Electro-optical imaging of electric fields in silicon sensors
EElectro-optical imaging of electric fields in silicon sensors
R. Klanner, A. Vauth ∗ Institute for Experimental Physics, University of Hamburg,Luruper Chaussee 147, D 22761, Hamburg, Germany.
Abstract
A conceptual set-up for measuring the electric field in silicon detectors by electro-optical imaging isproposed. It is based on the Franz-Keldysh effect which describes the electric field dependence of theabsorption of light with an energy close to the silicon band gap. Using published data, a measurementaccuracy of 1 to 4 kV/cm is estimated. The set-up is intended for determining the electric field in radiation-damaged silicon detectors as a function of irradiation fluence and particle type, temperature and bias voltage.The overall concept and the individual components of the set-up are presented.
1. Introduction
Segmented silicon detectors for precision tracking are the central detectors of most high-energy physicsexperiments. One of the major limitations of silicon sensors is radiation damage, in particular bulk radiationdamage from energetic particles. In order to understand radiation-damaged silicon sensors, simulate theirperformance and optimize their design, the knowledge of the electric field in the sensor is important.So far no satisfactory method for determining the electrical field in silicon sensors is available. Themethods Edge-TCT (Transient Current Technique) [1] and TCT-TPA (TCT-Two Photon Absorption) [2]provide precise information on the charge collection efficiency, however only an approximate estimate ofthe electric field [3]. One reason is that the information on the electric field is obtained from the initialsub-nanosecond rise of the pulse, which is strongly influenced by the time response of sensor and readout.In principle TCAD simulations can be used to estimate the electric field. However, the big number ofdifferent defects and defect clusters present a major problem, and so far no satisfactory description of allexperimental results on radiation-damaged silicon sensors has been achieved [4].The method proposed aims for a direct imaging of the electrical field. It is based on the theoreticallypredicted [5, 6] and experimentally observed [7] dependence of the absorption of photons with energiesclose to the silicon band gap (1.124 eV at 300 K) on the electric field. As a first step it is planned to applythe method to non-irradiated sensors, where the electric field is precisely known, and measure precisely theFranz-Keldysh effect. If successful, the method will then be used to study the electric field in irradiatedsensors as a function of bias voltage, temperature, irradiation fluence and particle type. In [8] it has beenobserved that the states in the band gap produced by radiation damage change the light absorption, whichcould be described as a reduction of the silicon band gap. With the proposed set-up, the temperature andradiation-damage dependence of this effect can be determined using pieces of silicon irradiated by differentparticles to different fluences.The idea of electro-optical imaging of solid-state detectors is not new. In [9] an extensive presentationof electro-optical imaging for CdS is given; however, nothing is said about silicon.It should be noted that silicon becomes birefringent at high electric fields [10], which may be used todetermine, in addition to the absolute value of the electric field, its direction by adding polarisation filtersto the proposed set-up. ∗ Corresponding author. Email address: [email protected], Tel.: +49 40 8998 4728
Preprint submitted to Elsevier July 31, 2020 a r X i v : . [ phy s i c s . i n s - d e t ] J u l . Measurement principle and set-up The position dependence of the intensity of light with photon energy E γ in a homogeneous medium can bedescribed by I ( x , E γ ) = I · e − α ( E γ )· x , with the initial intensity I and the absorption coefficient α . In Ref. [7]measurements of ∆ α ( E γ , E ) , the change of α ( E γ ) with electric field, E , for 1 .
020 eV ≤ E γ ≤ .
220 eV for40 kV / cm ≤ E ≤
100 kV / cm, temperatures between 72 K and 300 K for high-ohmic n - and p -type siliconare presented. Selected results at 23 ◦ C are shown on the left side side of Fig. 1. It is found that ∆ α increaseswith E with peaks at E γ = .
059 eV and 1.175 eV. The explanation of the increase in absorption, which waspredicted in Refs. [5, 6], is given on the right top side of the figure: The electric field causes tunneling statesclose to the band edges which result in a decrease of the effective band gap and therefore in an increase inthe absorption of light with energy close to the band gap (1.124 eV at 300 K). As silicon has an indirect bandgap, phonons are required to satisfy energy- and momentum conservation for the generation of electron-holepairs by photons with energies close to the band gap. The peak at E γ = .
059 eV is ascribed to phononabsorption and the one at 1 .
