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Semiconductor Detectors Part 1

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Semiconductor Detectors Part 1 1 Lectures on Detector Techniques Stanford Linear Accelerator Center September 1998 – February, 1999 Semiconductor Detectors Part 1 Helmuth Spieler Physics Division Lawrence Berkeley National Laboratory for more details see UC Berkeley Physics 198 course notes at http://www-physics.lbl.gov/~spieler Semiconductor Detectors Helmuth Spieler SLUO Lectures on Detector Techniques, October 23, 1998 LBNL 2 1. Principles Important excitations when radiation is absorbed in solids 1. atomic electrons ⇒ mobile charge carriers ⇒ lattice excitations (phonons) 2. elastic scattering on nuclei Recoil energy < order 10 eV ⇒ ionization ⇒ lattice excitations (phonons) at higher recoil energies ⇒ radiation damage (displacement of from lattice sites) Semiconductor Detectors Helmuth Spieler SLUO Lectures on Detector Techniques, October 23, 1998 LBNL 3 Most Semiconductor Detectors are Ionization Chambers Detection volume with electric field Energy deposited → positive and negative charge pairs Charges move in field → current in external circuit (continuity equation) Semiconductor Detectors Helmuth Spieler SLUO Lectures on Detector Techniques, October 23, 1998 LBNL 4 Ionization chambers can be made with any medium that allows charge collection to a pair of electrodes. Medium can be gas liquid solid Crude comparison of relevant properties gas liquid solid density low moderate high atomic number Z low moderate moderate ionization energy εi moderate moderate low signal speed moderate moderate fast Desirable properties: low ionization energy ⇒ 1. increased charge yield dq/dE 2. superior resolution ∆ E ∝ 1 ∝ 1 ∝ ε ε i E N E / i high field in detection volume ⇒ 1. fast response 2. improved charge collection efficiency (reduced trapping) Semiconductor Detectors Helmuth Spieler SLUO Lectures on Detector Techniques, October 23, 1998 LBNL 5 Ionization energy in solids is proportional to the band gap small band gap ⇒ ~ conductor electric field small DC current >> signal current large band gap ⇒ insulator high electric field small signal charge + small DC current example: diamond moderate band gap ⇒ semiconductor high electric field “large” signal charge small DC current, but “pn-junction” required. examples: Si, Ge, GaAs Semiconductor Detectors Helmuth Spieler SLUO Lectures on Detector Techniques, October 23, 1998 LBNL 6 Principle of Semiconductor Detector 1. Energy absorbed in semiconductor ⇒ formation of mobile electrons and holes number of electron-hole pairs in Si: E N = ε i E= absorbed energy, εi= 3.6 eV 2. Electric field in detection volume causes electrons and holes to move towards electrodes typical transit times: 10 – 20 ns for 300 µm 3. Induced signal current measured typical signal charge: 3.5 fC (22000 el) for minimum ionizing particles traversing 300 µm of Si Semiconductor technology allows micron-scale patterning of electrodes ⇒ strip and pixel detectors Semiconductor Detectors Helmuth Spieler SLUO Lectures on Detector Techniques, October 23, 1998 LBNL 7 Example: Strip Detectors Resolution determined by precision of micron scale patterning (e.g. strips on 50 µm pitch) Two options: Binary Readout Analog Readout Interpolation yields resolution < pitch Relies on transverse diffusion σ ∝ x tcoll to discriminators e.g. in Si Position resolution determined tcoll≈ 10 ns directly by pitch ⇒ σx= 5 µm σ = x pitch / 12 Interpolation precision depends on S/N and p p= 25 µm and S/N=50 ⇒ 3 – 4 µm resolution Semiconductor Detectors Helmuth Spieler SLUO Lectures on Detector Techniques, October 23, 1998 LBNL 8 Energy required for creation of an electron-hole pair Ionization Energy > Band Gap Formation of e-h pair requires both … 1. Conservation of energy 2. Conservation of momentum ⇒ additional energy excites phonons Remarkably, εi /Eg ≈ const for all materials, types of radiation. C.A. Klein, J. Applied Physics 39 (1968) 2029 Semiconductor Detectors Helmuth Spieler SLUO Lectures on Detector Techniques, October 23, 1998 LBNL 9 Fluctuations in the Signal Charge: the Fano Factor The mean ionization energy exceeds the bandgap for two reasons 1. Conservation of momentum requires excitation of lattice vibrations 2. Many modes are available for the energy transfer with an excitation energy less than the bandgap. Two types of collisions are possible: a) Lattice excitation, i.e. phonon production (with no formation of mobile charge). b) Ionization, i.e. formation of a mobile charge pair. Assume that in the course of energy deposition Nx excitations produce NP phonons and Ni ionization interactions form NQ charge pairs. On the average, the sum of the energies going into excitation and ionization is equal to the energy deposited by the incident radiation = + E0 Ei Ni Ex N x where Ei and Ex are the energies required for a single