A new approach to IR bimaterial detectors theory

Infrared Physics & Technology 50 (2007) 51–57 www.elsevier.com/locate/infrared

A new approach to IR bimaterial detectors theory Z. Djuric´ *, D. Randjelovic´, I. Jokic´, J. Matovic´, J. Lamovec IHTM—Institute of Microelectronic Technologies and Single Crystals, University of Belgrade, Njegosˇeva 12, 11000 Belgrade, Serbia Received 27 March 2006 Available online 21 August 2006

Abstract In this paper we propose a new theoretical approach to the analysis of bimaterial infrared thermal detector (BMD) performance. In order to determine the basic parameters of a BMD—sensitivity, noise equivalent power, detectivity—we considered the BMD as a mechanical oscillating system and introduced an equivalent ‘‘thermomechanical’’ excitation force. This force is proportional with temperature changes and causes mechanical oscillations of the bimaterial cantilever. After identifying all of the relevant noise mechanisms (temperature fluctuations, Brownian motion), we solved the appropriate Langevin stochastic equation and obtained the mean square deflection of the bimaterial cantilever oscillator. This enabled us to determine all of the important BMD parameters. These parameters depend on the relevant thermal, mechanical and geometrical properties of the constituent parts of the detector and the chosen materials, as well as on the gas type and pressure inside the housing. Our analysis is focused on the study of pressure influence to the BMD performance. We showed that detectivity can approach the ideal value with pressure decrease if other bimaterial microcantilever parameters are properly chosen. Finally, we applied our theory on a BMD fabricated at ORNL.  2006 Elsevier B.V. All rights reserved. PACS: 07.10.Cm; 07.57.K; 85.60.G; 46.30.M Keywords: IR bimaterial detector; Mechanical oscillator; Noise mechanisms; Thermomechanical force; Detectivity

1. Introduction Infrared photodetectors are usually divided into two principally different groups: thermal detectors and photon detectors. Historically, thermal detectors were the first detectors operating in the infrared (IR) range of electromagnetic spectrum. Since about 1930 the development of infrared technology has been dominated by narrow-gap semiconductor photodetectors. The devices exhibit both excellent signal-to-noise performance and very fast response. However, to achieve this, IR photodetectors require cryogenic cooling. This is necessary to prevent thermal generation of charge carriers. Thermal transitions compete with the optical ones, making non-cooled devices very noisy.

*

Corresponding author. Tel.: +381 11 638 188; fax: +381 11 182 995. E-mail address: [email protected] (Z. Djuric´).

1350-4495/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.infrared.2006.07.001

Cooling requirements are the main obstacle to the more widespread use of IR systems based on semiconductor photodetectors, making them bulky, heavy, expensive and inconvenient to use. In contrast, thermal detectors are typically operated at room temperature, with performance varying only slightly with temperature. Currently, there is a considerable interest in 2D thermal detector arrays [1–12] for low-cost thermal imagers, whose moderate sensitivity can be compensated by a large number of elements. The advent of large 2D arrays of uncooled IR sensors allows staring arrays. The appearance of MEMS technologies enabled development of new generation of thermal detectors—bolometertype. Today, bolometric focal plane arrays (FPA) with 320 · 240 elements are already in use [9]. Average noise equivalent temperature difference (NETD) of these FPA is about 25 mK with f/1 optics and their thermal time constant is approximately 20 ms. The detection mechanism in these

