^{1,a)}

### Abstract

This review discusses detector physics and Monte Carlo techniques for cryogenic, radiation detectors that utilize combined phonon and ionization readout. A general review of cryogenic phonon and charge transport is provided along with specific details of the Cryogenic Dark Matter Search detector instrumentation. In particular, this review covers quasidiffusive phonon transport, which includes phonon focusing, anharmonic decay, and isotope scattering. The interaction of phonons in the detector surface is discussed along with the downconversion of phonons in superconducting films. The charge transport physics include a mass tensor which results from the crystal band structure and is modeled with a Herring-Vogt transformation. Charge scattering processes involve the creation of Neganov-Luke phonons. Transition-edge-sensor (TES) simulations include a full electric circuit description and all thermal processes including Joule heating, cooling to the substrate, and thermal diffusion within the TES, the latter of which is necessary to model normal-superconducting phase separation. Relevant numerical constants are provided for these physical processes in germanium,silicon,aluminum, and tungsten. Random number sampling methods including inverse cumulative distribution function(CDF) and rejection techniques are reviewed. To improve the efficiency of charge transport modeling, an additional second order inverse CDF method is developed here along with an efficient barycentric coordinate sampling method of electric fields. Results are provided in a manner that is convenient for use in Monte Carlo and references are provided for validation of these models.

I would like to thank the entire CDMS collaborators for valuable discussions over the years, especially Adam Anderson, Paul Brink, Blas Cabrera, Enectali Figueroa-Feliciano, Scott Hertel, Peter Kim, Kevin McCarthy, Matt Pyle, Bernard Sadoulet, Kyle Sundqvist, and Betty Young. Blas Cabrera generously provided his note on “electron-phonon scattering,” which was adapted for this review. Kevin McCarthy has been an invaluable contributor to the CDMS Monte Carlo studies cited in the references and Peter Kim was invaluable in tirelessly running Monte Carlo on the SLAC computing farm.

I. INTRODUCTION

A. The CDMS experiment and detectors

II. RADIATION MODELING

III. PHONON SIMULATION

A. Introduction

B. Prompt phonon distributions

C. Phase velocities and polarization vectors

D. Group velocities

E. Anisotropic isotope scattering

F. Anharmonic decay

1. General case

2. Isotropic approximation

G. Phonon losses at surfaces

H. Time steps

I. Random number sampling

J. Numerical constants for phonon simulations

IV. QUASIPARTICLE DOWN CONVERSION

A. Monte Carlo process ordering

V. CHARGE MONTE CARLO

A. Introduction

B. Holes: Propagation and scattering with isotropic bands and isotropic phonon velocity

C. Electrons: Propagation and scattering with anisotropic bands and isotropic phonon velocity

D. Charge time steps, first order

E. Charge time steps, second order

F. Select constants for charge Monte Carlo

VI. ELECTRIC-FIELD LOOKUP

A. Electric-field lookup from triangulated mesh

B. Barycentric coordinates

C. Barycentric coordinate formulae

D. Barycentric coordinate procedures and shortcuts

E. Determining if a charge leaves a triangle

VII. TRANSITION EDGE SENSOR SIMULATIONS

A. Introduction

B. Electrical circuit modeling

1. TES voltage modeling

C. Thermal processes

1. Phonon heat

2. Joule heating

3. Substrate cooling

4. Diffusion within the TES

D. Parsing phonons into the TESs

E. Numerical constants for TES simulation

VIII. FINAL REMARKS

### Key Topics

- Phonons
- 155.0
- Collective excitations
- 25.0
- Aluminium
- 24.0
- Ionization
- 18.0
- Cumulative distribution functions
- 14.0

##### G01T

## Figures

(a) A CDMS “iZIP” detector with photolithographically defined phonon sensors. The crystal is 3 in. in diameter and mounted in its copper housing. The top surface contains an outer, guard phonon sensor, and three inner phonon sensors from which an event's position estimate can be made. The opposite face (not shown) has a similar channel design, but rotated 60°. (b) Close-up view of the iZIP phonon channel and ionization channel (thin lines in between the phonon sensors). The phonon channel is held at ground and the ionization channel is held at ∼±2 V for the top (bottom) surfaces.