175 eV to phonon emission. At the temperatures of interest for the proposedmeasurements the latter dominates.From now on the wavelength λ [ µ m] ≈ . / E γ [eV] is used instead of E γ , which is more commonfor optical measurements. Fig. 1 right bottom shows the total absorption coefficient α tot ( λ, E ) = α ( λ ) + ∆ α ( λ, E ) at 23 ◦ C. The values of α are taken from Ref. [11] and the ones for ∆ α are obtained by digitizingthe plot of the left side of the figure and a cubic spline interpolation. Figure 1: Electric field dependent light absorption in silicon. Left: Field induced change of the absorption coefficient ∆ α ( λ, E ) (Fig. 3of [7]). Right top: Physics explanation of ∆ α , (Fig. 1 of [7]). Right bottom: Total absorption coefficient α ( λ, E ) = α ( λ ) + ∆ α ( λ, E ) ,with α from Ref. [11] and ∆ α from the plot at the left. Fig. 2 shows a schematic drawing of the proposed set-up.As light source a tungsten-halogen lamp with a computer-controlled monochromator generates lightwith a bandwidth ∆ λ ≈ . µ m < λ < . µ m. An optical system produces aparallel beam with an angular divergence in the y direction of (cid:46) (cid:46) . is required for measuringthe transmission of the DUT with a 1 % accuracy in 1 second. Such light sources are available from severalvendors. 2 igure 2: Schematic layout of the proposed set-up for the electro-optical measurement of the electric field in silicon sensors. The photon beam is directed onto an assembly of the DUT (
Device under Test ) and the
Reference .Initially, for running-in the apparatus, a non-irradiated pad sensor of dimension 5 mm × × µ mpolished on the entrance and exit side will be used as DUT. The assembly can be moved vertically underremote control with sub-micron accuracy. For the precise alignment of the DUT onto the beam, it can alsobe rotated around the x - and z -axes. The intensity distribution for light with λ (cid:38) . µ m, where siliconis practically transparent, can be used to check the alignment, the angular dispersion of the light beam andthe quality of the DUT polishing.The Reference serves as a monitor for the measurements: With a polished SiO slab the light flux andthe alignment can be monitored, and a non-irradiated or an irradiated pad sensor at zero voltage serves asnormalisation for the voltage dependence of the absorption in the DUT.For the photo-detection a high speed NIR (near infrared) camera with an InGaAs CCD will be used.Cameras with 320 ×
256 pixels with 20 µ m pitch, a quantum efficiency exceeding 50 % for wavelengthsbetween 0.95 and 1.7 µ m and 14 bit readout resolution are commercially available. If an imaging resolutionbeyond the pixel pitch is required, an optical system can be installed in front of the camera.
3. Simulations
In order to investigate the feasibility of the proposed method, the following calculation has beenperformed. The ∆ α plot of Fig. 1 has been digitized for the four E -values shown. To obtain ∆ α ( λ, E ) , splineinterpolations in λ of the digitised ∆ α -values for E =
40 kV/cm and E =
100 kV/cm and interpolations in E using a ( λ ) · E b ( λ ) have been used. Values for b between 2 to 3 are found, which approximately agrees withthe expectation of b ≈
2. Using the ∆ α -values at 75 kV/cm and 100 kV/cm gives very different values for a ( λ ) and b ( λ ) , however the results for the E dependence of the transmission are not too different. Precisedata for ∆ α ( λ, E ) , which are required for the analysis, will be obtained with the proposed set-up usingnon-irradiated pad sensors. It is noted that the parametrisation a ( λ ) · E b ( λ ) will have to be changed, as thepeak of ∆ α shifts towards lower photon energies with increasing electric field. For the total absorption, α tot ( λ, E ) = α ( λ ) + ∆ α ( λ, E ) , the parametrisation of the zero-field absorption coefficient, α ( λ ) fromRef. [11] is used.For the calculation of the transmission, the classical formulae have been used. The Fresnel formulae atnormal incidence for the transmission Tra and the reflection
Ref at the interface between media of refractiveindex 1 and n ( λ ) Tra ( λ ) = · n ( λ )/( n ( λ ) + ) and Ref ( λ ) = − Tra ( λ ) , (1)and for the transmission of a slab of thickness d and absorption coefficient α Tr ( λ, α, d ) = Tra ( λ ) · e − α · d − ( Ref ( λ ) · e − α ( λ )· d ) . (2)For n ( λ ) the data from Ref. [11] are used.Fig. 3a shows the absolute transmission Tr ( λ, E , d ) calculated for d = . µ m and 1 . µ m, and electric fields between 0 and 100 kV/cm, and Fig. 3b the relative3 a) (b) Figure 3: Simulated transmission, Tr ( λ, E ) , for 5 mm silicon as a function of wavelength, λ , for selected values of the electric field, E . (a) Absolute transmission, and (b) transmission normalised to the value for E = transmission Tr rel ( λ, E ) = Tr ( λ, E )/ Tr ( λ, E = ) ). To estimate δ E , the accuracy of the E determination, E is reconstructed from Tr rel ( λ, E ) assuming an uncorrelated uncertainty of 1 % for the two Tr -values. Theresults for δ E ( E ) for three λ -values are shown in Fig. 4. For E >
