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Semiconductor Detectors (Solid State Detectors) energy, position particles & photons energy loss conversion electron-hole pairs energetic “cheap” improved resolution Semiconductor Detectors (Solid State Detectors) conduction band electrons 6 eV 1 eV holes valence band insulator semi conductor metal −E g −E g 3/2 ni = N c N v exp = AT exp 2kT 2kT NC,v Eg A number of states conduction,valence band gap at 0 K constant independent from T for pure Si: ni=1.45x1010 cm-3 Semiconductor Detectors how to get a signal ? particle passing through a thin layer of Si: 300 µm mimimum ionizing particle will produce: about 3.2x104 e-h pairs but 4.5x108 free charge carriers in the same volume ! reduce number of free charge carriers, i.e. deplete material ! Semiconductor Detectors “doped” n-type donor level n-type + e- add donor elements: Vth group elements like P, As, Sb extra electrons resides at discrete energy level in the energy gap, close to the CB (separated 0.01 - 0.05 eV) conductivity enhanced by electrons => n-type semiconductors Semiconductor Detectors “doped” p-type acceptor level p-type + holes add acceptor elements: IIIrd group elements like Ga, B, In extra electron states close to the VB, electrons easily excited into these states, creating hole states in the VB. conductivity enhanced by holes states => p-type semiconductors Semiconductor Detectors “doped” Typical doping levels for detector silicon: 1012 atoms/cm-3 which has to be compared with 1022 atoms/cm-3 density Ge & Si heavily doped semiconductors: (“+” sign after the material) 1020 atoms/cm-3 Semiconductor Detectors Si, characteristics band gap: Eg =1.12 V. E(e--hole pair) = 3.6 eV (~30 eV for gas detectors) high specific density (2.33 g/cm3) ΔE/track length for M.I.P.’s.: 390 eV/mm, ~108 e-h/ high mobility me=1450 cm2/Vs, mh = 450 cm2/Vs small dimensions fast charge collection (<10 ns) rigidity of silicon thin self supporting structures typical thickness 300 µm ~3.2 x104 e-h (average) µm (average) Semiconductor Detectors “np-junction” n p n p initial diffusion: holes -> n-region, electrons -> p-region charge building up: n-region-> positive, p-region -> negative emerging electric field (contact potential) stops diffusion region of changing potential: depletion zone, space charge region Semiconductor Detectors increasing depletion depletion zone - + n reversed bias junction (bias voltage about 100 V) apply negative voltage to p-side -> attract holes apply positive voltage to n-side -> attract electrons -> increase depletion zone p Semiconductor Detectors signal read out depletion zone n+ n bias p metal contacts to prevent depletion zone between metal and semi conductor -> layer of highly doped material p+ Semiconductor Detectors characteristics depletion depth V0 eND -xp xn -eNA Poissons' s equation : -xp xn d 2V ρ (x) =− 2 dx ε charge density : ρ(x) = eN D 0 < x < x n = −eN A − x p < x < 0 charge conservation : N A x p = N D xn Semiconductor Detectors characteristics depletion depth d integration yields V(x): 0 < x < xn eN D x 2 V (x) = − − x n x + C ε 2 −x p < x < 0 eN A x 2 V (x) = − x p x + C' ε 2 matching at x = 0 => C = C' eN C = A x p2 2ε Semiconductor Detectors characteristics depletion depth d integration yields contact potential V(xn)=V0: V0 = e N D x n2 + N A x p2 ) ( 2ε and xn = 2εV0 eN D (1 + N D / N A ) xp = 2εV0 eN A (1 + N A / N D ) depletion zone extends farther into the lighter-doped side € Semiconductor Detectors characteristics depletion depth d under assumption NA >> ND : 2εV0 d = xn + x p ≈ eN D resistivity of n-type material: € 1 ρn ≈ eN D µ e µe electron mobility d ≈ 2ερ n µ eV0 [cm 2 /Vs] Semiconductor Detectors characteristics “an application” ρ=20 kΩcm for heavily doped n-type Si and V0=100 V. The dielectric constant ε/ε0=12 Assume and µe (300K)=1350 cm2/Vs. How thick is the depletion zone ? Semiconductor Detectors characteristics junction capacitance for planar geometry: A C =ε d for ND>>NA or NA>>ND, respectively Si: € 2.2 C/A = pF / mm 2 ρ N V0 n - type 3.7 pF / mm 2 ρ N V0 p - type C/A = € Semiconductor Detectors characteristics pulse shape, rise time charge collection time for planar geometry, heavily doped p-type p n+ +V + 0 x0 d E Semiconductor Detectors characteristics pulse shape, rise time charge collection time for planar geometry, heavily doped p-type: qdx dQ = d integration of Poisson' s equation : eN A dV x =E =− x=− dx ε µhτ τ = ρε € with 1/ ρ = eN A µ h Semiconductor Detectors characteristics dx e µe x ve = = −µ e E = dt µh τ if mobility independent of E µet x e (t) = x 0 exp µ h τ electron reaches eletrode at x = d : µh d t=τ ln µ e x0 € Semiconductor Detectors characteristics µet e dx e Qe (t) = − ∫ dt = x 0 1 − exp d dt d µh analogue for holes, with x vh = µ h E = − τ e −t Qh (t) = − x 0 1 − exp d τ Semiconductor Detectors characteristics Q(t) Qtot Qe Qh t Semiconductor Detectors characteristics “an application” Estimate the rise time of the solid state detector discussed above !