(a) A CDMS “iZIP” detector with photolithographically defined phonon sensors. The crystal is 3 in. in diameter and mounted in its copper housing. The top surface contains an outer, guard phonon sensor, and three inner phonon sensors from which an event's position estimate can be made. The opposite face (not shown) has a similar channel design, but rotated 60°. (b) Close-up view of the iZIP phonon channel and ionization channel (thin lines in between the phonon sensors). The phonon channel is held at ground and the ionization channel is held at ∼±2 V for the top (bottom) surfaces.

Description of ion-electron potential interaction process which shows that for ion velocity much less than the Fermi velocity (*v* ≪ *v* _{ f }) the number of electron states that an ion can interact with scales like *v*.

Description of ion-electron potential interaction process which shows that for ion velocity much less than the Fermi velocity (*v* ≪ *v* _{ f }) the number of electron states that an ion can interact with scales like *v*.

Phase velocities in silicon. The distance from the origin represents the phase velocity speed (in m/s). The first three plots are views from the north pole. The fourth view containing all three surfaces is offset from the north pole. In the plots all modes are equally populated, which does not reflect the actual mode populations.

Phase velocities in silicon. The distance from the origin represents the phase velocity speed (in m/s). The first three plots are views from the north pole. The fourth view containing all three surfaces is offset from the north pole. In the plots all modes are equally populated, which does not reflect the actual mode populations.

Group velocities in silicon. Energy is focused in the direction of heavy banding and leads to the term *phonon focusing*. The distance from the origin represents the speed that phonon energy is carried through the crystal (in m/s). In the plots all modes are equally populated, which does not reflect the actual mode populations.

Group velocities in silicon. Energy is focused in the direction of heavy banding and leads to the term *phonon focusing*. The distance from the origin represents the speed that phonon energy is carried through the crystal (in m/s). In the plots all modes are equally populated, which does not reflect the actual mode populations.

Phonons isotope scatter off mass defects in the crystal. Equation (7) gives the individual phonon scatter rates. is the group velocity and is the polarization vector.

Phonons isotope scatter off mass defects in the crystal. Equation (7) gives the individual phonon scatter rates. is the group velocity and is the polarization vector.

Longitudinal phonons decay due to nonlinear terms in the elastic coupling constants. is the group velocity and is the polarization vector.

Longitudinal phonons decay due to nonlinear terms in the elastic coupling constants. is the group velocity and is the polarization vector.

Resultant energies in longitudinal phonon decay in germanium. The two branches *L* → *L* + *T* and *L* → *T* + *T* are shown; the plotted distribution is indicated in bold face in the legend.

Resultant energies in longitudinal phonon decay in germanium. The two branches *L* → *L* + *T* and *L* → *T* + *T* are shown; the plotted distribution is indicated in bold face in the legend.

Resultant angles in longitudinal phonon decay in germanium. The two branches *L* → *L* + *T* and *L* → *T* + *T* are shown; the plotted distribution is indicated in bold face in the legend.

Resultant angles in longitudinal phonon decay in germanium. The two branches *L* → *L* + *T* and *L* → *T* + *T* are shown; the plotted distribution is indicated in bold face in the legend.

Energy band structure of germanium, showing the L-valleys at ⟨111⟩, the Γ valley at ⟨000⟩, and the X-valley at ⟨100⟩. Symmetry results in 8 total L-valleys and 6 total X-valleys.

Energy band structure of germanium, showing the L-valleys at ⟨111⟩, the Γ valley at ⟨000⟩, and the X-valley at ⟨100⟩. Symmetry results in 8 total L-valleys and 6 total X-valleys.