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uncooled imaging arrays is based on electrical signal readout from each pixel. Complex fabrication of interconnections between pixels as well as the complexity of the readout circuits are the main reasons for high price which still represents the main obstacle for a wider use of these thermovision devices in a number of commercial applications. Besides this, thermal detectors which require electrical connection between each pixel and the readout circuit cannot achieve thermal isolation close to the radiation limit. Finally, electronically based systems such as bolometers inherently generate 1/f and Johnson noise which limits their detectivity far below radiation limit (1.8 · 1010 cm Hz1/2 W1). In recent years a number of papers appeared which suggest the use of optomechanical or capacitive-mechanical imaging systems in order to overcome the above stated problems [10–13]. Most of these papers are inspired by the fact that microcantilevers built of two different materials and such optical detection of displacement as that used in AFM systems can be applied for detection of small temperature variations. Small temperature changes of the bimaterial cantilever cause its deflection which is measured with a laser optical lever and a position sensitive silicon photodiode. Using this method deflections of the order 1012 m are readily measured. It is interesting to note that as early as in 1959 Jones et al. [14] constructed a small optical lever capable of detecting displacements even smaller than 1012 m. This system has also been applied to an infrared detector using the linear expansion of a metallic strip. For a 1 Hz bandwidth, according to the data given in the paper (active area 5 mm · 0.2 mm) it can be concluded that detectivity of approximately 1010 cm Hz1/2 W1 was achieved. In this paper, we present expressions for basic parameters of a bimaterial IR detector (sensitivity, noise equivalent power (NEP), detectivity) which describe their dependence on the relevant thermal and mechanical properties of the detector structure and also the influence of the conditions inside the detector housing (gas type and pressure). In this theoretical analysis we have neglected the influence of ambient temperature variations. Detailed studies of ambient temperature influence and methods of suppressing the same are given in [7,12,15,16]. Section 2 of this paper presents a short theory of the bimaterial effect. In order to find the sensitivity of this detector, in Section 3 we introduce the equivalent ‘‘thermomechanical’’ driving force and the solve equation of a mechanical oscillator assuming only the first mode vibrations. Section 4 is dedicated to the determination of noise mechanisms in a bimaterial detector. After identifying all of the pertinent noise mechanisms, the appropriate Langevin stochastic equation is solved. Thus, the spectral density of the mean square value of the bimaterial cantilever deflection is deduced. Using the results from the Section 3, we determine the detectivity of bimaterial IR detector.

In Section 5, for a given structure of bimaterial detector (BMD), its basic parameters (resonant frequency, Q-factor, thermal conductivity and thermal time constant) are analysed as a function of gas pressure inside the housing. In Section 6 we apply our theory on a chosen BMD structure and give calculation results of detector parameters. 2. Bimaterial effect Bimaterial effect is based on the bending of a microcantilever with a ‘‘sandwich’’ structure due to temperature increase induced by absorption of infrared radiation. In order to achieve temperature response the bimaterial cantilever is composed of two materials with different thermal expansion coefficients a1 and a2. The structure of a typical BMD is shown in Fig. 1. It consists of a relatively large absorbing area whose position is controlled by two active bimaterial cantilevers. The whole structure is connected to the support via two thermally isolating beams. It is convenient to define the following parameters: ratio of bimaterial layers thicknesses, n = t1/t2, and ratio of their Young’s moduli, / = E1/E2. By applying the theory of static deformation it can be shown that the deflection of the free end of a microcantilever at a temperature T is given by [17] xðL; T Þ ¼

3L2 ða1  a2 Þ ð1 þ nÞðT  T 0 Þ : t2 4 þ 6n þ 4n2 þ n3 / þ 1=n/

ð1Þ

It can be seen that the deflection of the free end is proportional to the temperature difference, DT = T  T0, the square of the cantilever length, L, and the difference in thermal expansion coefficients of the materials used, a1  a 2.

Active area

High thermal resistance beam

Bi-material beam α2 α

1

T0 t2 t1

L

x

T0+ΔT; α1 > α2 Fig. 1. IR detector based on the bimaterial concept.

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3. Bimaterial microcantilever with external thermal drive Bimaterial microcantilevers belong to the class of movable micromechanical structures. In the first approximation such a structure can be modelled as a mechanical oscillator with a certain damping. If we analyse a bimaterial cantilever with a spring constant kBM, resonant frequency x0, effective mass characteristic for the first harmonic meff, the equation of vibrational motion can be written as 2

dx R dx k BM F þ þ x¼ ; dt2 meff dt meff meff

ð2Þ

where x is the cantilever deflection, R is the mechanical resistance (R(x) = (x0/Q)meff)) and Q is quality factor. F is thermal driving force proportional to the temperature difference, F = KDT. Taking into account that for all types of thermal detectors the increase of the temperature of the active area in the first approximation is given by DT ¼ Rth eP ð1 þ x2 s2th Þ1=2 ;

