# The Tobin Coefficient: A Relevant Photodetector Performance Metric for IR Imaging

Olivier Gravrand, Alexandre Kerlain, Diane Sam-Giao, Maxence Soria, Johan Rothman

► To cite this version:

Olivier Gravrand, Alexandre Kerlain, Diane Sam-Giao, Maxence Soria, Johan Rothman. The Tobin Coefficient: A Relevant Photodetector Performance Metric for IR Imaging. Journal of Electronic Materials, 2024, 53 (10), pp.5874 - 5884. 10.1007/s11664-024-11302-7. cea-04765686

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Submitted on 4 Nov 2024

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# The Tobin Coefficient: A Relevant Photodetector Performance Metric for IR Imaging

Olivier Gravrand<fnref n="1" /> Ⓒ · Alexandre Kerlain<fnref n="2" /> · Diane Sam-Giao<fnref n="2" /> · Maxence Soria<fnref n="1" /> · Johan Rothman<fnref n="1" />

Received: 9 February 2024 / Accepted: 27 June 2024 / Published online: 26 July 2024
© The Author(s) 2024

## Abstract

The well-known Rule07 is a simple thus efficient way to compare available technologies for IR imaging detectors in terms of dark current. The noise is then often estimated using a shot noise approximation on the dark current. Both II–VI and III–V communities use this rule of thumb as a reference for well-performing IR photodiodes. For HOT applications, a dark current close to this rule07 is considered a necessary condition but not a sufficient one to obtain a high-performance IR imager. Indeed, when limited by shot noise, rule07 describes well the noise behavior of the considered device. However, when considering low-frequency noise, it fails to describe the expected performances. In this paper, we focus on another figure-of-merit, dedicated to detector low-frequency noise rather than dark current. Systemic 1/f noise investigation in an IR detector was first reported by Tobin et al. in 1980. There is today a relative consensus on the fact that measured 1/f noise is proportional to the dark current. The ratio between the amplitude of the 1/f noise and the dark current of the same devices may therefore be used as a figure-of-merit for a given technology. This ratio (called the Tobin factor $α_T$) therefore appears adequate to compare different technologies as a figure-of-merit qualifying 1/f noise properties. This dimensionless ratio can also be very useful for optimizing a particular technology or process. However, in order to be relevant, this figure-of-merit must be estimated carefully as it appears, for instance, pixel pitch-dependent. Different examples of Tobin coefficient extraction are presented in this paper. We show that, depending on the technologies, the values of the Tobin coefficient can spread over several orders of magnitude. However, only low values result in high-quality IR imagers. Today, the best results we obtained show that $α_T = 10^{-5}$ is a state-of-art value to be compared with.

**Keywords** HgCdTe · 1/f noise · infrared · diodes · focal-plane arrays

## Introduction

IR detector performance is usually qualified in terms of quantum efficiency (QE) and dark current. Detectivity or noise equivalent temperature difference (NETD) is then often calculated from these two parameters, assuming shot noise behavior on the photo and the dark current. In this classical view, a convenient reference for the dark current is the use of the rule07.<fnref n="1" /> Indeed, this very easy to calculate rule of thumb gives an easily accessible idea of the performance

of a state-of-the-art IR detector, given the wavelength cut-off and operating temperature considered. Detector manufacturers often use this rule as a reference to qualify the quality of their device and to compare themselves with the competition when the cut-off and temperature vary. Rule07 is also very useful for assessing the operating temperature at which an MCT high-performance detector will be limited by the dark current as the temperature rises in a given illumination scenario.