Charge carrier with initial wavevector and final wavevector scattering off of lattice at angle ϕ with respect to and emitting a phonon with wavevector at angle θ with respect to , where the vector momenta sum as shown on right.

Charge carrier with initial wavevector and final wavevector scattering off of lattice at angle ϕ with respect to and emitting a phonon with wavevector at angle θ with respect to , where the vector momenta sum as shown on right.

Mesh node points **r** _{1}, **r** _{2}, and **r** _{3} along with the probe point **r**. The areas *a* _{1}, *a* _{2}, and *a* _{3} are identically equal to the barycentric coordinates λ_{1}, λ_{2}, and λ_{3}.

Mesh node points **r** _{1}, **r** _{2}, and **r** _{3} along with the probe point **r**. The areas *a* _{1}, *a* _{2}, and *a* _{3} are identically equal to the barycentric coordinates λ_{1}, λ_{2}, and λ_{3}.

TES simulation flowchart.

TES simulation flowchart.

TES resistor interconnects as modeled using a finite element approximation.

TES resistor interconnects as modeled using a finite element approximation.

Surface and contour plots of *R* = *R*(*T*, *I*), , and for a high-Tc, inner iZIP channel. The colors in the surface plot indicate the value of resistance, alpha, and beta with blue representing 0 and red the highest value in the figure. The contour plots show the same information but over a limited current and temperature region. The black dot indicates a nominal bias region, which will affect noise and pulse shape after a radiation interaction in the detector. The gradient in resistance and temperature is generally along the temperature direction, whereas for β it is in a mixed −*T* + *I* direction.

Surface and contour plots of *R* = *R*(*T*, *I*), , and for a high-Tc, inner iZIP channel. The colors in the surface plot indicate the value of resistance, alpha, and beta with blue representing 0 and red the highest value in the figure. The contour plots show the same information but over a limited current and temperature region. The black dot indicates a nominal bias region, which will affect noise and pulse shape after a radiation interaction in the detector. The gradient in resistance and temperature is generally along the temperature direction, whereas for β it is in a mixed −*T* + *I* direction.

TES simulation biasing circuitry. Modeling reflects the biasing circuitry.

TES simulation biasing circuitry. Modeling reflects the biasing circuitry.

## Tables

Numerical constants for phonon simulations.

Numerical constants for phonon simulations.

Physical constants for Si and Ge crystals. The isotropic hole effective mass *m* _{ h }, and the anisotropic electron effective masses *m* _{∥} and *m* _{⊥} are ∥ and ⊥, respectively, to the conduction valley axes, and conductivity effective mass 3/*m* _{ c } = 1/*m* _{∥} + 2/*m* _{⊥}. The incident energy per final electron-hole pair is ε_{ eh }, *v* _{ L } the speed of sound, and *l* _{0} = πℏ^{4}ρ/(2*m* ^{3}Ξ^{2}) is the characteristic range for carrier scattering where Ξ_{1} (from Ref. 43) or Ξ_{ fit} (fit to data^{50}) is the deformation potential.

Physical constants for Si and Ge crystals. The isotropic hole effective mass *m* _{ h }, and the anisotropic electron effective masses *m* _{∥} and *m* _{⊥} are ∥ and ⊥, respectively, to the conduction valley axes, and conductivity effective mass 3/*m* _{ c } = 1/*m* _{∥} + 2/*m* _{⊥}. The incident energy per final electron-hole pair is ε_{ eh }, *v* _{ L } the speed of sound, and *l* _{0} = πℏ^{4}ρ/(2*m* ^{3}Ξ^{2}) is the characteristic range for carrier scattering where Ξ_{1} (from Ref. 43) or Ξ_{ fit} (fit to data^{50}) is the deformation potential.

Physical constants for tungsten TES and aluminum fin simulation, from Ref. 57.

Physical constants for tungsten TES and aluminum fin simulation, from Ref. 57.

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