ð3Þ

(where P is the incident optical power, e is active area emissivity and sth = RthCth is thermal time constant equal to the product of thermal resistance and thermal capacitance), solution of Eq. (2) in frequency domain for cantilever displacement is obtained X ¼

K Rth eP qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ððx20  x2 Þ2 þ ðxx0 =QÞ2 Þ1=2 : meff 2 1 þ ðxsth Þ

ð4Þ

BMD sensitivity is defined as the variation of displacement due to the variation of the power of the incident radiation dX dP KeRth 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ððx20  x2 Þ2 þ ðxx0 =QÞ2 Þ1=2 : ð5Þ ¼ meff 2 1 þ ðxsth Þ

S¼

On the other hand, it is well known that the force acting on a mechanical oscillating system is proportional to the displacement, F = kBMx(L). Combining the relation (1) for the microcantilever displacement with the starting assumption, F = KDT, the proportionality constant is obtained K¼

3k BM L2 ða1  a2 Þ 1þn ; 1 t2 4 þ 6n þ 4n2 þ n3 / þ n/

ð6Þ

where kBM is the spring constant for a bimaterial cantilever of width wBM which was calculated using [18] k BM ¼

wBM ðE1 t31 þ E2 t32 Þ  L3

3wBM ðE1 t21  E2 t22 Þ2 4L3 E1 t1 þ E2 t2

:

ð7Þ

The relation for constant K can be further simplified by putting relation (7) in (6):

3 wBM t22 n/ð1 þ nÞ : K ¼ ða1  a2 ÞE2 4 1 þ n/ L

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ð8Þ

By substituting the expression for K in Eq. (5) and using the relation x20 ¼ k BM =meff the expression for sensitivity is easily obtained. 4. Noise in BMD The performance of BMDs is influenced by various noise generation mechanisms [19,20]. These mechanisms include noise in the readout circuits and the intrinsic cantilever noise. We will analyse intrinsic cantilever noise dominated by two basic independent noise generation mechanisms. The first mechanism is related with cantilever displacements due to spontaneous fluctuations of its temperature during the heat exchange with the ambient. Since this type of cantilever displacement is not a result of the temperature variation due to absorption of IR signal, it is a parasitic effect, thus representing noise. This is the socalled temperature fluctuations noise. The second noise source is the quantized and stochastic exchange of the energy within the cantilever with the energy in the surrounding environment. This mechanism leads to spontaneous mechanical movement (oscillations) of the cantilever, when it is in thermodynamic equilibrium with the ambient. This kind of parasitic cantilever displacement is the socalled thermomechanical noise, mechanical analogue of the Johnson noise. If the influence of these two noise sources is represented with forces K Æ DTN (models temperature fluctuations) and FBraun (models the cause of thermomechanical noise) then the equation of parasitic cantilever vibrations has the form d2 xN x0 dxN K  DT N F Braun þ x20 xN ¼ þ þ ; 2 dt Q dt meff meff

ð9Þ

where xN is parasitic cantilever deflection, i.e. the noise. Solving this equation in the frequency domain the expression for the spectral density of mean square value of cantilever displacement fluctuations is obtained ! 1 K 2 DT 2N ðxÞ F 2Braun 2 X N ðxÞ ¼ þ 2 :  2 m2eff meff 2 ðx20  x2 Þ þ xQ0 x ð10Þ Spectral density of mean square value of temperature fluctuations is given by [17] DT 2N ðxÞ ¼ 4k B T 2 Rth =ð1 þ x2 s2th Þ;

ð11Þ

where kB is Boltzman’s constant, and spectral density of the mean square value of thermomechanical force is given by F 2Braun ðxÞ ¼ 4k B TR; where R represents mechanical resistance.

