Spatial Distribution of Electron-Hole Pairs Created by Photons in Detector Materials Fei Gao, Luke Campbell, Yulong Xie, Ram Devanathan, Anthony Peurrung, William Weber Pacific Northwest National Laboratory Supported by the Radiation Detection Materials Discovery Initiative at PNNL Outline • Introduction Spatial distribution of electron-hole pairs (e-h) – previous simulations Calculation of e-h pair yield Several issues to be addressed • Monte Carlo Model • Production of e-h pairs • Electron cascade and number distribution of e-h pairs W value and Fano Factor Spatial Distribution of Electron-Hole Pairs Conventional detailed MC method Spatial distributions of electron-hole pairs (in Ge) 2 Introduction Most previous simulations of e-h pairs – only consider track structure of secondary electrons created by photons to evaluate dE/dx [NIMA 563 (2006) 116] 106 photons Total deposited energy: 100 keV electron in CdZnTe – linear electron cascade ¾ e-h pair density (or number of e-h pairs) - ρ = 1 dE W dx W – mean energy required to create an electron-hole pair 3 Introduction An attempt to carry out the full e-h – only considers inner shell ionizations [J. Phys. D: Appl. Phys. 33 (2000) 1417] e-h pair distribution by a 140 keV photon in amorphous-Si Several issues – detailed distribution of e-h pairs ¾ e-h pair recombination ¾ Nonlinear electron cascade- more accurate calculations of intrinsic properties ¾ Nonlinearity observed in scientilators 4 Monte Carlo Model Model – interaction of photon with materials: ¾ Photoelectric absorption ¾ Compton scattering ¾ Electron-positron pair production Model – detailed energy-loss mechanisms for electrons (inelastic) ¾ Electron-Phonon interaction ¾ Valence Band Ionization ¾ Excitation of Plasmons and Decay ¾ Core shell ionization (K and L-shell…..) ¾ Shake-off electrons ¾ Bremsstrahlung emission – energy loss ¾ Band structure effects (for semiconductors) ¾ Stopping criteria for electrons – 2.5 eV Model – elastic interaction Mean free path between collisions; λ−1 = N ( σ el + σ in ) 5 Electron Cascades in Ge ¾ Photon energy – 50 eV ~ 2 MeV ¾ Number of simulations for each energy – 105 (statistics) ¾ Band gap – 0.74 for Ge Electron distribution in Ge 4 No. Events (x10 ) 3.0 50 eV 150 eV 600 eV 2.5 ¾ Asymmetric ¾ Not perfect Gaussian distribution ¾ No. electrons/event = 16.8, 54.3 and 224.6 for 50, 150 and 600 eV, respectively 2.0 1.5 1.0 0.5 0.0 0 50 100 150 200 250 No. Electrons (n) 6 Electron Cascades in Ge 1200 40 keV 662 keV No. Events 1000 662 keV gamma ray from 137Cs 800 600 400 200 0 1.500e+4 1.530e+4 2.496e+5 2.508e+5 No. Electrons (n) ¾ Symmetric distribution ¾ Approximate Gaussian distribution ¾ No. electrons ~ 557.4 and 2893.2 for 40 and 662 keV, respectively ¾ Height of the distribution decreases with increasing energy ¾ Width of the distribution increases with increasing energy 7 Electron Cascades in Ge W value and Fano Factor MII/MIII LII/LIII 3.0 LI 0.25 K Ge MI 0.20 0.15 F W (eV) 2.9 2.8 0.10 2.7 0.05 2.6 0.01 0.1 1 10 100 1000 0.00 10000 E (keV) ¾W: For E (>2 keV), ~ 2.64 eV (exp: 2.6 – 2.9 eV) ¾F: For E (>2 keV) 0.11 (exp: 0.09 ~ 0.195) 8 Electron Cascades in Several Materials ΔEin Intrinsic resolution: E 5.0 4.0 0.40 CZT Simulations Experiments 4.5 = 8 ln( 2 )WF / E 0.35 0.30 Si SiGe 0.20 Ge 3.0 0.15 2.5 2.0 0.6 F W 0.25 3.5 0.10 0.8 1.0 1.2 1.4 0.05 1.8 1.6 Band gap (eV) ⎡ 0.12 E g2 + 0.22 E g + 0.057 ⎤ ΔEin = 2.355 ⎢ ⎥ E E ⎢⎣ ⎥⎦ 1/ 2 9 Spatial Distribution of e-h Pairs Distribution of electron-hole pairs in Ge (2 keV) Photon Electron (interband) Electron (plasmon) Electron (ionization) Electron (relaxation) Hole ¾Electrons created by plasmon – along the track ¾Electrons created by interband transition – at periphery of cascade volume ¾There are some single electrons and holes. ¾Most electrons and holes are close to each other. ¾Thermalized electrons can move away from cascade volume. 