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Finite electron range and its implications for superconducting photon detectors
Physica B 169 (1991) 439-440 North-Holland
FINITE ELECTRON
RANGE AND ITS IMPLICATIONS
Kent S. WOOD, Michael
M. LOVELLETTE*,
Code 4120, US Naval Research
FOR SUPERCONDUCTING
and Deborah
Laboratory,
PHOTON DETECTORS
VAN VECHTEN
Washington,
D.C. 20375-5000
USA
We consider global transitions from superconducting to normal state of small grains induced by the absorption of energetic (l- 100 keV) photons. Initially the energy is transferred to a single photoelectron, which loses the energy as it scatters inelastically. To measure the energy of the photon, this collision cascade must be contained within the grain. Reconciling this requirement with the requirement that the dispersed energy must suffice to warm the grain above the transition temperature, leads to estimates of grain sizes and effective operating range of photon energies for various materials and T and H biasing conditions.
Superconducting devices are being developed as detectors1 of individual x-ray and T-ray photons and energetic particles. Two classes of devices are considered in this paper. Superconducting Granular Detectors (SGD) magnetically sense the phase transition which results from the bolometric response of an electrically isolated lump ("grain") of superconductor that has interacted with the incident particle. The grains must transition if there is to be a signal. The absorptive grains may be separately sensed, positioned in a spatially ordered array, or randomly positioned in a colloidal preparation. In Tunnel Junction Detectors (TJD) the single particle tunneling current has an excursion as the energy deposited by the incident particle degrades through the range corresponding to Cooper pair breaking. Since the statistical uncertainty in number of non-equilibrium quasiparticles which tunnel is inversely proportional to the energy gap, low T, junctions are inherently desirable. Processes associated with spatial variation in the gap, especially local phase transitions, degrade the energy resolution and should be avoided. On the last basis, materials that make good SGD make poor TJD and vice versa2. Thus in this article we focus on SGD used as photon detectors and quantify the minimum conditions necessary for grain "flipping' (transitions to the normal state) to result from photon interactions. The energy of the incident particle is always deposited over a significant volume. As the average energy per excitation decreases, the spatial extent and number of excitations increases to conserve energy. When an x-ray or y-ray interacts near some interior lattice site, the photoelectron or Compton electron carries 1 - 100 keV of kinetic energy. It progresses * National
Research
Council-
Naval Research
Laboratory
through the lattice ionizing other atoms, thereby producing secondary electrons, and otherwise inelastically scattering until it is ultimately captured. The total distance it travels is called the primary electron range, Rev typically less than 10 pm. Secondary electrons further enlarge the disturbed environment and produce additional lower energy electrons and plasmons. Stage 1 of the collision cascade refers to all those processes which occur before the primary electron stops. (Stage 2 designates the subsequent outward diffusion phase which is highly relevant to TJD optimization. However, for SGD it suffices to consider only stage 1, which we expect to take less than 1 ns.) At the end of stage 1, the energy is confined within an irregularly bending tubular region sparsely filled with a heavily branched network of electron tracks having a central spine. Similar structures result from incident massive, heavily ionizing particles such as alpha particles, the distinctions being that the primary particle's range is comparable to that of the secondary electrons and that the overall shape is more spherical. In order for the grain's behavior to provide a measure of the energy of the initial electron, and thus the incident photon, the entire stage 1 of the cascade must be confined within the grain. Thus assuming a spherical grain shape, the grain's radius should exceed Rmin = gRe* The factor g is N 1 for photons, where the absorption length often exceeds Re. while g w.5 for a particles. Specializing to photons, the range is material independent within a few percent and R, = r. R(l+P) (I) where r. (pm) = 3.5 p-1 and p= 0.754 for p in gm cmp3 and E in units of 10keV. Ignoring the Postdoctoral
Associate
in code 4120.
