History of Cryogenic Particle Detectors ~1845 Mayer, Joule heat is a form of energy 1876 Langley bolometer for ir-observations of sun 1903 Curie, Laborte calorimetric detection of radioactivity 1935 Simons cryogenic calorimeters 1939 Goetz superconducting transition calorimeter 1949 Andrew, Fowler detection of single α particles with a Williams superconducting calorimeter S.P. Langley (1834-1906) 1984 Fiorini, Niinikoski, modern type cryogenic particle Moseley, Mather, detectors McCammon today over 300 scientists are working in this field
Unique Characteristics Advantages No requirement for efficient charge transport no necessity to collect electrons large amount of impurities can be tolerated radioactive source can be embedded with detector flexible choice of detector materials large volume detector make detector out of source material Sensitive to exotic interaction that produce no ionization No Fano limit (compare to Si 118 ev at 6 kev) High resolution possible: 1.8 ev at 6 kev 0.15 ev at 4 ev
Calorimeter Principle particle Thermal relaxation time: thermometer t Thermal conductance absorber thermal bath weak thermal link : phonons electrons spins tunneling states quasi particles
Semiconducting Thermistors R Si ion-implanted (P,B) T Ge NTD ( Neutron-Transmutation-Doped) High impedance device particle I bias Thermistor + - JFET 100 K
Superconducting Transition Edge Sensors (TES) R self regulated working point T Materials Mo/Cu Ir/Au W Electro-thermal feedback V X-ray K. D. Irwin, Appl. Phys. Lett. 66, 1945 (1995) I shunt TES heat input: R TES goes up joule heating decreases fast response time SQUID t
Kinetic Inductance Detectors: KID
Motivations and Applications Dark Matter search for WIMP s,... Standard solar model: 0.4% p + e - + p 2 H + ν U 91+ e E ν = 1.44 MeV 99.6% p + p 2 H + e + + ν e E ν 0.42 MeV Solar Neutrino Problem spectrum of low energy neutrinos nuclear fusion of heavier elements X-Ray and γ-astronomy hot interstellar gases, black holes, quasars, pulsars, X-ray double stars,... Nuclear and Atomic Physics neutrino mass, double beta decay, Lamb shift, new elements X-Ray Fluorescence Analysis surface science, micro analysis sample (+25 kev) grid (0 x-ray V) optics deflection plates detector cryostat x-ray window Counts UV laser pump e - sample detector Biomedical applications mass spectroscopy of bio moleculs Crab-Nebula M82 Energy [kev]
Ideal Thermal Calorimetery
E gap = 1.2 ev w ~ 3.7 ev/e-h pair N = E photon /w ΔN = ΔE = wδn = fn fwe photon best possible resolution ~ 118 ev FWHM @ 6 kev
BRANCHING 100% Photoionization Time 30% 70% ionization (E min ~ 10 4 K) electron K.E. phonons (no E min ) Channels (places to put energy: ionization, electrons, phonons, spins, etc.)
Multiple Channels in Equilibrium: N total excitations τ Small system (thermometer) has fraction β on average, so fluctuations = βn, and relative fluctuations = 1/ βn. But reduced by τ τ meas. So fluctuations 0 as τ 0 In principle there is no limitation on the accuracy of an equilibrium measurement (except quantum limit ΔE Δt ).
Themal detectors are not all quasiequilibrium Phonon-mediated detectors can be non-equlibrium. For example, the dark matter detectors used by the CDMS experiment use phonon sensors on the surface of large Germanium and Silicon disks. These have superconducting collectors that are sensitive only to phonons with E >> kt. They are therefore susceptible to the similar statistical limitations to ionization detectors (but with much larger N, they are effectively limited by considerations such as positional uniformity of collection efficiency).