Such a shot noise assumption is usually relevant at sufficiently low temperatures, but it is often incorrect at high operating temperatures (HOT). In particular, for focal planes with a large number of pixels, this assumption no longer holds for all pixels. Some pixels may not be limited by shot noise, but tend to exhibit additional noise, usually low-frequency (LF) colored.<fnref n="2" />,³

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

<footnote>Olivier Gravrand
olivier.gravrand@cea.fr</footnote>

<footnote n="1">¹ UGA – CEA/Leti, 17 av des Martyrs, 38054 Grenoble, France</footnote>

<footnote n="2">² LYNRED, BP21, 38113 Veurey-Voroize, France</footnote>
These excess noise pixels then degrade the image quality of the resulting focal plane array (FPA). Furthermore, increasing the operating temperature of the FPA tends to increase the number of these excess noise pixels, further degrading its image quality.⁴ Ultimately, this LF noise becomes the main limiting factor as the temperature increases, sometimes well before it is limited by the increase in dark current.⁵ Therefore, there seems to be a strong need for a figure-of-merit (FOM) that qualifies this LF noise feature for any given IR detector structure in order to give a full picture of the detector performance.

A straightforward way of qualification would of course be a full characterization of a final FPA device. However, this is costly and time consuming. Therefore, in order to be efficient, we need a further FOM that quantifies the LF noise characteristics of a given detection device at an earlier stage of its development.

## The Tobin Factor as a Low Frequency Noise FOM

As mentioned previously, low operating temperature photodetectors (meaning $kT$ lower than the gap energy of the absorbing material) are usually limited by shot noise. This holds of course for high-quality photodetectors. When degraded (i.e., when a large density of electrically active defects is located in the narrow gap layer) this shot noise limit might not be respected anymore: some of those defects are trapping-detrapping carriers thus inducing LF fluctuations in the carrier flow, i.e., LF noise in the pixel output. Unfortunately, the electrical activity of those defects is generally thermally activated⁶⁻⁸ and the situation gets even worse at higher temperatures.⁴,⁹ Moreover, a larger variety of defects are able to express themselves, further degrading the LF noise. In other words, at higher temperatures, photodiodes tend to exhibit additional LF noise.⁵,¹⁰,¹¹ This starts with the appearance of noise distribution tails: some pixels making up this distribution tail are degraded by this LF noise whereas the others remain limited by shot noise. Further increasing the temperature, this distribution tail increases both in number of pixels as well as in amplitude of the additional noise (see Fig. 1 for an example of such noise distributions). For midwave IR (MW) and longer wavelengths, increasing the temperature up to room temperature usually ensures the obtaining of all the pixels limited by LF noise. In other words, the system then switches to a regime called systemic LF noise. Moreover, at high temperatures, this LF noise is usually more specifically 1/f noise.²,³,⁵,¹¹

The ability to obtain high-performance detection is therefore not only defined by QE or dark current, but also by this LF noise which degrades the imaging quality of the FPA. Still, the resulting LF noise may vary from one technology to

<figure><img src="image_4.png" /><figcaption>Fig. 1 Example evolution of noise distribution increasing the temperature of a n/p MW MCT array from LETI, VHg-doped (15-µm pitch, TV format). The noise $I_n[A/Hz]$ is normalized to the expected shot noise $\sqrt{2qI}$, where I stands for the diode current, and q stands for the electron charge.</figcaption></figure>

another, as shown in Fig. 2, from Ref. 12. This figure shows the evolution of dark current and associated noise spectrum of n/p MCT diodes from LETI, extrinsically P-doped. Two diode processes (A and B) are shown. Both structures have the same n/p structure and same dark current limited by diffusion but results in very different noise performances. Even if process B exhibits the same dark current as process A, it gives a much lower 1/f noise power spectral density (PSD) at high temperature. This LF performance tends to be an indicator of the image quality at lower temperatures (i.e., noise defective pixel density, blinkers, and correction stability, etc.).¹³⁻¹⁵ Therefore, there is a strong need for an FOM quantifying this LF noise in order to easily qualify and classify sensors as early as possible (before full packaging of the array, for instance).