excitation or ionization. Assuming gaussian statistics, the variance in the number of excitations σ = x N x and the variance in the number of ionizations σ = i Ni Semiconductor Detectors Helmuth Spieler SLUO Lectures on Detector Techniques, October 23, 1998 LBNL 10 For a single event, the energy E0 deposited in the detector is fixed (although this may vary from one event to the next). If the energy required for excitation Ex is much smaller than required for ionization Ei, sufficient degrees of freedom will exist for some combination of ionization and excitation processes to dissipate precisely the total energy. Hence, for a given energy deposited in the sample a fluctuation in excitation must be balanced by an equivalent fluctuation in ionization. ∆ + ∆ = Ex N x Ei Ni 0 If for a given event more energy goes into charge formation, less energy will be available for excitation. Averaging over many events this means that the variances in the energy allocated to the two types of processes must be equal σ = σ Ei i Ex x σ = Ex i N x Ei From the total energy Ei Ni + Ex Nx = E0 − = E0 Ei Ni N x Ex yielding σ = Ex E0 − Ei i Ni Ei Ex Ex Semiconductor Detectors Helmuth Spieler SLUO Lectures on Detector Techniques, October 23, 1998 LBNL 11 Since each ionization leads to a charge pair that contributes to the signal E N = N = 0 i Q ε i where εi is the average energy loss required to produce a charge pair, σ = Ex E0 − Ei E0 i ε Ei Ex Ex i E E ε σ = 0 ⋅ x i −1 i ε i Ei Ei The second factor on the right hand side is called the Fano factor F. Since σi is the variance in signal charge Q and the number of charge pairs is NQ=E0 /εi σ = Q FNQ In Silicon Ex= 0.037 eV Ei = Eg= 1.1 eV εi = 3.6 eV for which the above expression yields F= 0.08, in reasonable agreement with the measured value F= 0.1. ⇒ The variance of the signal charge is smaller than naively expected σ ≈ Q 0.3 NQ Semiconductor Detectors Helmuth Spieler SLUO Lectures on Detector Techniques, October 23, 1998 LBNL 12 A similar treatment can be applied if the degrees of freedom are much more limited and Poisson statistics are necessary. However, when applying Poisson statistics to the situation of a fixed energy deposition, which imposes an upper bound on the variance, one can not use the usual expression for the variance var N = N Instead, the variance is (N − N )2 = F N as shown by Fano [1] in the original paper. An accurate calculation of the Fano factor requires a detailed accounting of the energy dependent cross sections and the density of states of the phonon modes. This is discussed by Alkhazov [2] and van Roosbroeck [3]. References: 1. U. Fano, Phys. Rev. 72 ( 1947) 26 2. G.D. Alkhazov et al., NIM 48 (1967) 1 3. W. van Roosbroeck, Phys. Rev. 139 (1963) A1702 Semiconductor Detectors Helmuth Spieler SLUO Lectures on Detector Techniques, October 23, 1998 LBNL 13 Intrinsic Resolution of Semiconductor Detectors E ∆E = 2.35⋅ ε FN = 2.35⋅ε F = 2.35⋅ FEε i Q i w i Si: εi= 3.6 eV F= 0.1 Ge: εi= 2.9 eV F= 0.1 Intrinsic Resolution of Si and Ge Detectors 250 200 Si 150 Ge E FWHM [eV] 100 ∆ 50 0 0 5 10 15 20 25 E [keV] Detectors with good efficiency for this energy range have sufficiently small capacitance to allow electronic noise of ~100 eV FWHM, so the variance of the detector signal is a significant contribution. At energies >100 keV the detector sizes required tend to increase the electronic noise to dominant levels. Semiconductor Detectors Helmuth Spieler SLUO Lectures on Detector Techniques, October 23, 1998 LBNL 14 Semiconductor Crystals Lattice structure of diamond, Si, Ge (“diamond lattice”) (from Schockley) dimension a: lattice constant Diamond: 3.56 Å Ge: 5.65 Å Si: 5.43 Å Extent of wavefunctions of typical constituent atoms: (from Schockley) Semiconductor Detectors Helmuth Spieler SLUO Lectures on Detector Techniques, October 23, 1998 LBNL 15 When isolated atoms are brought together to form a lattice, the discrete atomic states shift to form energy bands: (from Harrison) Filled band formed by bonding states: Ψ= Ψa + Ψa (Ψa = wavefunction of individual atom) Empty band formed by anti-bonding states: Ψ= Ψa − Ψa (vanishing occupancy at mid-point between atoms) Semiconductor Detectors Helmuth Spieler SLUO Lectures on Detector Techniques, October 23, 1998 LBNL 16 Each atom in the lattice contributes its quantum states to each band: The number of quantum states in the band is equal to the number of states from which the band was formed. The bands are extended states, i.e. the state contributed by an individual atom extends throughout the crystal. Energy band structure (from Schockley) Typical band gaps (valence – conduction band) Ge 0.7 eV GaAs 1.4 eV Si 1.1 eV Diamond 5.5 eV Semiconductor Detectors Helmuth Spieler SLUO Lectures on Detector Techniques, October 23, 1998 LBNL 17 At 0K all electrons occupy bonding states, completely filling the valence band.
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