Switch? (Close to reset similar to optical reset on solid state detector) Random transport of energy between heat sink and detector over thermal link G produces fluctuations in the energy content of C. The magnitude of these can easily be calculated from the fundamental assumption and definitions of statistical mechanics: σ E 2 = k T 2 C Thermodynamic Fluctuation Noise (TFN) (Can think of as Poisson fluctuations in number of energy carriers in C N CT kt, ΔE rms = N kt with mean carrier energy kt:.) Not a limit on resolution, but sets the scale... ( ) = kt 2 C
Noise in different frequency bins uncorrelated Each frequency bin gives independent estimate of signal amplitude S/N square root (number of bins) ΔE = 2πf c Δf 1/2 k T 2 C
Optimal Filtering General problem: Linear system: V(t)=E 0 F(t), F(t) independent of E 0 Know F(t) Best estimate of E 0 in presence of noise described by e n (ω). Can be solved quite generally. V(t)=E 0 F(t) F(t) t Time domain: Each time bin provides an estimate of E 0. So we would like to do some kind of weighted average to get a best estimate. This is complicated because the noise is correlated between time bins unless it is white. For the special case of white noise, a least squares fit of E F(t) to V(t) will give the optimum estimate of E 0.
Optimal Filtering s(ω) = E 0 f (ω) f (ω) noise ω Frequency domain: Each frequency bin provides an estimate of E 0. The noise in different frequency bins is uncorrelated between frequency bins if the noise is stationary. This requires only that its statistical properties do not change with time, which should be true for most linear systems.
Optimal Filtering Frequency domain: in i th frequency bin signal = s i noise = n i weight = w i : E = w i s i ΔE rms = w i n i i=1 i=1 Choose w i to maximize E / ΔE r.m.s : ( ) 2 1 2 w k i=1 i=1 w i s i ( ) 2 w i n i 1 2 = 0 w k = s k n k 2 ( w i n i ) 2 i=1 i=1 1 w i s i Common factor on w i only affects normalization, so set = 1.
Optimal Filtering E ΔE = w i s i ( ( w i n i ) 2 ) 1 2 s i are Fourier transform of V(t) complex. n i have random phase denominator independent of phase of w i So choose w i to maximize numerator (w i s i pure real): s w i = ˆ i 2 n i, and the frequency - domain filtered pulse = ˆ s i s i. i=1 n i 2 Pure real (cosine) series time-domain filtered symmetrical, peak at t = 0: filtered V(t) t
Optimal Filtering Expected Energy Resolution ΔE 2 = 4 0 1 S V ( f ) 2 p( f ) 2 df e 2 n ( f ) (Use this version if P(t) is not a delta function) If input power P(t) is a delta function at t = 0, p(f ) = 1, and NEP( f ) e n ( f ) S V ( f ), ΔE rms = 0 4df NEP 2 1 2
Limitations of linear theory Heat capacity and thermal conductivities change with temperature Thermometer sensitivity may change with temperature TES detectors often severely limited by nonlinearity An optimal filtering scheme for nonlinear detectors with nonstationary noise has been developed by Fixsen et al. D.J. Fixsen et al., in Low Temperature Detectors, (F.S. Porter et al. Eds.), Proc LTD- Madison, Wisconsin, July 2001 (AIP, New York 2002), p 339 Fixsen, D. J., Moseley, S. H., Cabrera, B., Figueroa-Feliciano, E., Nucl. Inst. and Meth. A, 520 (2004) 555
Common complications Internal time constants Thermometer noise Other noise sources Metastable states You have to make the thing Absorber Detector Heat Sink Bias Power Thermometer You have to read it out (maybe a lot of them).
Detectors with ideal resistive thermometers
Thermistor: Resistance = R(T) Use as thermometer for a sensitive calorimeter A bad idea? Requires bias power to read out changes in R raises temperature and increases fluctuations Dissipative element unavoidable Johnson noise Saved by high sensitivity, convenience.
Detector Optimization Assume: C = C 0 t γ (t T DET /T 0 ) G = G 0 t β C 0 and G 0 are values at the heatsink temperature available Parameters are : T 0, C 0, α, G 0, R, R L, P BIAS Optimization depends on its starting assumptions. We will assume: T 0, C 0, and α (thermometer sensitivity) are limited by technology, stopping efficiency and area requirements, so these are given. NEP TFN and NEP Johnson do not depend on R nor R L, so we are free to chose these to minimize other noise sources. 1. Assume that if we choose R close enough to the noise resistance of the amplifier, amplifer noise will be neglibible. 2. Chose R L sufficiently larger (or smaller) than R to make load resistor noise negligible. We are left with free parameters G 0 and P BIAS to minimize the sum of thermodynamic fluctuation noise plus Johnson noise.