40 nm 10 Spatial Distribution of e-h Pairs Distribution of electron-hole pairs in Ge (2 keV) ¾Distribution of electron-hole pairs is different for different cascade (same energy) ¾Density of electron-hole pairs along main path is high ¾Density of electron-hole pairs at the periphery of cascade volume is low. 11 Spatial Distribution of e-h Pairs Distribution of electron-hole pairs in Ge (10 keV) Photon Electron (interband) Electron (plasmon) Electron (ionization) Electron (relaxation) Electron (shake off) Hole ¾Nonlinear electron cascade 12 Spatial Distribution of e-h Pairs Dynamic process of electron-hole pair creation (1 keV) 1 keV photon 13 Conclusions A Monte Carlo code has been developed to simulate electron cascade, production of e-h pairs and their number distribution in detector materials. Intrinsic resolution in semiconductors follows; ΔEin ⎡ 0.12 E g2 = 2.355 ⎢ E ⎢⎣ + 0.22 E g + 0.057 ⎤ ⎥ E ⎥⎦ 1/ 2 Plasmon and interband transition are major mechanisms to generate e-h pairs in semiconductors. Spatial distribution of e-h pairs indicates that electrons created by plasmon are along the track, while electrons created by interband transition distribute at the periphery of cascade volume - nonlinear electron cascade 14 Thank You 15 Monte Carlo Model Cross sections – based on the generalized oscillator strength model, a new model has been developed. 1e-15 Total 1e-16 Interband transition σ (cm2) 1e-17 LII/LIII 1e-18 Plasmon excitation 1e-19 1e-20 LI 1e-21 1e-22 K Si 1e-23 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6 1e+7 1e+8 1e+9 Ee (eV) ¾ Total cross section – within 2%-8% of experimental measurements ¾ Band structure effects on cross sections at low energy (<20 eV) must be calculated 16 First -Principles Approach First-Principles Differential electronic scattering cross section determined by dielectric function [ ] ∂σ (q, ω ) 8π ℑ ε −1 (q, ω ) =− 2 vn ∂q ∂ω q ab initio data model Dielectric function is a function of band energies and orbitals ε ( q ,ω ) = f ( En ,k ,< r | n ,k > ) Need band energies En, k, and orbitals r | n, k • Use ABINIT code to get band structure - Plane wave basis set, pseudopotentials • Correct DFT band energies using GW method. • Use DFT Kohn-Sham eigenfunctions for orbitals 17 First -Principles Approach First-Principles Our calculated dielectric function for silicon is similar to experiments Our calculations Philipp & Ehrenreich, Phys. Rev., 129, 1550, (1963) Experiment ¾With differential cross section known, we can integrate over q , ω to find total cross section & electron mean free paths. ¾Determine secondary particle (electron, hole) spectrum from plasmon decay Walter & Cohen, Phys. Rev. B, 5, 3101, (1972) Theory 18 First -Principles Approach First-Principles Cross section – ab initio data model 1e-15 Si σ (cm2) 1e-16 Total 1e-17 1e-18 Plasmon excitation 1e-19 Ab Initio data model Interband transition 1e-20 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6 1e+7 1e+8 1e+9 Ee (eV) ¾ Total cross section is in reasonable agreement with that from ab initio data model. ¾ We will apply the ab initio data model to known and unknown materials. 19 Spatial Distribution Elastic DCS – Walker, Adv. Phys. 20, 257 (1971) ¾ Solving the partial-wave expanded Dirac equation for the motion of the projectile in the field of the target atom dσ el 2 2 = f ( θ ) + g( θ ) dθ f (θ ) = 1 ∞ ∑ {(l + 1)[exp(2iδ l − ) − 1] + l [exp(2iδ l − ) − 1]}Pl (cosθ ) 2ik l =0 g( θ ) = 1 ∞ 1 ∑ {exp( 2iδ l − + exp( 2iδ l + }Pl (cos θ ) 2ik l =0 ¾ DCS δl – phase shifts Pl(cosθ) – Legendre polynomials 1e-14 Ge σ el dσ = ∫ el dθ σ(cm2) 1e-15 1e-16 1e-17 1e-18 1e-19 Inelastic Elastic 1e-20 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6 1e+7 1e+8 1e+9 E (eV) 20 Spatial Distribution Approach – conventional detailed MC method ¾ Each simulated path of a particle – characterized by a series of states; {rn , En ,cos θ n } λ−1 = N ( σ el + σ in ) ¾ Mean free path between collisions; ¾ Length s of the free path to the next collision is obtained by random sampling from the distribution; p( s ) = λ−1 exp( − s / λ ) rn+1 = rn + s cos θ n 1 dσ el p( θ ) = el σ dθ ¾ If elastic collision, θ is sampled from the distribution; ¾ If inelastic collision, θ is fixed by energy and momentum conservation Q( Q + 2 mc 2 ) = c 2 ( p 2 + p' 2 −2 pp' cosθ ) Depends on type of inelastic collisions Momentum transfer – q=p-p’ Recoil energy, Q – Q=[(cq)2+mc2]1/2-mc2 ¾ Azimuthal scattering angle, φ – φ=2πrand() 21