When limited by shot noise, the current PSD $S_I$ (expressed in $A^2/Hz$) evolves proportionally with the current as expressed by the classical shot noise expression:

$$S_{I\text{shot}} = 2qI \qquad (1)$$

For pure 1/f LF noise, the current noise PSD then appears more likely proportional to the dark current $I_d$ squared¹¹,¹⁶,¹⁷

$$S_{I_{1/f}} = \alpha_H \frac{I_d^2}{N_o f} \qquad (2)$$

with $N_o$ the number of charge carriers in the probe volume and $I_d$ the diffusion dark current. When dealing with simple resistors, the proportionality factor $\alpha_H$ is often known as the Hooge parameter.¹⁸ This name is also often used in various papers dealing with 1/f noise in photodiodes.¹⁹⁻²¹ In his early paper about 1/f noise in MCT photodiodes, Tobin² expressed
<figure><img src="image_5.png" /><figcaption>Fig. 2 Example of IV and noise measurements performed on n/p diodes from LETI, extrinsically P-doped. Dark current IVs are given for process A (a) and process B (b), respectively, and associated low-</figcaption></figure>

frequency noise PSD for process A (c) and process B (d), respectively, from Ref. 12.

the noise in current ($I_n$ expressed in $A/\sqrt{Hz}$) rather than in power, so that he ended with:

$$I_{n/f} = \sqrt{S_{I/f}} = \alpha_T \frac{I_d}{\sqrt{f}} \quad (3)$$

where $\alpha_T$ refers to a dimensionless parameter called the Tobin factor. This notation has then been adopted by the IR community to qualify 1/f noise in IR detectors.3,12,22,23 Note that the two alpha notations can be confusing as, by definition, $\alpha_H = \alpha_T^2$.

From the measurement point of view, this Tobin parameter may be conveniently estimated by normalizing the noise current density at 1 Hz $I_n(1 Hz)$ by the corresponding current:

$$\alpha_T = \frac{I_n(1 \text{ Hz})}{I_d} \quad (4)$$

## Practical Ways to Measure the Tobin Coeff

Because 1/f fluctuations are mostly affecting dark current rather than photocurrent, the Tobin factor should be estimated in dark conditions, making sure that the noise is

limited by proper 1/f noise. Those two conditions are usually fulfilled, raising the detector temperature. Ultimately, for IR devices, a 300-K measurement is a very efficient way to ensure dark conditions as well as systemic LF noise.

When dealing with a single diode (as opposed to an array of diodes), a convenient approach is to perform the acquisition of a time sequence $I_d(t)$. The mean value gives an estimate of the dark current. The PSD $S_I(f)$ may then be computed using Fourier calculations to obtain the noise spectrum. The 1-Hz value normalized with the dark current then gives the Tobin coefficient (see Eq. 4).

When the current measurement is carried out by integration (as in most of the IR ROICs for instance), the Tobin factor may be computed in a slightly different (simpler) way. Indeed, the root mean square (RMS) noise estimator integrates the noise onto a bandwidth given by the integration time $T_{int}$ and the total measurement time $T_t = N_f \times T_f$, where $N_f$ is the number of samples in the time series and $T_f$ is the time between each sample. The RMS noise is therefore written as:

$$I_{\text{rms}} = \sqrt{\int_{\frac{1}{T_f}}^{\frac{1}{2T_{\text{int}}}} S_I(f) df} \quad (5)$$
When limited by 1/f noise, i.e., using Eq. 3, this expression becomes:

$$I_{\text{rms}} = \sqrt{\int_{\frac{1}{T_{\text{t}}}}^{\frac{1}{2T_{\text{int}}}} \left[ \frac{\alpha_{\text{T}}^2 I_{\text{d}}^2}{f} \right] df} = \alpha_{\text{T}} I_{\text{d}} \sqrt{\ln\left(\frac{T_{\text{t}}}{2T_{\text{int}}}\right)} \quad (6)$$

with $I_d$ the dark current estimated by the mean value of the time series.