ΔE RMS = k B T 2 0 C 0 ( )t γ + 2 [ ] 4 β +1 α 2 ( 1 t ( β +1) ) Detector Optimization ( ) F LINK (t,β) 1+ α2 t β +1 1 ( β +1) t 2β + 3 1 2 ξ(t,α,β,γ ) k B T 0 2 C 0 Note that G 0 does not appear this expression, so we are free to chose the thermal conductance to get the desired speed or power dissipation (sort of...), and the temperature increase t due to the bias power is the only optimization parameter. Also, see that if α is large enough that the 1 can be neglected, ξ 1/α, giving the approximation that: ΔE RMS k B T 0 2 C 0 /α
γ=3 γ=1 γ=3 γ=1 ΔT BIAS /T 0 The optimum bias raises the temperature by 10 20%, almost independent of alpha for moderate to high alphas. ΔE RMS 4 α k B T 0 2 C 0
Electrothermal Feedback Bias circuits: if we have a resistive thermometer, we need to run a current through it so we can use the measured I and V to determine R. Usual detector bias circuits
Electrothermal Feedback R L V B R L I V R(T) R(T) Electrical Circuit C P in P ext G I V dissipation in R(T) adds to P ext G C T Real Device So P in = P ext + IV Thermal Circuit
Electrical β Feedback R L Electro-thermal β V ext + A V in =V ext + βv β V in V ) Vext A dv /dv in dv = AdV in = A(dV ext + βdv ) (1- βa)dv = AdV ext A CL dv = A 1 dv ext 1 βa V C R(T) G P ext + IV β P in V ) Pext = I dv dv + V di dv A dv dp in = Loop gain: βa = L 0 = αp GT P ext L 0 I 1+ iωτ ( ) L 0 1 + iωτ ( ) A + V dp in =dp ext + βdv R L R L + R R L R R L + R = dimensionless gain
Electrothermal Feedback Consequences in linear case Has same effect on signal and noise for any noise source ahead of or within feedback loop. Initial condition: filtered pulse : i=1 ˆ s i s i n i 2 = i=1 s i 2 n i 2 Detector Output Optimal Filter Output Increase Alpha: Turn on Electro- Thermal Feedback: t t Feedback has little effect in theory for the linear case BUT as pointed out by Kent Irwin is essential for stabilization of very high gain systems increases count rate capabilities for nonlinear systems (most!)
Current Output vs. Voltage Output Since we want negative feedback, we want to maximize βa, where βa = αp GT 1+ iωτ ( ) R L R R L + R For α < 0 (negative temperature coefficient), need R L >> R. Load resistor voltage noise 0 as R L, so this is good. But for α > 0 (as in TES), need R L >> R, so noise high and voltage signal small. Solution: use current as signal instead of voltage. current noise 0 as R L 0, current signal is maximized. The solution to this is to use current through the thermistor, rather than the voltage across it, as the output signal. Does this mean we need to re-derive all the equations?
