An optical transition-edge sensor with high energy resolution

In this study, we tested two TESs, one that achieved the high energy resolution (TES2 in Tab.1) and other that showed typical resolution (TES1 in Tab.1). They have Ti/Au thickness of 20/10 nm. Their size was 8 ${\rm\mu m}\times 8$ ${\rm\mu m}$. The critical temperature ($T_{\rm c}$) of TES2 was 115 mK. $T_{\rm c}$ of TES1 was also fairly low, 143 mK. The TESs were cooled down in a dilution refrigerator. The bath temperature was set to 7 mK. The TESs were embedded in optical cavities optimized to maximize the detection efficiency at wavelengths of interest (950 nm for TES1 and 1550 nm for TES2).

The energy resolution was measured using pulsed lasers. Figure 1 showed response of TES2 to a pulsed laser at 1550 nm (0.8 eV). The energy resolution was 67 meV. Since TES1 was designed to detect photons at around 950 nm, an 850 nm (1.46 eV) pulsed laser was used. The measured energy resolution was 156 meV.

We measured the complex impedance to extract parameters characterizing the TESs and to calculate the theoretical values for current noise and the energy resolution. The complex impedance is determined by a thermal model of a TES. The simplest model is a single-block model consisting of a TES and the thermal bath, as shown in Fig. 2(a). This model described behavior of optical TESs well ^18 19. The complex impedance of a single-block TES at an angular frequency $\omega$ can be written by ¹, ²⁰

where $\cal{L}$ is the constant-current loop gain given by ${\cal{L}}=RI^{2}\alpha_{I}/G_{\rm tes,b}T$, $G_{\rm tes,b}$ is the thermal conductance between the TES and the thermal bath, and $\tau_{I}$ is a time constant given by $\tau_{I{\rm,s}}=\tau_{0}/(1-{\cal{L}})$, where $\tau_{0}$ is an intrinsic time constant, $C/G_{\rm tes,b}$.

The complex impedance was measured by injecting small signals from a network analyzer into a voltage-biased TES on a cold stage at 7 mK. The current flowing through the TES was measured using a SQUID. As shown in Fig. 3(a), the complex impedance of TES1 can be well described by Eq. 2. Figure 4(a) shows the extracted $\cal{L}$ and $\beta$.

On the other hand, the responses of TES2 apparently deviated from that expected from the single-block model, as shown in Fig. 3(b)(solid lines). Instead, a two-block model (Fig. 2(b)) was adopted for TES2. The two-block model includes an additional thermal. When the thermal conductance between the additional body and the thermal bath is zero, the model represents a hanging model.This model describes a TES with an absorber ^15 21. When the thermal conductance between the TES and the thermal bath is zero, it represents an intermediate model ²². We will show that behavior of TES2 can be explained by the two-block model in Sec.2.3. The complex impedance given by the model is written as ²²

where $\tau_{1}$ is the time constant of the additional body, $\cal{L_{\rm eff}}$ is the effective loop gain and $g_{\rm two-body}$ is a term as a function of thermal conductance in the system. These parameters can be written as

where $C_{1}$ and $T_{1}$ are the heat capacity and the temperature of the additional body, respectively, $G_{\rm tes,1}$ is the thermal conductance between the TES and the additional body, and $G_{\rm 1,b}$ is the thermal conductance between the additional body and the thermal bath.

As shown in Fig. 3(b)(dashed lines), the measured complex impedance of TES2 was fitted well with Eq.3. Assuming that $g_{\rm two-body}$ and $\tau_{1}$ were independent of a bias point, they were 0.326 and 11.4 $\mu$s, respectively, as extracted from the measured complex impedance shown in Fig. 3(b). $\tau_{1}$ was similar to $\tau_{0}$ (13.9 $\mu$s at 0.5 $R_{\rm n}$).

