region 4, we have

$$\begin{array}{l} \displaystyle\frac{{dV_3 }}{{dx}} = 0 \,\,{\rm{for}}\,\, x = x_n , \,\,{\rm{and\, therefore \,\,\,\,C}}_{\rm{1}} = \frac{{eN_d }}{\varepsilon }. \,\,x_n \\ V_3 (x) = \frac{{eN_d }}{\varepsilon } \left( - \frac{{x^2 }}{2} + x.\, x_n \right) + C_2 \\ \end{array}$$

By a similar calculation we find for the potential V ₂(x) in region 2:

$$V_2 (x) = \frac{{eN_a }}{\varepsilon } \left(\frac{{x^2 }}{2} + x. \,x_p \right) + C^\prime_2 \,\,\,\,{\rm{for}}\,\,\,\, - x_p < x < 0$$

Using the continuity condition at x = 0, we have V ₂(0) = V ₃(0) and hence C ₂ = C'₂. The potential difference V ₀ over the junction is therefore given by

$$V_0 = V_4 - V_1 = V_3 (x_n ) - V_2 (- x_p ) = \frac{{eN_d }}{\varepsilon }\frac{{x_n^2 }}{2} + \frac{{eN_a }}{\varepsilon }\frac{{x_p^2 }}{2} \approx \frac{{eN_d }}{\varepsilon }\frac{{x_n^2 }}{2}$$

In our example where N _a>>N _d, we have x _n>>x _p and the thickness d of the depletion layer is to a good approximation d ≈ x _n. We see that the depletion region extends entirely towards the low dopant concentration region of the silicon. The thickness of the depletion layer depends on the smaller of the two dopant concentrations, in our example N _d. We obtain the following three useful relations: Thickness of the depletion layer:

$$d = \sqrt {\frac{{2\varepsilon V_0 }}{{eN}}}$$

((5.5))

In this and the following equations N represents the smaller of the two dopant concentrations. The capacitance of a parallel plate capacitor is given by

$$ C = \frac{{\varepsilon A}}{d}$$

where ‘A’ represents the area and d the distance between the plates. We therefore have the following expression for the capacitance per unit area in a junction diode:

$$\frac{C}{A} = \frac{\varepsilon }{d} = \sqrt {\frac{{\varepsilon eN}}{{2V_0 }}}$$

The capacitance of the junction is important because it represents an important source of noise (see Chap. 8). Capacitances well below 1 pF/mm² can be obtained.

From our calculation, it also follows that the electric field in the depletion layer has a triangular shape as shown in Fig. 5.7. The field reaches its maximum value E _max at the boundary between the two types of silicon. We therefore have the following expression for the value of the electric field at the point where it reaches its maximum value:

$$E_{\max } = \frac{{2V_0 }}{d} = \frac{{d\, eN}}{\varepsilon }$$

((5.6))

Equation (5.5) relates the thickness of the depletion layer to the voltage difference over the junction and to the dopant concentration of the part of the junction with the smallest dopant concentration. The thickness of the depletion layer is important, because in a junction only the charges induced in the depletion layer are collected. The smallest dopant concentration that can be used is essentially determined by the purity of the starting silicon material. The purer the silicon, the lower the dopant concentration that can be used. Modern n-type high purity silicon typically has a resistivity of 20,000 Ω cm. From Eq. (5.4) we find that this corresponds to a donor concentration of 2.3×10¹¹/cm³.

Fig. 5.7

Shape of the electric field over the junction

If no external voltage is applied, the potential difference over the depletion layer is of the order of 0.7 V and the corresponding thickness of the depletion layer is about 64 μm. To increase this thickness one can apply a reverse bias voltage over the diode. However, as the reverse bias voltage is increased, the maximum field over the junction also increases, and at some point this field is so strong that electrons and holes acquire sufficient energy to produce further electron–hole pairs, i.e. we have charge amplification. If the reverse bias voltage is increased further, eventually breakdown will occur.

