2.6 TES Bolometers
2.6.2 Bolometer Theory
values forβ, which describes the strength of the power law of the thermal conductance, are 2.0-2.5. The derivatives of Eqn. 2.4 are quantities that are can be measured in the SPIDER system:
dPlegs dTT ES
=G
TT ES T0
β
. (2.5)
When the TES is on transition, by definition the temperature of the TES is the same as its transition temperature, TT ES = Tc. If we choose T0 = Tc, then the above equation simplifies to
Gc= dPlegs
dTT ES. (2.6)
The quantity dPlegs/dTT ES is can be measured in the SPIDER system by varying the temperature of the focal plane and using Eqn. 2.2 to calculate the corresponding power through the legs. This measurement then lets us determine the thermal conductance of the island legs. Note that the transition temperature of the TESs is a property of the thin films used to make the detectors, not of the thermal conductance of the legs. Because of this, we instead generally choose T0 = 450mK, which allows us to quote a value of the island thermal conductance that is independent of the TES.
2.6.2.2 Electrical Model
None of the discussion in§2.6.2.1 is specific to the TES detector. A transition edge sensor is a type of detector that exploits the strongly temperature-dependent resistance of supercon- ducting materials. The TES detector operates in the very narrow transition region where the metal goes from its normal resistance to zero, which is an extremely steep function of the temperature. The sharp phase transition makes the TES an extremely sensitive thermometer.
Although the TES detector was invented in the 1940s [11], it did not become widely used until techniques to achieve stable operation in the transition region were developed in the 1990s. (An additional problem was matching the noise of the TES to the noise of readout amplifiers. This was largely eliminated through the use of SQUIDs, which are covered in the next section.) The SPIDER TESs are maintained at their transition temperature via voltage biasing, which takes advantage of negative electrothermal feedback. This technique was proposed by Kent Irwin [50] and the following discussion follows the seminal Irwin and Hilton review paper [51].
Figure 2.16: (a) A schematic of a simplified TES circuit. The shunt resistor RSH voltage biases the TES resistor RT ES, which is in series with the inductor L, which includes the SQUID and any stray inductance, and a parasitic resistanceRP AR. In the SPIDERsystem, the shunt resistor is located on a separate Nyquist chip, along with an additional inductor.
(b) The Thevenin-equivalent circuit model used for the equations in this section. A bias voltageV is applied to the load resistorRL=RSH+RP AR, the inductance L, and the TES resistance R. Figure from [51].
The response of the TES is governed by two coupled differential equations that describe the electrical and thermal response of the circuit. The electrical differential equation that characterizes the TES circuit is
LdI
dt =V −IRL−IR(T, I). (2.7) Here,Lis the electrical inductance,I is the current,V represents the bias voltageIbiasRSH, RLis the resistance of the inductor, andR(T, I) is the resistance of the TES, which generally depends on both the temperature and current. A schematic of the TES circuit is shown in Fig. 2.16.
The corresponding thermal differential equation is CdT
dt =−Pbath+PJ oule+Q, (2.8)
where C is the heat capacity, Pbath is the power lost to the bath, PJ oule is the joule power dissipated in the TES, and Qis the optical power incident on the detector.
The coupling between these two equations comes from the term for joule power:
PJ oule =I2R(T, I). (2.9)
For small signals, the resistance of the TES can be expanded to first order as R(T, I)≈R0+ ∂R
∂T I0
δT + ∂R
∂I T0
δI, (2.10)
whereδI =I−I0.
Using the expression for the unitless logarithmic temperature sensitivity, αI= ∂logR
∂logT I0
= T0 R0
∂R
∂T I0
(2.11) and current sensitivity
βI = ∂logR
∂logI T0
= I0
R0
∂R
∂I T0
, (2.12)
we can rewrite Eqn. 2.10 as
R(T, I)≈R0+αIR0
T0δT +βIR0
I0δI. (2.13)
This equation shows the dependence of the TES resistance on both the temperature and the electrical current. A change in temperature on the TES island is transformed into an electrical current via the change in the resistance of the TES. The electric current in the TES is transformed into a temperature signal via Joule power dissipation in the TES. This process is known as electrothermal feedback (ETF) and it arises from the cross-terms in Eqn. 2.7 and Eqn. 2.8.
Electrothermal feedback (ETF) can be either positive or negative. When the circuit is voltage biased (R0 >> RL), the joule power PJ oule = V2/R decreases with increasing resistance and so the feedback is negative. There are several advantages to being in this regime. Negative ETF linearizes the detector response and increases the dynamic range.
The TES is stable against thermal runaway even when the loop gain is high. The TES self-biases in temperature within its transition (over a certain range of signal powers and biases). Negative ETF also makes the bolometers much faster than the natural thermal time constant, which allows the detectors to recover quickly from cosmic ray hits and electronic
glitches.
The natural thermal time constant is given by τ = C
G, (2.14)
while the electrical time constant (in the limit of low loop gain) is given by τel= L
RL+Rdyn, (2.15)
whereRdyn is the constant-temperature dynamic resistance of the TES:
Rdyn≡ ∂V
∂I T0
=R0(1 +βI). (2.16)
Approximate values for the thermal, electrical, and ETF time constants are τ ∼50 ms, τel∼0.5 ms, and τET F ∼3 ms.