Semiconductor Detectors Calorimetry and Tracking with High Precision

Transcription

1 Semiconductor Detectors Calorimetry and Tracking with High Precision

2 Applications 1. Photon spectroscopy with high energy resolution. Vertex detection with high spatial resolution 3. Energy measurement of charged particles [few MeV] Main advantages: (i) Possibility to produce small structures using micro-chip technology; 10 μm precision; relatively low costs... (ii) Comparably low energy deposition per detectable electron-hole pair required... e.g. Silicon : 3.6 ev per electron-hole pair Ionization (LAr): O(30 ev) for a single ion; see later Scintillators : O(100 ev) depending on light yield [typical 1-10%]

3 Applications Examples ATLAS Pixel Detector

4 Applications Examples ATLAS Pixel Module ATLAS Pixel Detector [Details] Pixel Sensor

5 Applications Examples CMS Inner Barrel

6 Applications Examples CALICE SiW ECAL Event Display Structure.8 Structure 1.4 Structure 4. Front End Electronics ECAL Physics Prototype [CALICE] Active area [Pads: 10x10 mm ]

7 Applications Examples pnccd Camera X-ray astronomy satellite XMM-Newton

8 Basic Semiconductor Properties Conduction band Conduction band Electrons Conduction band Energy gap Eg 6 ev Eg 1 ev Holes Gap Valence band Valence band Valence band Insulator Semiconductor Metal

9 Basic Semiconductor Properties Intrinsic semiconductor: Very pure material; charge carriers are created by thermal, optical or other excitations of electron-hole pairs; Nelectrons = Nholes holds... Commonly used: Silicon (Si) or Germanium (Ge); four valence electrons... Doped or extrinsic semiconductor: Majority of charge carriers provided by donors (impurities; doping) n-type : majority carriers are electrons (pentavalent dopants) p-type : majority carriers are positive holes (trivalent dopants) Pentavalent dopants (electron donors): P, As, Sb,... [5 th electron only weakly bound; easily excited into conduction band] Trivalent dopants (electron acceptors): Al, B, Ga, In,... [One unsaturated binding; easily excepts valence electron leaving hole]

10 Intrinsic Semiconductors Basic Semiconductor Properties Conduction band Energy bands: Regions of many discrete energy levels with very close spacing Valence band Energy Gap Ec Ev Arise from interaction of electrons with the very many atoms of the crystalline/solid material... Energies within a band treated like particles in a box... [Fermi gas model] Yields: E( k)= k m e = m e (k x + k y + k z) g(e) me 3 E E [Dispersion relation] [Density of states]

11 Intrinsic Semiconductors Basic Semiconductor Properties Derivation of g(e): V = L 3 [Volume of solid] E( k)= k m e = m e (k x + k y + k z) = m e π L with k i = n iπ L n with n = n x + n y + n z [i = x, y, z] [... quantized due to boundary conditions] We have to fill all states within a sphere of radius nmax or, alternatively, kmax: Number of electrons Each state occupied twice N = 4 3 πn3 max 3N 8π n max = 1 3 E max = m e π L E max = n max m e 3π N V 3

12 Intrinsic Semiconductors Basic Semiconductor Properties Derivation of g(e): E max = m e 3π N V 3 N(E) =E 3 me 3 V 3π Emax is also called Fermi Energy EF; it can be associated with the highest kinetic energy of electrons in a solid at T = 0 K... Remark: Characteristics of solids determined by location of Fermi Energy Metal: EF below top of an energy band Insulator: EF at top of valence band; large gap Semiconductor: EF at top of valence band; smaller gap Density of states follows from: g(e) = dn de = E 1 me 3 V π Density of states g(e) E E F E

13 Intrinsic Semiconductors Basic Semiconductor Properties At a temperature T the occupation probability of the available states is given by the Fermi-Dirac distribution... f(e,t)= 1 e (E µ)/k BT +1 with chemical potential μ [metals: EF = μ; often identified with EF] T = 0: Step function; only states below μ are occupied... T > 0: 1 f(e) T = 0 Fermi-Dirac distribution Fermi-Dirac distributions develops a 'soft zone' T > 0 Notice: EF ~ several ev soft zone: K μ (= EF) E F

14 Intrinsic Semiconductors Basic Semiconductor Properties Electron density is given by: f(e) n(e) n(e) =g(e)f(e) ½ g(e) Sometimes extra factor if g(e) does not account for spins... 1 Total number of electrons up to an energy Emax: N = Emax 0 g(e)f(e)de 0 f(e) n(e) E F μ (= EF) T = 0 E Electron density: n = 1 V Emax 0 g(e)f(e)de T > 0 μ (= EF)