In TES2, while most of important parameters can be extracted from the complex impedance, some parameters such as the ratio between $G_{\rm tes,b}$ and $G_{\rm tes,1}$ remain unknown. Therefore, $G_{\rm tes,b}$ is unknown and $\alpha_{I}$ cannot be derived. However, $C/\alpha_{I}$, which is an important parameter for the energy resolution, can be directly extracted from the complex impedance. In the two-block model, $C/\alpha_{I}$ can be calculated using Eqs. 4, 5, ${{\cal L}}_{\rm eff}$ and $\tau_{I}$. In the single-block model, from Eq. 1, the energy resolution is proportional to $\sqrt{C/\alpha_{I}}$, if assuming that the readout noise is zero. This is also the same applies to the two-block model, as shown in Appendix A. The values of $C/\alpha_{I}$ extracted from the complex impedance measurements are shown in Tab.1. TES2 showed smaller $C/\alpha_{I}$ than TES1. This is consistent with the fact that TES2 exhibited better energy resolution. Using Eq. 3, the current sensitivity and the effective loop gain of TES2 were extracted from the complex impedance, as shown in Fig. 4(b).

In this section, we will compare the measured noise with the theoretical noise level to evaluate the excess Johnson noise. The results of noise measurements for TES1 and TES2 are shown in Figs.5 and 6.

The theoretical noise requires the thermal conductance between a TES and the heat bath as an input parameter. The parameter is used for calculation of the thermal fluctuation noise (TFN) ¹. The thermal conductance is associated with Joule power dissipation of the a TES. The power dissipation is equal to the power flow to the heat bath and can be written as

where $n$ is the exponent determined by the nature of the thermal link to the heat bath. The parameter $K$ is associated with the thermal conductance by $G=nKT^{n-1}$. The cool-down mechanism of the TESs can be explored by measuring the Joule power dissipation as a function of the bath temperature. The measured $P_{J}$ for TES1 and TES2, when they were biased at 0.5 $R_{\rm n}$, were fitted with Eq.8 and the resultant exponents were $n=5.07$ and $4.95$, respectively. This implies that the TESs were cooled down by electron-phonon coupling.

The measured thermal conductance of TES1 ($G_{\rm tes,b}$) was $9.9\times 10^{-12}$ W/K. For TES2, which agreed well with the two-block model, the measured value of $6.3\times 10^{-12}$ W/K was a function of thermal conductance in the system ($G_{\rm tes,b}$, $G_{\rm 1,b}$ and $G_{\rm tes,1}$).

In the following, we will show that the current noise in TES2 was fitted well with the two-block model. In this model, we can define the effective thermal conductance $G_{\rm eff}$, which was equal to the thermal conductance extracted from Joule power dissipation. We will show that we can calculate the theoretical current noise without knowing $G_{\rm tes,b}$ or the exact form of $G_{\rm eff}$.

In order to calculate the theoretical current noise, we also need the responsivity of the TES to small signals. The responsivity $s_{I}(\omega)$ can be associated with the complex impedance $Z_{\rm TES}$ by ²²

where $R_{s}h$ is the shunt resistance in the bias circuit and 18 m$\rm\Omega$ in our setup. The Johnson noise is written as

The Johnson noise due to the shunt resistor in the bias circuit is $|I_{\omega}|^{2}_{\rm sh}=4kT_{\rm sh}R_{\rm sh}/|Z_{\rm tes}+R_{\rm L}+i\omega L|^{2}$, where $T_{\rm sh}$ is the temperature of the shunt resistor, $L$ is the SQUID input impedance and was 18 nH in our readout. The contribution of $|I_{\omega}|^{2}_{\rm sh}$ to the total current noise was negligible as shown in Figs. 5(a) and 6(a). It should be noted that the temperature at the shunt resistor are assumed to be same as the bath temperature.

Next, we consider the TFN current noise of the single-block model. It is written as

where $P_{\rm tes,b}$ is the power spectral density of the TFN. $F(T_{\rm tes},T_{\rm bath})$ depends on the nature of the thermal link. In the radiative limit, it becomes $F(T_{\rm tes},T_{\rm bath})=\left[(T_{\rm bath}/T_{\rm tes})^{n+1}+1\right]/2$ ²³.