ð12Þ

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Using these expressions and after including relation for mechanical resistance the spectral density of the mean square value of displacement is  2  1 K 4k B T 2 Rth 4k B T x0 2 X N ðxÞ ¼  þ  :  2 m2eff 1 þ x2 s2th m2eff Q ðx20  x2 Þ2 þ xQ0 x ð13Þ If we define noise equivalent power (NEP) as the incident power which results in cantilever displacement equal to the root mean square value of the spectral density of noise induced cantilever displacement fluctuations, then NEP [W/Hz1/2] is given by relation qffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2N ðxÞ : ð14Þ NEP ¼ S Substituting the above derived expressions (5) and (13) in (14) we finally obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4k B T 2d 4k B T d x0 meff NEP ¼ ð1 þ x2 s2th Þ: þ 2 2 2 ð15Þ e2 Rth e K Rth Q For a detector with an active area A the specific detectivity D* is given by pffiffiffi pffiffiffi A A  ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; D ¼ ð16Þ 2 NEP 4k B T d 4k B T d x0 meff 2 s2 Þ þ ð1 þ x th e2 Rth e2 K 2 R2 Q th

where Rth is the total thermal resistance of the detector comprising losses via conduction through supporting legs, Gleg, and surrounding gas Ggas and radiation losses Grad: 2

1=Rth ¼ Gleg þ Ggas þ 2eAeff rk B ðT 3d þ ðT 0 =T d Þ T 30 Þ:

ð17Þ

The above expressions take into account the emissivity of the upper surface of active area, e, and also the possibility that the detector temperature, Td, differs from the background temperature, T0. Aeff is the effective active area which takes into account the difference of emissivities of the front and back side. In our analysis of a BMD performance we assume e = 1, and Td  T0.

intrinsic—low pressure region, where intrinsic damping of the vibrating structure is much higher than air damping, (2) molecular—dominated by interaction between air molecules and vibrating structure, (3) viscous—around atmospheric pressure and higher, when air should be considered as a viscous fluid. A typical pressure dependence of Q-factor of a microcantilever is shown in Fig. 2; this curve is adapted from [21]. At low pressures (intrinsic region), below 1 Pa, Q-factor is constant and has a maximum value, Qint. This value is determined by various physical mechanisms which cause intrinsic damping. Among those are thermoelastic dissipation, phonon–phonon scattering, defect migration, losses in the junction microcantilever-substrate. The largest change in quality factor occurs in the molecular region (1 Pa 6 p 6 103 Pa). In this region Q-factor is inversely proportional with pressure  1=2 cmol k 2n  t 2 qs E ; cmol ¼ Qmol ¼ ; ð18Þ 12 p km l where t and l are thickness and length of cantilever, respectively, qs is density of the material used and E is its Young modulus. Factor km is given by relation rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 32 M mol km ¼ ; ð19Þ 9pRT where Mmol is the molar mass of gas molecules, R is gas constant and T is absolute temperature. For air molecules (Mmol = 28.964 g/mol) at room temperature km = 3.6 · 103 s/m. The factor kn is related with resonant frequency [19] and for the first resonant mode it is k1 = 1.875. For higher pressures, over 103 Pa, viscosity effects dominate (Qvisc). Q-factor is again constant around 104 Pa and after that drops with p1/2. Now, taking into account the above mentioned damping mechanisms in all three pressure regions and using the superposition rule, Q-factor can be written as 1 1 1 1 ¼ þ þ : Q Qint Qmol Qvisc

ð20Þ

5. Gas pressure influence on BMD performance

5000 2000

Q

As previously stated, a BMD can be regarded as a mechanical oscillator characterized by its resonant frequency x0 and the Q-factor [19,20]. On the other hand, its operation is based on temperature increase induced by heat absorption which depends on thermal conductance of the structure, Gth. The response time of a BMD depends both on the thermal conductance and the thermal capacitance (Cth) of the structure. All these basic parameters are dependent on the gas pressure inside the housing. Sensitivity and specific detectivity are functions of these parameters and are therefore also pressure-dependent. Depending on the dominant damping mechanisms, it is convenient to analyse three pressure regions [21–24]: (1)

1000 500 200 10

-1

10

0

10

1

10

2

10

3

10

4

10

5

p [Pa] Fig. 2. Typical pressure dependence of Q-factor for a cantilever.