15 Intrinsic Semiconductors Basic Semiconductor Properties Carrier concentration in conduction and valence band: Conduction band g(e) n = 1 V g c (E)f(E,T)dE Ec Chemical Potential μ Eg E c p = 1 V Ev g v (E)[1 f(e,t)]de [using symmetry of f(e,t)] Ev Valence band n(e) f(e,t>0) E = 0 Eg is generally large compared to 'soft zone', i.e. (E μ)» kbt, such that... f(e,t)= 1 e (E µ)/k BT +1 e (E µ)/k BT... for calculation of the electron density. Band structure and electron density Using above conventions (see Fig.): n = 1 3 V me E Eg V π e (E µ)/kbt de E g = (m e) 3 π 3 eµ/k BT E g E Eg e E/k BT de...

16 Intrinsic Semiconductors Basic Semiconductor Properties... calculation continued... Conduction band g(e) n = (m e) 3 π 3 eµ/k BT E g E Eg e E/k BT de Ec Eg = (m e) 3 π 3 (k BT ) 3 e (E g µ)/k B T using 0 with X g =(E E g )/k B T [substitution] X 1 g e X g π dx g =... 0 X 1 g e X g dx g Ev Chemical Potential μ Valence band n(e) f(e,t>0) E = 0 n = π (m e k B T ) 3 π 3 e (E g µ)/k B T Band structure and electron density n = π (m* e k B T ) 3 π 3 e (E C µ)/k B T = N C e (E C µ)/k B T m*: effective mass [electrons in crystal] p = π (m* h k B T ) 3 π 3 e (µ E V )/k B T = N V e (µ E V )/k B T μ EC EV

17 Intrinsic Semiconductors Basic Semiconductor Properties Carrier concentration in conduction and valence band: n = N C e (E C µ)/k B T p = N V e (µ E V )/k B T NC,V ~ (m*t) 3/ T dependent NC: effective density of electrons at edge of conduction band NV: effective density of holes at edge of valence band Pure semiconductors: carrier concentration depends on separation of conduction/valence band from chemical potential or Fermi level... Location of Fermi level determines n and p... But, product is independent of location of Fermi level... np = N C N V e (E V E C )/k B T (m e m h ) 3 Law of mass action [holds more generally] At given temperature characterized by effective mass and band gap.

18 Intrinsic Semiconductors Basic Semiconductor Properties Intrinsic semiconductors; no impurities number of electrons in conduction band is equal to number of holes in valence band. n = p or n i = p i to characterize that this holds for intrinsic semiconductors only The expressions for n,p then yield: µ = E C + E V k BT ln NC N V = E C + E V 3 4 k BT ln m e m h At T = 0: Fermi-level (EF = μ) lies in the middle between valence and conduction band... At T > 0: In case the effective masses of electrons and holes are non-equal, i.e. NC NV the Fermi-level changes with temperature... EC EV μ = EF

19 Intrinsic Semiconductors Basic Semiconductor Properties Some properties of intrinsic semiconductors Si Ge GaAs [III-V Semiconductor] Egap [ev] K [m -3 ] K [m -3 ] me/me * mh/me * Energy/e-hole-pair [ev] at 77 K

20 Doped Semiconductors Basic Semiconductor Properties Introducing impurities (doping) balance between holes and electrons in conduction band can be changed; yields higher carrier concentrations. n-doping n-doping: extra electron resides in discrete energy level close to conduction band... p-doping: additional state close to the valence band can accept electrons... n-doping: p-doping: majority carriers = electrons [holes don't contribute much; minority carriers] majority carriers = holes [electrons are minority carriers] p-doping n-doping: Sb, P, As... p-doping: B, Al, Ga...

21 Doped Semiconductors Basic Semiconductor Properties Sb P As E c ev ev ev ev ev ev μ (= EF) E F [pure Si] B Al Ga E V Energy levels for silicon with different dopants

22 Doped Semiconductors Basic Semiconductor Properties Position of chemical potential for n-doped semiconductor: High temperature (intrinsic) Intermediate temp. (extrinsic) Low temp. (freeze-out) All donors and some intrinsic carriers ionized Almost all donors; very few intrinsic carriers ionized Only few donors are ionized