In such a case, an easy access Tobin factor estimation is then given by:

$$\alpha_{\text{T}} = \frac{I_{\text{rms}}}{I_{\text{d}} \sqrt{\ln\left(\frac{T_{\text{t}}}{2T_{\text{int}}}\right)}} \quad (7)$$

In a practical way, the 1/f noise behavior of the detector can be assessed by verifying the $\sqrt{\ln(\frac{T_t}{2T_{\text{int}}})}$ dependence of the noise. Being less computationally intensive than the previous spectral method, this estimator is well-suited when dealing with arrays with many pixels, using an integration input stage, such as direct injection or a capacitance trans-impedance amplifier ROIC. This method has been successfully used to evaluate Tobin factors using p/n MCT arrays in the MW blue (MWb) spectral range⁷ (4.1-µm cutoff). Figure 3 shows an example of such a time series and associated noise spectrum, for a 15-µm pitch pixel from such an array. In this example, the two computational methods for the Tobin factor give very similar values.

## Physical Meaning of this Tobin Parameter

The LF noise and therefore the Tobin factor is relevant to a given diode area so that we can rewrite as:

$$I_n = \alpha_T \frac{J_d A}{\sqrt{f}} \quad (8)$$

where $J_d$ is the dark current density (assumed diffusion limited at high temperature) associated with the surface A.

Existing models for 1/f noise involve a fluctuation of the number of charges (density $N_T$) at a given interface with a given area ($A_T$), which can be lower than the total pixel surface A. This charge fluctuation then modulates the surface recombination velocity and/or the space charge region volume and therefore modulates the dark current.²⁴,²⁵ In this frame, the dimensionless Tobin factor may then be rewritten as²⁶:

$$\alpha_T = \frac{\sqrt{N_T \times A_T}}{N_{\text{dop}} \times V_{\text{dark}}} = \frac{1}{\text{th}} \frac{\sqrt{N_T}}{N_{\text{dop}}} \times \frac{\sqrt{A_T}}{A} \quad (9)$$

where $N_{\text{dop}}$ stands for the doping density, $V_{\text{dark}} = \text{th} \times A$ is the volume considered for dark current generation, and th is

<figure><img src="image_6.png" /><figcaption>Fig. 3 Example of estimation of the Tobin factor for MW blue p/n MCT diodes at 296 K, 15-µm pitch low: time series and RMS estimation up: extraction from the PSD.</figcaption></figure>

the narrow gap layer thickness. In this view, the Tobin factor can be seen as the ratio between the square root of the total number of surface fluctuating charges (charge trapping centers) that can be found in the device divided by $N_o$ the number of charge carriers in the probe volume (as in the case of the Hooge formulation of 1/f noise¹⁶).

In practice, in several papers,¹²,²⁵⁻²⁷ the LF noise performance is normalized by the pixel dimension This results in a noise figure written as $I_{1/f} = \sqrt{S_t \times A} = I_n \sqrt{A}$ which is then plotted against the dark current density $J_d$. Figure 4 gathers such data from different references¹²,²⁵⁻²⁷ for the MW spectral band. In this representation, the lower limit is the shot noise, represented by a plain line. As expected, low dark current devices follow this shot noise limit. Furthermore, increasing dark current (for instance increasing the temperature) tends to increase LF noise and the data then tend to stand above this shot noise limit.

Going back to previous noise modeling, within the assumption that the fluctuating surface is the pixel surface itself ($A_T = A$), Eqs. 3 and 9 give a linear relationship between the noise figure $I_{1/f}$ and the dark current density $J_d$:

$$I_{1/f} = I_n \sqrt{A} = \frac{1}{\text{th}} \frac{\sqrt{N_T}}{N_{\text{dop}}} J_d = \beta_T \times J_d \quad (10)$$
<figure><img src="image_7.png" /><figcaption>Fig. 4 Literature data about 1/f noise figures for MCT MW photodiodes.¹²,²⁵⁻²⁷</figcaption></figure>

where the slope of the $I_{1/f}(J_d)$ curve is written as:

$$ \beta_T = \alpha_T \times \sqrt{A} = \frac{\sqrt{N_T}}{\text{th}N_{\text{dop}}} \quad (11) $$

The resulting slope $\beta_T$ (in cm) does not depend on the pixel area and is in fact the Tobin factor normalized by the pixel dimension.