From Eq. (5.6) we obtain a relation between the maximum thickness of the depletion layer and the breakdown voltage

$$d_{\max } = \frac{{\varepsilon \,E_{\rm breakdown} }}{{e\,N}}$$

This equation shows that the maximum thickness of the depletion layer is inversely proportional to the dopant concentration. The smallest dopant concentration that can be used depends on the amount of impurities in the starting material. A thick depletion layer is only possible if extremely pure materials are used. The breakdown voltage is about 16,000 V/mm in high resistivity n-type silicon, allowing a depletion layer of up to10 mm to be obtained in silicon. In germanium, a depletion layer of up to 10 cm can be achieved.

To have a useful radiation detector it is essential that all the metal contacts on the silicon are realised in such a way as to have a negligible resistance. These are called ohmic contacts. Figure 5.8 shows how this is realised in the case of a metal contact on n-type silicon. Over the contact layer between the metal and the silicon there is a potential difference equal to the difference in the work functions between the two materials. The work function is the energy needed to move an electron from a point inside the material at the Fermi level, to a point outside the material. A region of space charges and a depletion region in the silicon are associated with this potential difference. In a metal, the region with non-zero space charge is extremely thin. If the concentration of donors is large in the contact region of the silicon, the total depletion region is very thin and the electrons can tunnel through this potential barrier. Such a thin layer of heavily doped n-type or p-type silicon is denoted as n+ or p+ layers in the literature. Ohmic contacts allow the current to pass in both directions with a resistance that is small compared to the bulk resistance of the silicon.

Fig. 5.8

Energy levels in a metal contact on a piece of n-type silicon. The notation eΦ designates the work function. Figure (a) shows the levels in the absence of contact between the materials. Figure (b) shows the situation when there is contact

5.3 Silicon Semiconductor Detectors

Because of its rather long radiation length (93.6 mm), silicon is mainly used as a detector for charged particles. It can be used to track minimum ionising particles, and it is an almost ideal detector for alpha particles. The range of alpha particles of nuclear origin never exceeds 1 mm in silicon, and the amount of ionisation collected is therefore proportional to the energy of the alpha particle. In order to accurately measure the energy of the alpha particles it is important that the dead layer on the entrance side of the particle in the silicon is as thin as feasible. The amount of energy lost in this dead layer depends on the angle of penetration of the alpha particle and this will degrade the energy resolution. Usually, surface barrier detectors are used for alpha particle detection because this allows to obtain a very thin dead layer. In this type of detector, the junction is formed between the metal and the silicon. The resulting depletion layer behaves very much in the same way as discussed earlier. The metal contact also needs to be kept as thin as possible. Instead of using gas phase diffusion, ion implantation is often used as a method for achieving carefully controlled dopant layers. In ion implantation the surface of the silicon is exposed to a beam of ions produced by an accelerator. With this technique, entrance windows as thin as 34 nm of silicon equivalent can be achieved.

Figure 5.9 shows the energy spectrum of alpha particles recorded with a surface barrier detector. The energy resolution that can be achieved with this type of detector can be derived as follows. Let N be the number of electron–hole pairs created by the alpha particle. The energy of the alpha particle is proportional to the number of electron–hole pairs created: E[eV] = 3.62N. The r.m.s. deviation of the measured energy is given by

$$\sigma \{ E\} = 3.62\ \sqrt {NF} = \sqrt {3.62\,\, E\,\, F}$$

As before, the Fano factor ‘F’ is due to the fact that energy conservation reduces the fluctuations on the number of charges produced to be less than what it would be for a Poisson distribution. The energy resolution, expressed as full width at half maximum (FWHM) is therefore given by

$$FWHM[{\rm eV}] = 2.35 \sqrt {3.62 \,\,F\,\, E[{\rm eV}]}$$

Fig. 5.9

Alpha particle spectrum of ²³⁴U recorded by a high-resolution surface barrier detector. Figure from [6] by courtesy of ORTEC

The experimentally measured energy resolution is somewhat larger than what is predicted by the formula above. For example, for an alpha particle of 5.5 MeV, the formula predicts an energy resolution of 3.7 keV, but only about 10 keV is achieved in the best detectors. The main reason for this discrepancy is the energy loss of alpha particles due to elastic collisions with silicon nuclei. The recoil nuclei in these collisions are usually too slow to produce any ionisation as silicon and the corresponding energy is lost.