The measured current noise in TES1 shown in Fig.5 agreed well with the sum of the theoretical noise and the readout noise. The theoretical noise was calculated using the measured thermal conductivity and parameters extracted from the complex impedance. The theoretical noise was sum of the TFN noise, the Johnson noise including the excess term $M$, and the shunt resistor Johnson noise. $M$ was calculated at each bias point as shown in Fig. 7.

The TFN of the two-block model can be written as ²²

where $P_{tes,1}$ is the TFN between the TES and the additional thermal block, and $P_{1,b}$ is that between the additional block and the the thermal bath.

As shown in Fig. 6(b), the measured noise was fitted well to Eq. 12. The parameters which can be extracted from the fit are $P_{1}\equiv P_{\rm tes,b}^{2}+P_{\rm tes,1}^{2}$ and $P_{2}\equiv P_{\rm tes,1}^{2}\left(\frac{\left(\frac{P_{\rm 1,b}G_{\rm tes,1}(T_{1})}{P_{\rm tes,1}}\right)^{2}+G_{\rm 1,b}^{2}}{(G_{\rm tes,1}(T_{1})+G_{1,b})^{2}}-1\right)$. We found $P_{1}/P_{\rm TFN,0}^{2}=4.12$ and $P_{2}/P_{\rm TFN,0}^{2}=-1.15$, where $P_{\rm TFN,0}=\sqrt{2kT^{2}G_{\rm eff}((T_{\rm bath}/T)^{n+1})+1)}$ is the TFN assuming that TES2 follows the single-block model and its thermal conductance to the heat bath is $G_{\rm eff}$. It should be noted that $\tau_{1}$ extracted from the complex impedance was used for the fits. To determine $P_{\rm tes,b}$, $P_{\rm tes,1}$, $G_{1,b}$, $T_{1}$ individually, additional experiments may be necessary. Without knowing these values, the measured noise was able to be fitted to Eq. 12, and the excess Johnson noise term was obtained as shown in Fig. 7.

In this section, we will discuss possible models that could explain the behavior of the TESs based on detector physics. In complex-impedance measurements, response of TES1 was effectively explained by the single-block model as shown in Fig. 2(a). The more detail model of TES1 is presented by a thin metal film directly deposited on an insulated substrate, as shown in Fig. 8(a) ²⁴. The relaxation time of phonons in the TES was thought to be much faster than probe signals for measuring the complex impedance. Therefore, the phonon system can be effectively treated as the heat bath, and the detector response was described well by the single-block model.

In case of TES2, its thermal model became more complicated. From the measured complex impedance and the current noise, thermodynamic model was determined to be the two-block model. However, the components of the additional block (electrons or photons, etc) remained unknown. Here, we present a possible model that could explain the behavior of TES2.

Considering that the time constant of the additional block ($\tau_{1}$) was similar to that of the TES ($\tau_{I}$) when the loop gain was zero, it is straightforward to assume that the block could be an electron system. When the system is cooled by electron-phonon weak coupling, the time constant is determined by the temperature. Therefore, the temperature must be similar to $T_{\rm TES}$.

To maintain the temperature above the bath temperature, the electron system must be heated. In our setup, the only mechanism that could heat the electron system was Joule dissipation. Thus, the additional block should have the resistance $R_{1}$. To maintain consistency with the two-block model, the current flowing through the block must be negligible and did not affect electro-thermal feedback (ETF) of the TES. This implies that $R_{1}$ must be much larger than $R_{\rm TES}$ and was connected in parallel with the TES. The model could have the form shown in Fig. 8(b). The large resistance could be created by a partial volume in the TES. The volume should be normal regardless of the voltage applying to the TES. The large resistance may imply that the cross section of the volume was small.

We will show that large resistance in parallel with a TES does not affect ETF and the set of equations are reduced to that derived by the two-block model. When a TES is in parallel with resistance $R_{1}$, the electrical differential equation is

where $I_{1}$ are current flowing through $R_{1}$ and $V$ is the voltage across the bias circuit. Here we assume that $R_{1}$ is constant.