440
Table 1: Energies
KS. Wood et al. I Finite electron range
and grain radii that allow grain transitions.
element name primary e- range in /Am. 6keV incident, R, lattice/electron specific heat at T,(H=O) max. capturable energy E' in keV, H=O corresp. max. radius R(E') in /Im. H=O operating range: E2 in keV, H=O El in keV, H=O Rl2 in pm. H=O max. capturable energy E' keV, H=.5H,(T=O) corresp. max radius R(E') km, H=.5Hc(T=O) operating range: E2 in keV, H=.SH,(T=O) El in keV, H=.5H,(T=O) Rl2 in km, H=.SH,(T=O)
Cd 0.17 0.034 66.313 11.23 46.42 10.15 6.01 31.48 3.04 22.03 4.82 1.63
possibility of superheating and states that are not uniform spatially, the grain will transition only if the initial energy E exceeds the energy required to raise the grain 7 0 a final temperature Tf > Tc. The maximum grain radius is thus l/3 Tc R = 3 ET/ 4rj c,(T) dT (2) max 1 Tb For H=O there is no latent heat contribution. As Table 1 indicates, the lattice contribution to the specific heat is smaller than the electronic in the vicinity of T, for most of the elemental superconductors. Inspecting e uations 1 and 2, it becomes clear Rmin a ET P l+P), while Rrnax a E7 l/3, Plotted, these two curves intersect at an energy E' which is the maximum photon energy that can flip grains of this material starting from Tb. Since most SGD are biased close to T, in order to minimize the threshold energy, it is generally sufficient to approximate the specific heat integral by the product of c,(T,) and AT. We find E' and R(E') are only weak functions of cs, namely E' a cs0.23 and R(E') a ~~O.41. Values of E' and R(E') representative of Tb/Tc = .95 are included in Table 1. It is necessary to reduce the maximum energy to which the grain will respond and shrink its radius to achieve any dynamic range. (See the El, E2, R12 entries in Table 1. calculated such that E2 = .7E' for the same Tb.) The above assumes H=O, yet an applied field is required to sense magnetically the phase transition of the grains. Applying a magnetic field will suppress the transition temperature, thereby decreasing both the lattice and the electronic contributions to the specific heat. However, the introduction of a field also requires the addition of a latent heat term AQL where in general (3) AQL = Tc(H)(Sn - S,) = T(Hc(T)/4r)(dHc/dT). Eq. (3) must be evaluated using a definite approximation to the function H,(T). Because of the weak dependence of E' and R(E') on cs, the empirical relation H,(T) = H,(l - (T/Tc)2) (4)
1
0.14 0.004 42.381 4.34 29.67 6.49 2.32 21.52 1.32 15.07 3.30 0.71
All entries assume Tb/Tc(H) = 0.95.
In 0.20 4.30 14.095 0.88 9.87 2.16 0.47 9.31 0.42 6.51 1.43 0.23
Al 0.53 0.011 16.039 3.00 11.23 2.46 1.60 a.03 0.89 5.62 1.23 0.48
Pb 0.13 15.1 9.332 0.28 6.53 1.43 0.15 7.61 0.19 5.32 1.16 0.10
Sn 0.25 0.840 15.380 1.30 10.77 2.35 0.70 7.48 0.37 5.24 1.15 0.20
Nb 0.17 0.418 8.583 0.31 6.01 1.31 0.17 4.57 0.10 3.12 0.70 0.06
is adequately accurate. The values in Table 1 illustrating the special case H = .5 H,(T=O), show that suppressing the operating temperature by applying a field does not allow significantly larger grains to be flipped. The above theoretical discussion was derived to treat the cases of incident photons and high energy electrons. Thus it is difficult to test it by comparision with published experiments since most of these have been done by irradiating superheated grains with alpha particles. Detectors involving superheated grains cannot reset themselves without outside intervention. In addition, the experiments show evidence of transitions of only part of the volume, a possibil't i y assumed forbidden in the above which considers only transition of whole grains. In summary, Table 1 indicates grain sizes that should provide workable photon detectors and their maximum energies. In only one of the 28 entries is the grain radius over 1Opm. Photolithographic fabrication of arrays will be difficult below 1 pm. Many other small size effects pose additional problems. Materials with higher density and lower T, are preferable. Preliminary investigations suggest that biasing in the intermediate state might increase the maximum grain size and energy range beyond the tabulated values. A more detailed discussion of these topics is available from the authors as a preprint. FOOTNOTES AND REFERENCES (1) See the following conference proceedings and references therein: A. Barone, ed.. Superconductive Particle Detectors (World Scientific, Singapore, 196s); L. GonalezMestres and D. Perret-Gallix, ed., Low Temperature Detectors for Neutrinos and Dark Matter II (Editions Frontieres, Gifsur-Yvette, 1988); and G. Waysand and G. Chardin, ed., Superconducting and Low Temperature Particle Detectors (North Holland, Amsterdam, 1989).