20 kV/cm δ E -values below 4 kV/cmare found. The high accuracy of the determination of E from the relative transmission measurement mayappear surprising. It can be understood from Eq. 2: The term ( Ref · e − α · d ) in the denominator, whichdescribes the contribution from the light reflected inside of the DUT, can be ignored relative to 1. As aresult, Tr rel ( λ, E ) = e − ∆ α ( λ, E )· d , which shows a strong E dependence for sufficiently large d -values, does notdepend on the absolute Tr -values. In addition, many systematic uncertainties cancel when E is determinedfrom Tr rel .From the simulations the required photon flux, Fl γ ( λ ) , and the current density, J γ ( z ) , from the photonsconverting in the detector are estimated. For a statistical uncertainty δ Tr , a measurement time t meas , aphoto-detector area A = ∆ x × ∆ y for the measurement of the position dependence of the electric field, and4 igure 4: Uncertainty, δ E , of the determination of E for selected values of λ obtained from the ratio Tr ( λ, E )/ Tr ( λ, E = ) assuminguncorrelated uncertainties δ Tr ( λ, E ) = δ Tr ( λ, E = ) = a photon-detection efficiency, pde ( λ ) , the required photon flux is: Fl γ ( λ ) = (cid:0) δ Tr · pde ( λ ) · A · Tr ( λ, E max , d ) · t meas (cid:1) − , (3)with E max =
100 kV/cm, the maximum field for the planned measurements. The corresponding powerdensity for the selected ∆ λ interval of the measurement is P γ ( λ ) = Fl γ ( λ ) · E γ ( λ ) . Fig. 5a shows P γ ( λ ) for δ Tr = pde ( λ ) =
50 %, ∆ x × ∆ y = × µ m, and t meas = ∆ y is thetypical pixel size of a NIR CCD. The choice of the value of ∆ x is adequate for determining the electricfield in a pad detector, which does not depend on x , but has to be reduced for strip- or pixel detectors. Thestrong dependence of P γ on λ necessitates the use of filters, as shown in Fig. 2, and/or different choices of t meas to adapt to the dynamic range of the photo-detector. (a) (b) Figure 5: (a) Required light power density, P γ ( λ ) , and (b) resulting position-dependent photo-current density, J γ ( z , λ ) , for thetransmission measurement with an accuracy of 1 % in 1 second for a 5 mm light path. For estimating the photo-current density, J γ ( z ) , Fl γ ( λ ) has to be multiplied with Tra ( λ ) (Eq. 1) to obtainthe photon flux entering the silicon at x =
0. The z -dependent current density is approximately J γ ( z , λ ) = q · Fl γ ( λ ) · Tra ( λ ) · ∫ d (cid:16) α ( λ, E ( y )) · e − α ( λ, E ( y ))· z (cid:17) d y , (4)with q the elementary charge. Fig. 5b shows J γ ( z , λ ) for several values of λ for the photon flux requiredto achieve a 1 % measurement accuracy for 1 second measurements. Even the maximum of J γ at z = ∆ α ( λ, E ) the results can be considered onlyan estimate. However, they indicate that a precise determination of the electric field in silicon sensors withthe proposed method appears feasible. 5 . Summary and Conclusions A method for measuring the electric field in silicon sensors by electro-optical imaging is proposed.It makes use of the Franz-Keldysh effect, the increase with electric field of the absorption coefficient ofphotons with energies close to the silicon band gap. A simulation using published data shows that for anedge-on illumination of silicon sensors an accurate field measurement appears to be possible: The estimateduncertainty is in the range of 1 to 4 kV/cm for fields exceeding 20 kV/cm and a 5 mm light path.A schematic set-up is presented and possible choices for its components are discussed. After running-in the set-up with non-irradiated pad sensors and a precise measurement of the Franz-Keldysh effect, itis intended to determine the electric fields in different radiation-damaged pad and segmented sensors.Although the knowledge of the electric field in radiation-damaged silicon sensors is important for theirunderstanding and optimisation in the harsh environment of collider experiments, no satisfactory methodfor its determination exists so far; the proposed experiment may be such a method. The experiment willalso provide valuable data on the change of the light absorption as a function of irradiation. As an extensionof the method, the experimentally observed birefringence of silicon at high electric fields (Ref. [10]) couldpossibly be used for determining the direction of the electric field in addition to its absolute value.First steps towards the realisation of a set-up for the electro-optical imaging of electric fields in silicondetectors are underway at the Detector Lab of Hamburg University. In addition, the work on more detailedsimulations and analysis methods, which include the angular spread of the photon beam and the effects ofdiffraction, has started.
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