Silicon detectors are also commonly used to localise charged particle trajectories. If the particles are minimum ionising particles, about 30,000 electron–hole pairs are produced in a silicon slice of only 300 μm thick. That is a small signal but sufficient to be detected by modern low-noise electronics. To provide particle localisation, the electrode on one side of the silicon is subdivided into strips and each strip is connected to an amplifier (see Figs. 5.10 and 5.11). The distance between the strips is typically 200 μm. A detector of 300 μm thick needs a few 100 V to become fully depleted. Standard silicon wafers are round disks of silicon 5 or 8 inches (12.5 or 20 cm) in diameter. The largest detector that can be made is about 14 × 14 cm². To reduce the number of readout channels, often some kind of interpolation between the strips is used. If all the channels are equipped with electronics reading out the amplitude of the signals on all the strips, a position resolution of a few 10 μm can be achieved. The charges are collected in about 10 ns, making this indeed a very fast device.

Fig. 5.10

Schematic representation of the structure of a silicon strip detector

Fig. 5.11

Charged particle tracking detector used in the CMS experiment. The detector consists of two wafers of silicon put side to side. Four amplifying chips with 512 amplifying channels each are visible in the top-left side of the picture. Each silicon wafer has strips as shown in Fig. 5.10, with a pitch of 180 μm. The r.m.s. spatial resolution is ≈25 μm. Photograph copyright CERN

Figure 5.12 shows a silicon strip detector with a readout allowing both the x- and the y-coordinate to be determined with the same detector. Silicon strip detectors are fast detectors since the charge is typically collected in 10 ns, but the small signal-to-noise ratio makes it difficult to obtain very good timing measurements with silicon.

Fig. 5.12

To obtain the two coordinates x and y from one slice of silicon the detector can be equipped with mutual perpendicular strips on both sides of the silicon. In this example a resistor network was used to reduce the number of electronics channels

Silicon strip detectors are prone to leakage currents, both in the bulk and on the surface, and particular care is needed to obtain reliable devices. These detectors are also sensitive to radiation damage.

5.4 Germanium Semiconductor Detectors

The long radiation length of silicon (94 mm) and the fact that it is difficult to produce a depletion layer of much more than a few mm, makes silicon unattractive for gamma detection. Germanium has a radiation length of 23 mm, making it much more suitable for this purpose. Furthermore, modern germanium detectors are made from high purity germanium containing less than 10¹⁰ impurity atoms per cubic centimetre, making it possible to reach a depletion thickness of several centimetres. High purity germanium is grown in cylinder-shaped ingots. To achieve maximum use of the expensive material, germanium detectors are usually made in a cylindrical geometry as shown in Fig. 5.13(a) and Fig. 5.14.

Fig. 5.13

(a) Germanium detectors usually have a coaxial geometry to make optimal use of the expensive germanium material grown in cylindrical boules. (b) To suppress the leakage current the germanium detector must be used at liquid nitrogen temperature. It is often directly mounted on a Dewar as shown

Fig. 5.14

Germanium ingot and Ge detector elements. Figure by courtesy of CANBERRA – an AREVA company

Germanium has a band gap of 0.7 eV, giving rise to a large bulk leakage current caused by thermal electron–hole pair creation in the depletion region. This current gives rise to an unacceptable noise and therefore germanium detectors must be used at reduced temperature, usually liquid nitrogen (77.2 K). The typical geometry of a liquid nitrogen-cooled germanium detector is shown in Fig. 5.13(b).

Before modern high-purity germanium was available, it was common to rely on compensated germanium, which was obtained using the lithium drift method. These detectors must always (i.e. also when not in use) be maintained at liquid nitrogen temperature to maintain the proper compensation.