FINITE ELECTRON
RANGE AND ITS IMPLICATIONS
Kent S. WOOD, Michael
M. LOVELLETTE*,
Code 4120, US Naval Research
FOR SUPERCONDUCTING
and Deborah
Laboratory,
PHOTON DETECTORS
VAN VECHTEN
Washington,
D.C. 20375-5000
USA
We consider global transitions from superconducting to normal state of small grains induced by the absorption of energetic (l- 100 keV) photons. Initially the energy is transferred to a single photoelectron, which loses the energy as it scatters inelastically. To measure the energy of the photon, this collision cascade must be contained within the grain. Reconciling this requirement with the requirement that the dispersed energy must suffice to warm the grain above the transition temperature, leads to estimates of grain sizes and effective operating range of photon energies for various materials and T and H biasing conditions.
Superconducting devices are being developed as detectors1 of individual x-ray and T-ray photons and energetic particles. Two classes of devices are considered in this paper. Superconducting Granular Detectors (SGD) magnetically sense the phase transition which results from the bolometric response of an electrically isolated lump ("grain") of superconductor that has interacted with the incident particle. The grains must transition if there is to be a signal. The absorptive grains may be separately sensed, positioned in a spatially ordered array, or randomly positioned in a colloidal preparation. In Tunnel Junction Detectors (TJD) the single particle tunneling current has an excursion as the energy deposited by the incident particle degrades through the range corresponding to Cooper pair breaking. Since the statistical uncertainty in number of non-equilibrium quasiparticles which tunnel is inversely proportional to the energy gap, low T, junctions are inherently desirable. Processes associated with spatial variation in the gap, especially local phase transitions, degrade the energy resolution and should be avoided. On the last basis, materials that make good SGD make poor TJD and vice versa2. Thus in this article we focus on SGD used as photon detectors and quantify the minimum conditions necessary for grain "flipping' (transitions to the normal state) to result from photon interactions. The energy of the incident particle is always deposited over a significant volume. As the average energy per excitation decreases, the spatial extent and number of excitations increases to conserve energy. When an x-ray or y-ray interacts near some interior lattice site, the photoelectron or Compton electron carries 1 - 100 keV of kinetic energy. It progresses * National
Research
Council-
Naval Research
Laboratory
through the lattice ionizing other atoms, thereby producing secondary electrons, and otherwise inelastically scattering until it is ultimately captured. The total distance it travels is called the primary electron range, Rev typically less than 10 pm. Secondary electrons further enlarge the disturbed environment and produce additional lower energy electrons and plasmons. Stage 1 of the collision cascade refers to all those processes which occur before the primary electron stops. (Stage 2 designates the subsequent outward diffusion phase which is highly relevant to TJD optimization. However, for SGD it suffices to consider only stage 1, which we expect to take less than 1 ns.) At the end of stage 1, the energy is confined within an irregularly bending tubular region sparsely filled with a heavily branched network of electron tracks having a central spine. Similar structures result from incident massive, heavily ionizing particles such as alpha particles, the distinctions being that the primary particle's range is comparable to that of the secondary electrons and that the overall shape is more spherical. In order for the grain's behavior to provide a measure of the energy of the initial electron, and thus the incident photon, the entire stage 1 of the cascade must be confined within the grain. Thus assuming a spherical grain shape, the grain's radius should exceed Rmin = gRe* The factor g is N 1 for photons, where the absorption length often exceeds Re. while g w.5 for a particles. Specializing to photons, the range is material independent within a few percent and R, = r. R(l+P) (I) where r. (pm) = 3.5 p-1 and p= 0.754 for p in gm cmp3 and E in units of 10keV. Ignoring the Postdoctoral
Associate
in code 4120.
