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In a previous issue, we saw how photoconductors have a cutoff wavelength. This represents a serious limitation of using radiometers, that is, radiation detectors based on photoconductors. The cutoff wavelength usually occurs in the far infrared spectrum. For longer wavelengths (infrared, microwaves) thermal detectors (devices measuring temperature variations) are used. As one can imagine, semiconductors are also used here due to their sensitive temperature dependence. Among these are, in fact, the bolometers.
Figure 1 shows the thermal diagram of a semiconductor cryogenic bolometer. Thanks to a bias circuit, a semiconductor is traversed by current i. R is the resistance of the semiconductor body (not to be confused with the dynamic or differential resistance), for which it dissipates a power W due to the Joule effect. The environment in which the device is located generates a radiative background of power Wγ. The semiconductor is in thermal contact with a cryostat (thermostat at T0 close to absolute zero) through a metal wire of thermal conductivity G. As we will see, very low temperatures (reached through cryogenic techniques) are essential to reduce the thermal noise. Furthermore, a stable thermal reference T0 is necessary to measure temperature fluctuations, unlike what would happen if we used the semiconductor since the latter has a low thermal capacity due to its small size.
The corresponding electric bias circuit is shown in Figure 2. A low noise battery supplies a voltage V0 and the corresponding current is as follows:
where RL is the load resistance. The output voltage is given by:
The operating principle of the bolometer is very simple. Since in a semiconductor R = R (T ), i.e., the resistance depends on the operating temperature, and a variation δWγ of the incident radiative power (possibly due to a thermal source) determines a variation δR of the resistance following the consequent variation in temperature δT. By differentiating equation (2) with respect to R and after some steps we obtain:
where:
Equation (3) expresses the link between the output voltage fluctuation (signal to be measured) and the temperature fluctuations. To achieve a good value for δVout (which in any case must be amplified), we must maximize the dimensionless parameter F (i.e., F ≃ 1), and this occurs for values of the load resistance RL ≫ R.
Rewriting the equation (3) in a more compact form, we have:
Here we see that, barring an unessential scale factor K, measuring δVout is equivalent to measuring the temperature fluctuation δT due to the variation δWγ of the radiative power incident on the bolometer.
We used a cube-shaped germanium sample with 200 µm, equipped with sapphire as a radiation absorber. The cryostat (helium 3) was at T0 = 0.3 K.
Recall that at absolute zero any semiconductor is an insulator. In correspondence with a temperature variation δT, hopping conduction1 is activated, while the other conduction modes are “frozen”. However, in the range [0, δT], the conductivity is still low and the resistance R of the semiconductor body is relatively high (in our case, R = 1 MΩ). As the temperature increases, the conductivity increases, and therefore R decreases. In other words, in the low-temperature range, dR/dt < 0 and α < 0. This is expressed by saying that the semiconductor cryogenic bolometer is negative. In the case under consideration: α = −10 K−1.
From what has been said, we have used a high load resistance (R), thus obtaining F = 1, with a bias potential such as to produce i = 1 nA.
The operationally useful quantity for calibrating the bubble meter is the responsivity:
which can be determined by measuring the δVout produced by a δWγ of a known source, such as a black body. Since it is difficult to make such a source at infrared wavelengths, determining the electrical responsivity is preferred. Basically, we consider a virtual source with an emission spectrum such as to produce a (δWγ) (t) decomposable into sinusoidal elementary oscillations, i.e., developable in Fourier integral:
where the following:
is the Fourier transform of (δWγ) (t).
Let us now resume the thermal diagram of Figure 1. The following stationary equation describes the background+system+semiconductor+cryostat at equilibrium:
where T > T0 is the equilibrium temperature, and Gs is the static thermal conductivity. A fluctuation δWγ determines a temperature fluctuation δT, accompanied by a further fluctuation δW of the power dissipated by the semiconductor due to the Joule effect. These fluctuations are obviously functions of time, so we will use a pedantic but unambiguous notation below:
From equation (9) we can write:
where δWb is an effect due to the thermal capacity of the semiconductor (small, but not zero):
After which:
By substituting into equation (11) after having obtained the fluctuation δW due to the Joule effect, the equation of the bolometer is obtained after the calculations have been performed:
In the intermediate steps, we first introduced dynamic conductivity:
and therefore, the effective conductivity:
since the bolometer is negative. This result has the following physical interpretation: α < 0 improves the thermal contact between the semiconductor and cryostat, as −αBW > 0 and as such increases the dynamic conductivity. We call αBW the electrothermal interaction term. We note that since α < 0, for δW > 0, the temperature increases, and therefore R and the dissipated power W decrease.
The equation (14) is a first-order linear ordinary differential equation. Considering equation (7) and developing the generic solution in Fourier integral, we obtain:
where we have considered a peaked spectral density around a dominant frequency ω0. The individual Fourier components are out of phase by an angle χ = arctan ωτe with respect to the corresponding Fourier component of the radiative source. It follows that the quantity τe = C/Ge defines the response time of the bolometer. This can be interpreted as a time constant because τe defines a transient term destined to fade away. This behavior is a characteristic of dynamical systems represented by a linear and inhomogeneous ODE2. In the case of the bolometer, under the theorem proved in the cited article, we have:
since the transient term is exp (−t/τe), it fades exponentially. Put simply, for t ≫ τe the bolometer “instantaneously tracks” the fluctuations of the radiative power. That’s the reason why we need to minimize τe; otherwise, the device would be “slow”. From equation (18) we see that to have a measurable δT we have to minimize Ge. Since this is equivalent to increasing the response time τe, we have to look for a compromise situation. In our experiment, we have:
Remark: In this framework, the bolometer belongs to the class of dynamic systems described here2.
The results we have obtained allow us to define the spectral responsivity:
where:
From the above equation, we see that we must use a low-conductivity thermal conductor to have appreciable responsivity. However, this increases the response time of the bolometer. This corroborates the previous arguments on the need to seek a compromise situation. Other parameters affecting the responsiveness are the bias current, α, and the resistance R. Increasing the current and R too much is not advisable as this promotes noise.
Operationally, the procedure opposite to the previous issue is carried out, where we determined δT for a given δWγ (in any case left unexpressed via Fourier integral). Now, from a measurement of δVout = KδT, we have to “reconstruct” the radiative signal δWγ. For this purpose, we have equipped the bolometer with an electric motor to rotate the device around a vertical axis (zenith). The signal δVout coming from the bias circuit, after being amplified (by a low noise amplifier) was acquired in a .dat file via software (Octave) and then reprocessed in the Mathematica calculation environment. Precisely, through the Interpolation instruction we obtained a function δT (ρ, θ, t) where (ρ, θ) are the polar coordinates in the plane tangent to the earth’s surface at the point where the device is located. Specifically, the range:
where ρmax is the maximum distance, which depends on the humidity of the air since water vapor molecules strongly absorb infrared. For example, on a rainy day, we have that ρmax = 0. Some of the results are shown in Figure 3.