23 Doped Semiconductors Basic Semiconductor Properties Carrier density depends on doping and temperature... Neutrality condition: Intrinsic ND + p = NA + n ND: donor concentration NA: acceptor concentration Extrinsic region: n-type: n ND [NA = 0; n» p] p-type: p NA [ND = 0; p» n] n e [cm 3 ] 1 n-doped Si Extrinsic pure Si Typical concentrations: Dopants: atoms/cm 3 [Strong doping: 10 0 atoms/cm 3 ; n + or p + ] Compare to Si-density: /cm Electrons in conduction band [ ] T K

24 The np-junction Function of semiconductor detectors depends on formation of a junction between n- and p-type semiconductors... Thermodynamic equilibrium Fermi energies should become equal... E0 moves up when forming junction EC μ (= EF) μ (= Ei ) μ (= EF) p-type n-type EV moves down when forming junction

25 The np-junction p-type E0 n-type EC E0 μ (= Ei ) EV EC μ (= EF) EV acceptors donors

26 The np-junction Equilibration process: E C np ( ) Electrons diffuse from n to p-type semiconductor and recombine... Holes diffuse from p to n-type semiconductor and recombine... E V N A pp ( ) acceptors Löcher holes eu D + N D + donors nn () + Resulting electric field counteracts and stops diffusion process... eu D = E pot = E (p) C E(n) C ln npn,,, N D A p-type n-type pn () N D = k B T ln n n type n p type = k B T ln N DN A n i n p n i [using n = N C e (E C µ)/k B T,p=... ] x N A At the boundary concentration of mobile carriers is depleted... [depletion layer] ρ( x) + x

27 The np-junction Depletion depth: N A x p = N D x n d V dx = ρ(x) ρ(x) = { en D en A 0 <x<x n x p <x<0 dv dx = { en D / x + C n en A / x + C p 0 <x<x n x p <x<0 = { en D / (x x n ) en A / (x + x p ) 0 <x<x n x p <x<0 p-type n-type as electric field E = -dv/dx must vanish for x=xn and x= -xp Model for calculating depletion zone

28 The np-junction Depletion depth: N A x p = N D x n V (x) = { en D / (x / x n x)+c en A / (x /+x p x)+c 0 <x<x n x p <x<0 Solution must be continuous at x=0; thus C = C'; Also V(-xp) = 0 and V(xn) = V0 with V0 contact potential... Thus: C = en A x p V 0 = en D x n + en A x p Using NAxp = NDxn yields: d = x n + x p = V 0 (N A + N D ) en A N D With: V 0 x n = and x p = en D 1+ N D N A V 0 en A 1+ N A N D Remark: If one side more heavily doped, depletion zone will extend to lighter doped side; e.g. NA» ND, xn» xp... p-type n-type Model for calculating depletion zone

29 The np-junction Depletion depth: d = x n + x p = If e.g. NA» ND [as in figure]... V 0 (N A + N D ) en A N D For large depth choose asymmetric doping! N A x p = N D x n d x n V0 en D ρ n µ e V 0 using conductivity σ = 1/ρ = e(n μe + p μh), with n = ND and mobility μ = v/e. Depletion depth determined by mobility of charge carriers... p-type n-type Typical values: Silicon: Germanium: 0.53 (ρnv0) 0.5 μm (n-type); 0.3 (ρpv0) 0.5 μm (p-type) 1.00 (ρnv0) 0.5 μm (n-type); 0.65 (ρpv0) 0.5 μm (p-type) [Typical ρ 0000 Ωcm and V0 = 1V d = 75 μm] Model for calculating depletion zone

30 The np-junction Application of an external voltage: Here: consider only electrons [similar for holes] + + No voltage Forward bias Reverse bias Equilibrium: drift of minority electrons from p-side compensates diffusion current from n-side which have to move against E-field Voltage drop over depletion zone; diffusion current higher due to shift of chemical potential; current increases exponentially with bias Voltage drop over depletion zone; diffusion current smaller due to shift of chemical potential; widening of the depletion zone I = I0 (e ev/kt -1) I = I0 (e -ev/kt -1)

31 The np-junction Characteristic I(V) curve of a diode U I n p Forward bias U B Leakage current U Reverse bias I / 100

32 Basic Semiconductor Detector Requirement: Large sensitive region... p + dead layer n + Bias We know: d x p U en A Typical: NA = /cm 3 n + region highly doped: ND» NA 1 μm p [ρ = 10 kωcm; NA] 300 μm 1 μm Signal U = e N Ad =100V Sensitive volume Metal contact Electric field: E = U d = 100 V m V m [Safe. Breakdown limit at 10 7 V/m] n + and p + needed to allow metallic contacts... [High doping = small depletion zone] Bias voltage supplied through series resistor...