Germanium makes a very good detector for gamma rays because of its excellent energy resolution as is discussed below. For X-rays of the order of 10 keV, silicon is more appropriate. However, germanium detectors are very expensive and need to be cooled at liquid nitrogen temperatures for proper operation. Moreover, because the signal formation depends on the drift of the charges over rather long distances, the detector is not very fast. For all these reasons, it is often preferable to use scintillators as is discussed in Chap. 6.

The energy resolution of a germanium detector depends on the fluctuations on the number of charges. The number of charges produced by a gamma ray of energy E is given by

$$N = \frac{{E[{\rm eV}]}}{{2.96}}$$

The r.m.s. dispersion on this number is given by$\sqrt {N F}$ where F again represents the Fano factor. The FWHM energy resolution is therefore given by

$$FWHM[{\rm eV}] = 2{\rm{.35}}\,{\rm{.}}\,{\rm{2}}{\rm{.96}}\sqrt {\frac{{\rm{E}}}{{2.96}}F} = 2.35\sqrt {2.96\,E[{\rm eV}]\,F} $$

And the energy resolution FWHM in percent of the total energy is given by

$$FWHM{\rm{[\% ]}}\,{\rm{ = }}\,{\rm{235}}\sqrt {\frac{{{\rm{2}}{\rm{.96}}F}}{{E[{\rm eV}]\,}}} $$

For a gamma ray of 1.33 MeV, the above formula predicts an energy resolution of 1.33 eV. In practice, such a good energy resolution is never obtained. The difference is explained by incomplete charge collection.

5.5 Other Semiconductor Detector Materials

Silicon and germanium are by far the most commonly used materials in semiconductor detectors. However, there are several reasons to look for other materials. The properties of germanium make this an excellent material for making X-ray and gamma ray detectors, but the need to use the detector at the temperature of liquid nitrogen is a major complication. Moreover, detector grade germanium is expensive and the nuclear charge Z of germanium is only 32. Materials with a larger nuclear charge have a shorter radiation length and have a ratio of the photoelectric cross section over the Compton cross section that is larger. These are very desirable properties for a detector material used for detecting gamma rays.

Silicon is a good material for the tracking of charge particles, but the band gap in silicon is smaller than one would like and this causes a significant dark current and therefore a significant noise, when operating the detector at room temperature. A material with a somewhat larger band gap would be preferable. Moreover silicon is prone to radiation damage and possibly materials that perform better in this respect can be found. Furthermore, the good performance of silicon is only obtained with silicon monocrystals and these are limited in size to 5 or 8 inch. Materials that can be deposited in thin layers on large surfaces are needed in certain applications.

Table 5.2 lists a few materials that are being considered as alternatives to silicon or germanium. For all these materials the mean free path of the charge carriers before they are trapped, and therefore charge collection efficiency, is much lower than in silicon or germanium. This results in a significant reduction of the performance of the detector.

Table 5.2 Properties of some semiconductor materials for particle detection. All properties are given at room temperature, unless otherwise noted. Most of the entries in this table are taken from reference http://4.5

Of all the materials listed in Table 5.2 only CdTe (CT) and CdZnTe (CZT), and to less an extend HgI, have found significant applications as alternatives to germanium. CdTe suffers from the ‘polarisation’ of the detector material. This ‘polarisation’ is a change in the material caused by the previous interactions of gamma rays in the detector. This causes the detector response to become time dependent and rate dependent, a very undesirable property. The addition of a small amount of zinc reduces the dislocation density in CdTe and improves the performance. The concentration of zinc in CZT commonly used varies between 4 and 20%. CZT is less prone to polarisation and has a larger intrinsic resistivity than CdTe.

The production technology of CT and (CZT) has been improving steadily over the years. However, it remains difficult to produce large monocrystals and the materials suffer from poor hole collection efficiency. As a result of the hole trapping the pulse height spectra develop important ‘tails’, below the photopeak as shown in Fig. 5.15. These tails are due to ionisation events close to the anode. For these events the holes produced have to travel a long distance to the cathode. Since the holes are trapped before reaching the cathode, the signal is correspondingly reduced.