In Fig. 4, data from the same device tend to nicely follow such a given slope (represented by dotted lines) corresponding to a given Tobin factor. Depending on the diodes, data from this figure suggest Tobin factors between $10^{-5}$ and $10^{-6}$ (green shaded area) for the best diodes, and between $10^{-4}$ and $10^{-5}$ (red shaded area) for the worst ones. For instance, data from Fig. 2 fit with $\alpha_T \sim 10^{-5}$ for process B and mid-$10^{-6}$ for process A.

In case several different interfaces are contributing with their relative areas $A_{T_i}$ and their relative trap densities $N_{T_i}$, the previous equation may be rewritten as the quadratic sum of uncorrelated noise contributions:

$$ \alpha_T = \frac{1}{\text{th}} \frac{\sqrt{\sum_I N_{T_i} A_{T_i}}}{N_{\text{dop}} A} \quad (12) $$

Figure 5 represents an example of such interfaces potentially contributing to the 1/f noise for the case of a photodiode in array configuration. When all the diodes are collecting with a suitable diode design (optimal fill factor), the pixel area is then given by $A = \text{pitch}^2$. For planar photodiodes (upper drawing), contributors to 1/f noise might be the space charge region surface $N_{T_i}/A_{T_i}$ or the interpixel area $N_{T_2}/A_{T_2}$. For those two contributions, the $A_{T_i}$ area is lower than $A$. For instance, considering the rear interface

<figure><img src="image_8.png" /><figcaption>Fig. 5 Schematic of the possible different interfaces playing in the 1/f model considered.</figcaption></figure>

$N_{T_3}/A_{T_3}$ the fluctuating area is in this case identical to the pixel area $A_{T_3} = A$.

As a conclusion, the fine understanding of physical meaning of the 1/f noise factor requires a precise knowledge of the charge collection area and of the fluctuating surface area.

## Evolution with the Sensitive Area

A first interesting example is the study of different pixel pitches, i.e., different sensitive areas. This particular example deals with $p/n$ diodes, implanted in a MW MCT absorbing layer. The diode under test (DUT) is located within an array of identical diodes with a given pitch. Surrounding diodes are identical to the DUT (process, geometry...) and biased the same way. Therefore, those surrounding diodes are collecting the carriers in the same way the central diode does, so that the collection area of the DUT is the pixel area only: $A = \text{pitch}^2$.

Figure 6 gathers Tobin factors measured with the RMS method at 300 K on such diodes with different pixel pitches. Values are increasing from low $10^{-5}$ for 15-µm pitch up to slightly above $10^{-6}$ for 5-µm pitch diodes. Those measured values are very consistent with a $\beta_T = \alpha_T \times \text{pitch} = 610^{-8}\text{cm}$ (dashed line plotted onto the graph), consistently with $A_T \propto A = \text{pitch}^2$ in this case.

When N devices are measured in parallel, as seen for example in Ref. 20 the noise measured on this diode bundle corresponds to $\sqrt{N}$ the noise of a single device, whereas, the current corresponds to N times the current of a single diode. Hence, from such a configuration, the ratio $\alpha_T(N)$ between 1 Hz noise and dark current may be written as follows:

$$ \alpha_T(N) = \frac{\sqrt{N} \cdot I_{\text{rms1/f}} \left(\frac{1}{f}\right)}{N \cdot I_d \sqrt{\ln\left(\frac{T_i}{2 \cdot T_{\text{int}}}\right)}} = \frac{1}{\sqrt{N}} \alpha_T \quad (13) $$
<figure><img src="image_9.png" /><figcaption>**Fig. 6** Example of evolution of the Tobin factor with pixel pitch for *n/p* MW MCT diodes at 300 K.</figcaption></figure>

Therefore, for a given pixel pitch, the estimation of the Tobin factor reduced to a single device $\alpha_T$ must take into account the number of pixels $N$ measured in parallel, simply by multiplying by $\sqrt{N}$ the factor $\alpha_T(N)$ measured on a bundle of $N$ diodes.