The linearized differential equation is

when $R_{1}>>R$, in frequency domain, $\Delta I_{1,\omega}=\frac{Z_{\rm tes}(\omega)}{R_{1}}\Delta I_{\omega}$ and thus $\Delta I_{1}$is much smaller than $\Delta I$. When $|\Delta I_{1}|<<|\Delta I|$, Eq.14 can be reduced to the equation without the resistance $R_{1}$. Therefore, the contribution of $I_{1}$ to electro-thermal feedback is negligible.

Next, we will show that $\Delta I_{1}$ does not affect the thermal response of the additional body. The linearized equations regarding to the additional body is

where $V_{\rm tes}=RI=R_{1}I_{1}$ is the voltage across the TES. To show that $V_{\rm tes}\Delta I_{1}$ is negligible in Eq. 15, one needs to consider the linearized equation regarding to the time derivative of the change in temperature of the TES,

Comparing Eqs. 15 and 16, the terms $G_{\rm tes,1}(T_{\rm tes})\Delta T_{\rm tes}-G_{\rm tes,1}(T_{1})\Delta T_{1}$ should be the same order of $V_{\rm tes}(2+\beta)\Delta I$, which is much larger than $V\Delta I_{1}$. Thus, the term regarding to $\Delta I_{1}$ in Eq. 15 is negligible. Therefore, the behavior of a TES in parallel with large resistance can be described by the two-block model which assumes that an additional body is not heated by current flowing through it. The Johnson noise of $R_{1}$ is also negligible, which is obvious from Eq.10.

We examined the measured current noise of the TESs and found that it matched with the sum of the TFN and the Johnson noise with the excess noise term implemented. Now we can readily calculate the expected energy resolution based on the current noise and can compare it to the measured energy resolution. In the small-signal limit, the FWHM energy resolution of a TES is associated with the current noise as follows ²³,

Using Eq. 17 and the measured current noise, we calculated the expected energy resolution as shown in Tab. 2. The calculated energy resolution in the presence of the excess noise ($E_{\rm c}$) was slightly worse than the energy resolution in the absence of the excess noise. Although the excess noise contributed significantly to the current noise, it could be excluded as a major source that limits the energy resolution. Note that the calculated energy resolution took into account the readout noise.

To compare these values with the measured energy resolution, we need to consider the energy collection efficiency. It is defined as the ratio of the energy absorbed by a TES to the energy of a photon. The collection efficiency is less than 100 % due to phonon escape from the detector. In a linear detector, the energy deposited on a TES should be removed by electrothermal feedback and is equal to $E_{\rm ETF}=-\int_{0}^{\infty}V_{\rm tes}(t)\delta I(t)dt$. The energy collection efficiency becomes $E_{\rm ETF}/E_{\rm photon}$, where $E_{\rm photon}$ is the energy of a photon. As shown in Tab. 2, TES2 showed better efficiency than TES1 did.

If the current noise is only the source for fluctuations in the measured energy of a photon, the measured energy resolution $\Delta E_{\rm m}$ should agree with the calculated energy resolution obtained by $\Delta E_{\rm c}/\eta$. However, as in Tab. 2, $\Delta E_{\rm c}/\eta$ is significantly smaller than $\Delta E$. The difference can be written as $\sqrt{\Delta E_{\rm m}^{2}-\Delta E_{\rm c}^{2}/\eta^{2}}\equiv\Delta E_{\rm other}$. There must be some unexplained degradation in the energy resolution. In TES2 whose energy resolution is less than 100 meV, $\Delta E_{\rm other}$ had a significant impact on the energy resolution.. Therefore, it is necessary to prioritize the reduction of $\Delta E_{\rm other}$ to obtain higher energy resolution.. To do this, we need to understand nature of the unexplained sources.