Fig. 5.15

Typical pulse height spectrum taken with a 5×5×2 mm CZT detector. Data taken at 21°C and with a bias voltage of 150 V over 2 mm. Figure from [8], with permission

A number of ways have been proposed to overcome this problem. Because of the difference in drift velocity of electrons and holes, the pulse shape varies with the position of the gamma interaction relative to the readout electrodes. It is therefore possible to derive a correction for the pulse height based on the pulse shape. A different approach for reducing the sensitivity to hole trapping was developed by Luke [7]. It uses two different but coplanar anodes in such a way that the signal only depends on the movement of the electrons. This can be achieved by replacing the planar anode electrode by a set of fine parallel strips. The strips are connected alternatively to two different amplifiers, forming therefore two independent readout electrodes. One set of strips, hereafter called anode A, is brought at a slightly larger positive potential than the other set of strips, hereafter called anode B. In this way the electrons only collect on anode A. The motion of the charges at a large distance from the plane of strips induces the same signal on both anodes. The motion of the electrons close to the strips induces most of the signal on anode A only. The difference between the signals on both anodes, A and B, therefore only depends on the motion of the electrons in the last few 100 microns close to the anode.

Diamond is considered as an alternative to silicon for tracking in high-energy physics because of its good radiation hardness. Diamond of course is not a semiconductor, but the properties of diamond detectors are somewhat similar to the properties of semiconductor detectors. Its discussion is therefore included here. The radiation hardness of diamond is reported in the literature as an order of magnitude better than silicon. Detectors using true diamonds have been tried, but these are limited to very small sizes for obvious reasons. Most diamond detectors use synthetic ‘diamond-like layers’ obtained by chemical vapour deposition, referred to in the literature as CVD diamond. The band gap in diamond is larger than in semiconductor devices and the detectors made with CVD diamonds do not need a junction to suppress the leakage current. It is sufficient to use metal contacts on both sides of the diamond and apply an electric field. Because of the large band gap, diamond detectors can be used at elevated temperature.

Amorphous selenium (a-Se) is a good candidate for replacing silicon in applications where one would like a larger detector area than what can be obtained with silicon monocrystals. An example of such an application is the direct conversion X-ray detector for medical imaging. The simple band gap model used throughout this chapter does not really apply to amorphous selenium. For a number of reasons, the amorphous selenium layer in such X-ray detectors has to be operated at very high electric fields (up to 10 V per micron). An important issue in the use of amorphous selenium is the minimisation of dark current. Reference [9] contains an extensive discussion of the use of selenium as a detector for X-ray imaging.

5.6 Exercises

• 1. Calculate the dopant concentration in n-type silicon with a resistivity of 2000 Ω cm.

• 2. For a silicon strip detector made starting from n-type silicon with a resistivity of 2000 Ω cm, calculate the voltage to fully deplete a silicon microstrip detector of 300 μm thickness.

• 3. Derive an expression for the energy resolution (FWHM and in %) of silicon as an X-ray detector at room temperature. How much will the energy resolution be for X-rays of 50 keV?

• 4. You deposit a very thin layer of 241 Am with an activity of 2 MBq on a surface barrier alpha particle detector. This isotope emits alpha particles, see annex 6 for the characteristics of the emission. Calculate the magnitude of the pulses in number of electrons and the current in the detector.

• 5. Calculate the number of charges produced in a silicon strip detector of 500 μm thick by a minimum ionising particle.

• 6. Calculate the density of states for an electron enclosed in an infinitely deep and cubic potential well. Use the expression below for the energy levels of the electron in such a potential well. The numbers n₁, n₂ and n₃ are positive integers.

  $$E = \frac{{p^2 }}{{2m}} = \frac{{\pi ^2 ({\rm h} c)^2 }}{{a^2 2mc^2 }} (n_1^2 + n_2^2 + n_3^2 )$$

• 7. Calculate the potential difference over a p–n junction if the dopant concentrations in the n-type silicon and the p-type silicon are N _D = 10¹²/cm³ and N _A = 10¹⁶ cm³, respectively.