In the previous configurations, the tested diodes are surrounded by collecting diodes (see insert in Fig. 6). The state of those guard diodes appears important for a good understanding of the measured Tobin factor. Indeed, as mentioned above, when these neighboring diodes collect charges (i.e. are biased), the central diode is confined, similar to the situation of an imaging focal plane array, and collects charges on an area roughly given by pitch²,²⁸ However, if the guard diodes are not biased, (i.e. not collecting), the current of the central diode becomes larger due to lateral collection: for long diffusion lengths, the central diode is able to collect charges over an area much larger than the pixel pitch area and the measured current increases. However, the noise might not increase simply linearly with the current as expected. In fact, we should consider different surface in the pixel that might have different LF fluctuation properties (for instance junction surface $A_{T_1}$ and interpixel surface $A_{T_2}$ for the planar case represented in Fig. 5). As an example, for 15-µm $p/n$ LW MCT diodes from Lynred (Scorpio product test devices), the measured Tobin factor goes from $2.5 \times 10^{-5}$ in confined configuration to $6 \times 10^{-6}$ when the confinement ring is not collecting. The difference might be explained by the fact that lateral diffusion current carries lower 1/f noise contributions than the central pixel current alone. In other word, going back to Eq. 12 formalism, $A_{T_1}N_{T_1}$ is lower for the lateral diffusion current contribution than for the central current contribution of such a single diode.

This kind of consideration about lateral contribution is also clearly observed on variable area single diodes, as the collecting area varies strongly from one diode to the other,

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 7** Case of a variable area single diode set (MCT MW *n/p* VHg diodes at 300 K): upper curve evolution of the measured current with diode radius; lower curve evolution of the measured 1/f noise at 1 Hz with diode radius.</figcaption></figure>

enabling us to study the LF noise evolution with area. The upper plot in Fig. 7 gives the evolution of the current measured at 300 K for a set of *n/p* VHg diodes measured at 300 K with radius $R_{junc}$ ranging from 2.5 µm to 100 µm. Diodes are not guarded so that lateral collection might extend up to the typical diffusion length $L_d$ in the P-type MCT material. For this round geometry, the junction surface simply writes $A_{junc} = \pi R_{junc}^2$ (red surface in the inserted drawing in Fig. 7), whereas the lateral collection area then writes $A_{lat} = \pi [(R_{junc} + L_d)^2 - R_{junc}^2]$ (blue surface in the inserted drawing in Fig. 7). The dotted line represents the expected geometrical evolution of the current, which fits nicely with measured data with extracted values of dark current density $J_d$ and lateral diffusion $L_d$.

The lower curve in Fig. 7 gives the corresponding noise normalized to area, previously noted as $I_{1/f} = I_n/\sqrt{A}$, measured with the RMS method in Eq. 6. Given the two fluctuating interfaces $A_{junc}$ and $A_{lat}$ with their respective Tobin
factors $\alpha_{T_{junc}}$ and $\alpha_{T_{lat}}$, we can write the noise using Eqs. 10 and 7:

$$I_{1/f} = J_d \sqrt{\ln\left(\frac{T_t}{2T_{\text{int}}}\right)} \times \sqrt{\left(\alpha_{T_{\text{junct}}A_{\text{junc}}}\right)^2 + \left(\alpha_{T_{\text{lat}}A_{\text{lat}}}\right)^2} \quad (14)$$

which may also be written using $\beta_{T_t}$ instead of $\alpha_{T_t}$:

$$I_{1/f} = J_d \sqrt{\ln\left(\frac{T_t}{2T_{\text{int}}}\right)} \times \sqrt{\beta_{T_{\text{junc}}}^2 A_{\text{junc}} + \beta_{T_{\text{lat}}}^2 A_{\text{lat}}} \quad (15)$$

