1. Development of a Toolbox for High Purity Germanium Detector Internal Fields Calculation
16 May 2017 1

2. Motivation
It is desirable to fabricate High Purity Germanium (HP-Ge) detectors to operate with internal electric fields exceeding 20 V/mm. This leads to saturation velocity of charge carriers in the detector and minimizes the charge collection time per gamma event.
In order to minimize leakage current and minimize the capacitance of the electrode, operation of the detector occurs at an applied reverse bias that is just beyond total depletion.
Electric field modeling allows for an understanding of the effects of physical dimensions of detector and impurity doping profile. This offers an opportunity for optimization of these parameters. A specific design for the physical dimensions of a detector can be determined for a given Ge crystal.
CoSSURF-LB workshop

3. Detector Technology Team
High purity germanium crystal growth
HPGe detector manufacturing
Special detector system assembly
Technology advancement - detector development
CoSSURF-LB workshop

4. HP Germanium detector production
Fabrication
Process HPGe detector
Status: Load
Load detector element into the selected cryostat configuration
Crystal Growth
Grow HPGe detector element
Shaping
Shape detector element
GE TEST
Test detector
Initial and Performance testing
PUMP
Pump/Bake
GE TEST
Cycle for stability
Pass/Fail? P
Pass/Fail? F P
LB? Y N F
Do LB Evaluation
LB Evaluation Y
Pump / Bake? N
Disposition:
Complete for Shipment
Cycle time is measured in days
CoSSURF-LB workshop

5. Poisson Equation Approximation
- Poisson’s Equation
∇ 2 𝑉=− 𝜌 𝜖
Laplace Operator in Cylindrical Coordinates
𝜕 2 𝑉 𝜕𝑧 𝜕 2 𝑉 𝜕𝑟 𝑟 𝜕𝑉 𝜕𝑟 + 1 𝑟 2 𝜕 2 𝑉 𝜕𝜃 2 =− 𝜌 𝜖
- Rotational Symmetry
𝜕 2 𝑉 𝜕𝑧 𝜕 2 𝑉 𝜕𝑟 𝑟 𝜕𝑉 𝜕𝑟 =− 𝜌 𝜖
- Finite Difference Approximations
𝑉 𝑖,𝑗+1 − 2𝑉 𝑖,𝑗 + 𝑉 𝑖,𝑗−1 Δ𝑧 𝑉 𝑖+1,𝑗 − 2𝑉 𝑖,𝑗 + 𝑉 𝑖−1,𝑗 Δ𝑟 𝑟 𝑖 𝑉 𝑖+1,𝑗 − 𝑉 𝑖−1,𝑗 2Δ𝑟 =− 𝜌 𝑖,𝑗 𝜖 𝑖,𝑗

6. Poisson Equation Approximation
- Linear System of Equations 𝐷 𝑈 ⋯ 𝐿 𝐷 ⋯ ⋮ ⋮ ⋱ 𝑉 1,1 𝑉 1,2 ⋮ = − 𝜌 1,1 𝜖 1,1 − 𝜌 1,1 𝜖 1,1 ⋮ 𝐴 𝑉=𝑏 - Decomposition of Matrix 𝐴 𝑉= 𝐷+𝑈+𝐿 𝑉=𝑏 𝐷+𝐿 𝑉 𝑛+1 =𝑏− 𝑈 𝑉 𝑛 - Iterative solution by point by point evaluation through the vector (V), proceed until the change from one iteration to the next is smaller than a predetermined increment (<1μV)

7. Boundary Conditions Boundary values have three possibilities:
Applied Reverse Bias (V=VApplied)
Ground (V=0)
Passivated/Insulated surface
Electric Field in normal direction is zero

8. Converged Solution (Full Depletion)

9. Convergence Acceleration
- Successive Overrelaxation Technique 𝜔 𝐷+𝐿 𝑉 𝑛+1 =𝜔 𝑏− 𝑈 𝑉 𝑛 𝐷+𝜔𝐿 𝑉 𝑛+1 =𝜔𝑏− 𝜔𝑈+ 𝜔−1 𝐷 𝑉 𝑛 - Adjustment of relaxation factor (ω) gives significantly faster convergence rate. Relaxation factor must be greater than zero and less than two for convergence.

10. Partial Depletion

11. Impurity Gradient

12. Pinch-Off

13. Inverted Coax Hole

14. Future Uses Estimation of impurity profile in the detector
Prediction of full depletion voltage
Custom physical dimension design for a given Ge crystal
Pulse shape determination
Segmented detectors