Z. Djuric´ et al. / Infrared Physics & Technology 50 (2007) 51–57

Resonant frequency of structures with bimaterial cantilevers is pressure dependent ([6,23]), but this effect is of minor importance for our analysis and will be neglected. In a BMD structure the absorbed heat is transferred by conduction through the bimaterial and the isolating beams, by conduction and convection through surrounding gas and by radiation mainly from the absorbing area. Thermal conductivities related with these processes will be denoted by Gbeam, Ggas and Grad, respectively. In order to find thermal conductivity of the beams, both the bimaterial and the isolating parts should be considered. For the structure shown in Fig. 1 total thermal conductance of one beam is given by   lBM li Gbeam ¼ þ ; ð21Þ k1 wBM t1 þ k2 wBM t2 ki wi ti where lBM and wBM are the length and width of bimaterial beam, k1,2 and t1,2 are thermal conductance and thickness of materials composing the bimaterial beam, and the index i denotes the same parameters for the isolating part of the beam. Regarding heat loss through the surrounding gas, it can be shown that convection can be neglected for BMDs. Thermal conductance via conduction is given by A ; d

ð22Þ

where kgas is thermal conductivity of the gas inside the housing, A is the absorber area and d is the distance between the absorbing plate and the substrate. Thermal conductivity of the gas shows strong dependence on pressure. According to the model given by Eriksson et al. [24] kgas is given by relation 1 1 1 : ¼ þ kgas khp clp pd

6. Calculation results The theory of BMDs presented in this paper was applied for a structure designed and fabricated at the ORNL (Fig. 3) [6]. This detector has an active area of 66 lm · 41 lm, and is made of silicon nitride 550 nm thick, covered with aluminium film of 100 nm thickness. Bimaterial beams are made of the same Al/SiNx combination and have width of 3 lm and length 101 lm. The thermal isolation parts of the beams are made of SiNx 250 nm thick, 30 lm long and 1.5 lm wide. The values of other relevant parameters used in calculation were taken from [6]. We assumed that the equivalent spring constant of the two legs equals 2 · kBM. First, the analysis of thermal conductance was performed. The total thermal conductance of the two legs, including their bimaterial and isolation parts, is Gleg = 2 · Gbeam and equals 3.85 · 107 W/K, the radiative part is equal Grad = 1.01 · 108 W/K and air conductance Ggas is pressure dependent. Summing all of these three components we obtained the pressure dependence of thermal conductance of the BMD as shown in Fig. 4.

ð23Þ

Parameters khp and clp are characteristic for high and low pressure regions, respectively and are given by the following expressions: sffiffiffiffiffiffiffiffiffiffiffi 2 c pffiffiffiffiffiffiffiffiffiffiffiffi c 8M khp ¼ pffiffiffi k B TM ; clp ¼ ; ð24Þ 3 pk B T 3 p r0 where c is the specific heat, M is the molecular weight and r0 is the scattering cross-section of the molecules. It can be seen that thermal conductivity of the gas is described with two different expressions at low and high pressures. The same is valid for the gas conductance which can be written as 1 1 1 ¼ þ ; Ggas Ghp Glp

take into account pressure dependences of the relevant BMD parameters given above, we can conclude that since thermal resistance and Q-factor increase with a pressure decrease, detectivity can be improved by pressure decrease and by a proper choice of other bimaterial microcantilever parameters (increase of K). In this way, detectivity approaches the background limited value, D* = 1.81 · 1010 cm Hz1/2/W.

ð25Þ

where conductances at high and low pressures are Ghp = khpA/d and Glp = clppA. From the theoretical analysis presented in previous sections we have showed that specific detectivity depends both on thermal and mechanical properties of the detector. If we

ACTIVE AREA

BIMATERIAL PART ISOLATION PART

Fig. 3. Simplified schematic drawing of the analysed BMD structure fabricated at ORNL.