Doing so, we made the assumption that the dark current density is identical in the two areas $J_d = J_{d_{junc}} = J_{d_{lat}}$. The lower plot in Fig. 7 gives the result of such a modeling in the dotted line using $\beta_{T_{junc}} = 10^{-7} \text{cm}$ and $\beta_{T_{lat}} = 1.5 \cdot 10^{-8} \text{cm}$. Note that the fact that one interface gives a much lower $\beta_T$ than the other demonstrates that the junction interface introduces more fluctuating $1/f$ sources than the lateral interface investigated by lateral diffusion. This seems consistent with the fact that the ion implantation process used here is susceptible to generating extra crystal defects that might serve as extra LF fluctuation sources. Therefore, for a classical 15-µm pitch pixel geometry junction occupying only half the pixel area, the expected value of the Tobin factor for such a pixel would be $\alpha_T = 5 \cdot 10^{-5}$. This value appears consistent with the $4 \cdot 10^{-5}$ extracted for 15-µm pitch in Fig. 6, when analyzing the pitch dependency with the same diode technology.

Therefore, for MCT MW diodes, the full understanding of the Tobin factor requires a combined analysis of dark current and noise versus diode geometry.

## Variation of the Diode Design/Process

Going back to Eq. 9, this Tobin factor is supposed to be sensitive to doping $N_{dop}$. It also appears sensitive to defect surface density $N_T$ onto a given area $A_T$ lower or equal to the total area of the considered detector. Indeed, such a Tobin analysis may be used to assess the difference in $1/f$ features between different diode processes. As an example, Fig. 8 gives the evolution of the measured Tobin factor for two different $n/p$ MCT processes in the MWr band, called Process-1 and Process-2. Note that both processes give the same dark current densities, but the $1/f$ noise may differ. Four different junction geometries (i.e., four different $A_T$ values) are studied, all of them contained in a 15-µm pitch. The figure shows an increase in the Tobin factor with $\sqrt{A_T}$ for Process-1

<figure><img src="image_12.png" /><figcaption>Fig. 8 Study of two different processes with $n/p$ VHg MCT diodes at 300 K and 15-µm pitch.</figcaption></figure>

whereas Process-2 does not show any clear dependence on $\sqrt{A_T}$.

Knowing the thickness and doping of the MCT layer, from Eq. 9, the slope in Fig. 8 gives an estimation of the density of fluctuating centers. We end up with $N_T = 2.4 \cdot 10^{12} \text{cm}^{-2}$ for Process-1. In the case of variant 2, the process was optimized to limit the number of process induced defects and the associated Tobin factor does not appear to be dependent on the previously incriminated surface, resulting in a much lower density estimation of the fluctuating sources: $N_T < 2e^{11} \text{cm}^{-2}$.

$1/f$ noise is also likely to decrease as doping increases, as shown by the $\alpha_T \propto 1/N_{dop}$ shown on Eq. 9. Indeed, increasing the total amount of charges in the system decreases the relative effect of a given number of fluctuating centers.¹¹ Such an effect is illustrated on Fig. 9 in the case of $p/n$ MWr MCT diodes. The left plot shows the Tobin factor measured on 15-µm pitch diodes with two different doping levels. As expected, the Tobin factor is larger for the lower doping than for the larger one. Considering $N_T$ and $A_T$ identical for both dopings, the modeling from Eq. 9 is represented with a dotted line. In this example, such a very simple modeling does not reproduce the experimental values very accurately, but still gives a good trend.

Another example is given in Fig. 9 (right), where the doping was kept constant (previous lower value) but the passivation was modified between two different $p/n$ diodes. Passivation B was supposed to be more effective than passivation A. As expected, going from passivation A to passivation B decreases significantly. This evolution can be modeled by a reduction of $N_T$ from $1.5 \cdot 10^{12} \text{cm}^{-2}$ to $10^{11} \text{cm}^{-2}$, confirming (and quantifying) the improvement in passivation.
<figure><img src="image_13.png" /><figcaption>**Fig. 9** Example of Tobin factor variation with doping (left plot) and passivation quality (right plot), for 15-µm pitch *p/n* MWr MCT diodes at 300 K.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 10** Example temperature study of the measured noise on 3 × 3 bundles 15-µm pitch *n/p* VHg MWr diodes.</figcaption></figure>

## Temperature Dependence of this Tobin Coefficient

As suggested by Soria et al.,¹² the Tobin coefficient may slightly depend on temperature. This can be attributed to a temperature dependence of the active trap density related to $N_T$, or even doping activation related to $N_{dop}$. Such effects have to be considered in the measurement uncertainty and repeatability when dealing with large temperature differences.