Changing the Equations Flat circuit theorem: (from symmetry of Kierkoff s laws. Works for any linear circuit that can be drawn without crossing lines.) Circuit Duals V (Volts) I (Amps) C (Farads) L (Henries) R (Ohms) 1/R (Siemens) (parallel series) voltage responsivity S V (ω) = For example: Transform by above rules: current responsivity S I (ω) = αp /GT I( 1+ iωτ) αp /GT V ( 1+ iωτ) R L K F volts/watt R + R L R K R L + R F amps/watt
Changing the Equations Dual circuit transform Thevenin equivalent bias (optional) L=C R S L=C R L R C V I B = V B R = R S = 1/R I=V V B = R I 1/R L I I B R S I That is, the current output of a voltage-biased (R L << R) positive temperature coefficient detector is identical to the voltage output of a current-biased (R L >> R) negative temperature coefficient detector. V B
Detector Impedance (a brief detour) I 1/R ω Im(Z) ω ω = 0 V Z 0 Re(Z) R C 1 Absorber Ideal Detector: C G Detector Bias Power Real Detector: G 1 C 2 Detector Bias Power Thermometer C 3 Heat Sink G 2 G 3 Heat Sink
Detector Impedance (a brief detour) f = C /2πG Simple detector C 1 Absorber X-rays G 1 Bias Power +p 0 sinωt C 2 Detector Thermometer G 3 C 3 More complex detector model from thermal cicuit at right. G 2 Heat Sink Z(ω) =?? As you might imagine, it s more algebra, but for above thermal circuit and others Z(ω) has been calculated analytically.* There is also a powerful matrix method that can calculate anything numerically POSTs, for example.**
END (spare slides follow)
Physics of doped semiconductor thermometers
T 0 is predictable for NTD Germanium and (somewhat) for ion-implanted Silicon: NTD Ge (Haller & Beeman) implanted Si (Wisconsin/GSFC)
Electric field effects: R=R(T,V)
Variable-Range Hopping Conductivity (Mott Shklovskii and Efros)
* electrons Bias Power lattice Heat Sink G T β Hot electrons? Looks like a duck
A: from I-V curve measurements C electrons τ = C/G G lattice Heat Sink B: from complex AC impedance I 1/R ω ω = 0 V Im(Z) Z 0 ω R C: open:c e = A B Walks like a duck filled: measured C tot
electrons 4kT 2 G G lattice Heat Sink Quacks like a duck
High Performance Silicon Hot Electron Bolometers H. Moseley 1 and D. McCammon 2 1 NASA/GSFC, 2 Dept. of Physics, Univ. of Wisconsin Abstract. High performance Si bolometers can be made using the electron-phonon conductance in the heavily doped Si to provide thermal isolation from the cryogenic bath. Thermal
Doped semiconductor thermistors: Well-understood and characterized (at least empirically) Can optimize design and predict performance. Modest sensitivity (practical α ~6 10) Slow at low temperatures: τ e-ph ~ 100 µs 0.1 K T Easy to fabricate as integrated part of silicon structure NTD Ge can be mass-produced with high reproducibility
Meta-stable States Time 30% 70% ionization (E min ~ 10 4 K) ~ TRAPS Channels phonons (no E min )
Major Noise Terms: TFN ΔE 2 = k B T 2 C : p TFN G 2 C ΔE 2 = k B T 2 C Δp TFN = 4k B T 2 G This was for thermal equilibrium, with the detector temperature equal to the heat sink temperature. Bias power produces a temperature gradient in G: P BIAS T 0 p TFN G C Assume: G = G 0 T β T = T 0 + ΔT t T 0 Two limiting cases have been solved: 1. Diffusive Conduction 2. Radiative Conduction λ M.F.P. << L λ M.F.P. >> L T 0 T 0 t T 0 t T 0 (Si x N y films are closer to this case!) 2 p TFN = 4k B T 0 2 G 0 (photons or phonons) β +1 2β + 3 t2β + 3 1 t β +1 1 (W 2 2 Hz) p TFN = 4k B T 2 0 G 0 tβ + 2 +1 2 (W 2 Hz)
TIPS You can consider the effects of most things on resolution with feedback turned off to make spectra simpler. Electrothermal feedback makes the spectra more complicated, but most lines move up or down together. Output of optimal filter = s * 2 ( f ) s( f ) s( f ) = n 2 ( f ) n( f ) = s n ( f ) 2 That is, it depends only on the S/N ratio as a function of frequency. This is unchanged by feedback. Many practical advantages to large negative ETF: -Detector cools faster, so due to nonlinear effects can have better resolution at high count rates. -Necessary for practical stable operation at very high α.
MORE TIPS Remember that energy resolution is proportional to useful bandwidth, meaning the part of the signal spectrum where signal to noise ratio is approximately constant. This can be limited by a) the noise spectrum flattening due to a new source with a flatter spectrum, or b) the signal spectrum steepening due to an additional pole from some source. Remember the exact symmetries between high impedance/ voltage output and low impedance/current output, series inductance vs parallel capacitance, etc.