-5

5x10 Gtot[W/K]

Ggas ¼ kgas

55

10-5 -6

5x10

-6

1x10 -7 5x10 10-1

100

101

102

103

104

105

p [Pa] Fig. 4. Total thermal conductance of the BMD as a function of pressure.

Z. Djuric´ et al. / Infrared Physics & Technology 50 (2007) 51–57

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Next we analysed the influence of pressure on the detector thermal time constant. Taking into account the thermal capacitance of the active area (CA = 3.13 · 109 J/K) and the total thermal conductance of the BMD structure we obtained that the time constant decreases with pressure. In the pressure range of interest it drops from about 8 ms to nearly 0.1 ms (Fig. 5). Although a variation of resonant frequency with pressure was observed for the analysed detector [6], in our calculations we assumed a constant resonant frequency of 9.3 kHz. Using this assumption and the above obtained

10 5

τth [ms]

2 1 0.5 0.2 0.1

10-1

100

101

102

103

104

105

p [Pa] Fig. 5. Thermal time constant of the BMD as a function of pressure.

0

10

S [m/W]

30 Hz

10

-1

0.5 kHz 1 kHz -2

10

10

0

-1

10

10

2

1

10 p [Pa]

4

3

10

10

10

5

Fig. 6. Sensitivity versus pressure in the detector housing of the BMD for three oscillation frequencies (30 Hz, 0.5 kHz and 1 kHz) and T = 300 K.

1x1010

D* [cmHz1/2/W]

5x109

30 Hz 0.5 kHz

2x109

1 kHz

109 5x108

7. Conclusion This paper presents a novel theoretical approach which enables the calculation of basic parameters of an IR BMD. In order to perform analysis of cantilever displacement induced by temperature variations, we have introduced equivalent thermomechanical force. We assumed that this excitation force is proportional with temperature change (proportionality constant K) and causes mechanical oscillations of the bimaterial cantilever. This approach allowed us to reduce our BMD analysis to a case of a mechanical oscillator. In order to analyse the detectivity of the BMD we have first identified all the relevant noise mechanisms (temperature and thermomechanical) and then modelled each of these with an equivalent force. The resultant of these two forces induces parasitic cantilever oscillations. In this way we obtained expressions which relate basic BMD parameters with thermal, mechanical and geometrical properties of the constituent parts of the detector and the chosen materials. We have also taken into account the fact that the thermal and the mechanical properties of the detector are dependent on the gas pressure inside the housing. Therefore an analysis of pressure dependence of basic BMD parameters was performed which allowed us to include this influence when analysing sensitivity and specific detectivity of the structure. We concluded that since both thermal resistance and Qfactor increase with a pressure decrease, detectivity can be improved through a pressure decrease and a proper choice of other bimaterial microcantilever parameters (increase of K and thermal resistivity). In this way, detectivity is approaching the background limited value, D* = 1.81 · 1010 cm Hz1/2/W. Finally, we applied out theory to a BMD fabricated at the ORNL [6]. It was concluded that high vacuum and low conduction heat losses are necessary for the detectivity of real structures to attain values close to the ideal ones. Acknowledgements

2x108 10

results for other relevant parameters we calculated the sensitivity and the specific detectivity of a BMD for several oscillation frequencies. Pressure dependence of sensitivity is shown in Fig. 6. Fig. 7. shows the dependence of the specific detectivity of the BMD on the pressure in the detector housing. It can be concluded that high vacuum and low conduction heat losses (Gleg) are necessary to reach near-background limited detectivities (1.81 · 1010 cm Hz1/2/W).

-1

10

0

10

1

2

10 p [Pa]

10

3

10

4

10

5

Fig. 7. Specific detectivity versus pressure in the detector housing of the BMD for three oscillation frequencies (30 Hz, 0.5 kHz and 1 kHz) at T = 300 K.

Authors would like to thank Dr. Slobodan Rajic, Dr. Panos G. Datskos and Dr. Dragoslav Grbovic for useful discussions and the authorization for implementing our theory on a detector fabricated at Oak Ridge National Laboratory.

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