From the noise measurements, we have found that the optical TESs have the excess Johnson noise. The effect of the excess noise on the energy resolution of TES2 was relatively small. It was strongly affected by fluctuations caused by unexplained sources.

It had been considered that the reduction of the current noise is crucial to improve the energy resolution of an optical TES. Therefore, a TES should have low $T_{\rm c}$ and $\beta$, a small heat capacity, high $\alpha_{I}$ and low excess noise. We have successfully fabricated such a TES.

The study of the excess Johnson noise may be important for improvement of the energy resolution, though its contribution was small as shown in Tab.2. For TESs designed to detect X-ray photons, the excess Johnson noise is known to be one of the limiting factors. Optical TESs have a different detector design than X-ray TES: They have no membrane or absorber and an order of magnitude smaller in size. Thus, the cause of the excess Johnson noise in optical TES could also be different. Many groups have studied the excess noise in X-ray TES, but few works have been done on the excess noise in optical TESs. This is an issue that needs to be addressed in the future.

To obtain higher energy resolution, the source of $\Delta E_{\rm other}$ should be investigated and be reduced. The factors of $\Delta E_{\rm other}$ could be (1) non-linear response, (2) error in the pulse height estimate of a signal by the optimal filtering method and (3) downconversion phonon noise ²⁵.

We discussed the energy resolution in the small-signal limit, assuming that the detector response is linear. If there is non-linearity in a TES, the energy resolution calculated based on the small-signal theory may deviate from the measured resolution.

In a linear detector, the pulse height should be proportional to the total energy of photons. When a linear TES is irradiated with a pulsed laser, the pulse height is proportional to the number of photons simultaneously absorbed there. In reality, as shown in Fig. 9, the measured pulse height was fitted to a quadratic function rather than a linear function. TES2 showed more obvious nonlinear responses than TES1. Note that the pulse height spectra including Fig. 1 were calibrated using a quadratic function, from which the energy resolution was obtained.

It is questionable whether it is optimal to fit with a quadratic function over the entire energy region. In Fig. 9, the region where TES2 responded linearly was unclear. There was a possibility that the TES responded linearly to small-energy deposition, such as a region from 0 eV to the single-photon peak. The single-photon peak in the pulse-height spectrum was fitted with Gaussian function and the resultant energy resolution was 62 meV. The change in the resolution was 10% from fit with the quadratic function, and implies uncertainty introduced by non-linearity. In the case of TES1, which exhibited better linearity, applying the same procedure to the measured pulse height spectrum resulted in the energy resolution of 150 meV. The change in the resolution was 4% and was smaller.

The non-linear responses also linked to optimal analysis of the pulse height. In the analysis, the static noise is assumed. However, a large change in the resistance (or a bias point) invalidate this assumption. This could lead to an error in the pulse height and deteriorate the energy resolution. For future work, to understand responses of a TES in more detail, we will study the nonlinear detector response between the small-signal limit and the large signal region where a TES is saturated.

To improve the energy resolution, lowering $T_{\rm c}$ furthermore could be still an option in fields where slow detector response is acceptable. In applications that require the high energy resolution, such as biological imaging and microscopic spectroscopy, the counting rate of incident photons are occasionally low, and slow detector response may be acceptable. For low-$T_{\rm c}$ TESs, with a given energy of a photon, a change in temperature and thus the resistance will be larger and their responses will be non-linear. Therefore, it is important to have a better understanding of the nonlinearity.

In this paper, we have focused on the current noise. Another important source of fluctuations in the measured energy is downconversion phonon noise. The noise is independent of the current noise and is originated from fluctuations in the number of phonons escaping into a substrate. The noise could be reduced by forming a TES on membrane to limit escape paths of phonons.

This membrane structure was implemented by other group ²⁶. Its purpose was to reduce energy loss and to enhance the energy collection efficiency $\eta$. We have achieved fairly high $\eta$ without membrane. However, non-uniformity in $\eta$ was quite high, i.e., typically $\eta$ was from 50 to 90 %. By adopting the membrane structure, we could improve $\eta$ and the non-uniformity.