440
Table 1: Energies
KS. Wood et al. I Finite electron range
and grain radii that allow grain transitions.
element name primary e- range in /Am. 6keV incident, R, lattice/electron specific heat at T,(H=O) max. capturable energy E' in keV, H=O corresp. max. radius R(E') in /Im. H=O operating range: E2 in keV, H=O El in keV, H=O Rl2 in pm. H=O max. capturable energy E' keV, H=.5H,(T=O) corresp. max radius R(E') km, H=.5Hc(T=O) operating range: E2 in keV, H=.SH,(T=O) El in keV, H=.5H,(T=O) Rl2 in km, H=.SH,(T=O)
Cd 0.17 0.034 66.313 11.23 46.42 10.15 6.01 31.48 3.04 22.03 4.82 1.63
possibility of superheating and states that are not uniform spatially, the grain will transition only if the initial energy E exceeds the energy required to raise the grain 7 0 a final temperature Tf > Tc. The maximum grain radius is thus l/3 Tc R = 3 ET/ 4rj c,(T) dT (2) max 1 Tb For H=O there is no latent heat contribution. As Table 1 indicates, the lattice contribution to the specific heat is smaller than the electronic in the vicinity of T, for most of the elemental superconductors. Inspecting e uations 1 and 2, it becomes clear Rmin a ET P l+P), while Rrnax a E7 l/3, Plotted, these two curves intersect at an energy E' which is the maximum photon energy that can flip grains of this material starting from Tb. Since most SGD are biased close to T, in order to minimize the threshold energy, it is generally sufficient to approximate the specific heat integral by the product of c,(T,) and AT. We find E' and R(E') are only weak functions of cs, namely E' a cs0.23 and R(E') a ~~O.41. Values of E' and R(E') representative of Tb/Tc = .95 are included in Table 1. It is necessary to reduce the maximum energy to which the grain will respond and shrink its radius to achieve any dynamic range. (See the El, E2, R12 entries in Table 1. calculated such that E2 = .7E' for the same Tb.) The above assumes H=O, yet an applied field is required to sense magnetically the phase transition of the grains. Applying a magnetic field will suppress the transition temperature, thereby decreasing both the lattice and the electronic contributions to the specific heat. However, the introduction of a field also requires the addition of a latent heat term AQL where in general (3) AQL = Tc(H)(Sn - S,) = T(Hc(T)/4r)(dHc/dT). Eq. (3) must be evaluated using a definite approximation to the function H,(T). Because of the weak dependence of E' and R(E') on cs, the empirical relation H,(T) = H,(l - (T/Tc)2) (4)
1
0.14 0.004 42.381 4.34 29.67 6.49 2.32 21.52 1.32 15.07 3.30 0.71
All entries assume Tb/Tc(H) = 0.95.
In 0.20 4.30 14.095 0.88 9.87 2.16 0.47 9.31 0.42 6.51 1.43 0.23
Al 0.53 0.011 16.039 3.00 11.23 2.46 1.60 a.03 0.89 5.62 1.23 0.48
Pb 0.13 15.1 9.332 0.28 6.53 1.43 0.15 7.61 0.19 5.32 1.16 0.10
Sn 0.25 0.840 15.380 1.30 10.77 2.35 0.70 7.48 0.37 5.24 1.15 0.20
Nb 0.17 0.418 8.583 0.31 6.01 1.31 0.17 4.57 0.10 3.12 0.70 0.06
is adequately accurate. The values in Table 1 illustrating the special case H = .5 H,(T=O), show that suppressing the operating temperature by applying a field does not allow significantly larger grains to be flipped. The above theoretical discussion was derived to treat the cases of incident photons and high energy electrons. Thus it is difficult to test it by comparision with published experiments since most of these have been done by irradiating superheated grains with alpha particles. Detectors involving superheated grains cannot reset themselves without outside intervention. In addition, the experiments show evidence of transitions of only part of the volume, a possibil't i y assumed forbidden in the above which considers only transition of whole grains. In summary, Table 1 indicates grain sizes that should provide workable photon detectors and their maximum energies. In only one of the 28 entries is the grain radius over 1Opm. Photolithographic fabrication of arrays will be difficult below 1 pm. Many other small size effects pose additional problems. Materials with higher density and lower T, are preferable. Preliminary investigations suggest that biasing in the intermediate state might increase the maximum grain size and energy range beyond the tabulated values. A more detailed discussion of these topics is available from the authors as a preprint. FOOTNOTES AND REFERENCES (1) See the following conference proceedings and references therein: A. Barone, ed.. Superconductive Particle Detectors (World Scientific, Singapore, 196s); L. GonalezMestres and D. Perret-Gallix, ed., Low Temperature Detectors for Neutrinos and Dark Matter II (Editions Frontieres, Gifsur-Yvette, 1988); and G. Waysand and G. Chardin, ed., Superconducting and Low Temperature Particle Detectors (North Holland, Amsterdam, 1989).