Figure 10 gives an example of diodes measured on the legacy *n/p* VHg MCT technology from Lynred. It shows the RMS measured noise at 1 Hz as a function of temperature

for a pair of 3 × 3 clusters of 15-µm pitch diodes with guard ring active. The trend is well fitted with a Tobin coefficient $\alpha_T = 7 \cdot 10^{-5}$ from 160 K up to 300 K. This means that the linear relationship (Eq. 9) remains relevant onto this temperature range and that the Tobin coefficient may be used to quantify the 1/f noise. Below 160 K, the measured noise differs from the Tobin model, and the diodes may no longer be limited by 1/f noise at 1 Hz.

## Tobin Parameter as an Image Quality Indicator

As seen previously, 1/f current noise tends to be proportional to the dark current, whereas shot noise is proportional to the square of the total current. All other things being equal, we can then understand that a diode limited by shot noise at low temperature will eventually switch to its 1/f regime as the dark current increases with temperature. Nevertheless, some diodes are more affected by these LF fluctuations than others, and therefore switch earlier into this 1/f noise regime (for instance a pixel with an additional extended defect such as a threading dislocation). This can result in excess noise for those given pixels, even at low operating temperature. Such diodes might therefore be detected as a noise defective pixels and degrade the overall imaging performance of the focal plane array. Ultimately, a key parameter for FPAs is the noise defectivity at operating temperature (i.e. the proportion of pixel which, at low temperature, exhibits higher noise than expected). A poor noise defectivity has also been shown to be usually correlated to RTS pixel rates or even stability issues.²⁹ In general, this noise distribution tail is often considered as a good figure-of-merit to qualify the image quality features, in a general manner. On the other hand, the Tobin factor is rather intended to quantify 1/f fluctuation of a given
<figure><img src="image_15.png" /><figcaption>Fig. 11 Example of correlation between Tobin factor measured at 300 K on side test chips and noise defectivity measured on ¼ of study VGA FPAs at 110 K, extracted from the same wafers. (defect criteria is defined as a RMS noise $> 2 \times \text{RMS mean}$, diodes are MW MCT $n/p$ VHg with a 15-µm pitch).</figcaption></figure>

technology (i.e. qualify a statistical phenomenon), even if measured on a limited set of single diodes.

Figure 11 plots the correlation between noise defect rates on FPAs and Tobin factors measured on test devices from the same wafers (with same design and process). Different processes and different wafers are gathered onto this plot. In the end, a very clear correlation was found between noise defectivity and Tobin coefficient. This shows that this Tobin factor can be used to qualify image quality prior to FPA hybridization, which is very helpful in optimizing the process development cycle.

## Brief Review of Available Literature Data

The idea of using 1/f noise as a figure-of-merit for photodiodes is not new for II-VI photodiodes. In 1980, Tobin found a value of $\alpha_T = 10^{-3}$.² However, in this work, the effective surface of the device under test was not given. In addition, the paper does not specify whether or not the device was confined, making it difficult to compare with more recent results.

More than 40 years later, technologies have improved, and previous examples from Lynred⁷,²⁶ and LETI¹¹ demonstrated a typical value in the range of low $10^{-5}$ for mature technologies (15-µm pitch with p/n technology or n/p VHg-doped technologies). Kinch⁵ reported the noise and dark current for DRS MCT N + P MW technology. Assuming a confined 15-µm pitch diode, the extracted Tobin coefficient is in the same range of nearly $\alpha_T = 2 \cdot 10^{-5}$.

In the extensive work by Zhu,²¹ on presumably unconfinemented 20-µm-diameter MCT diodes, the dark current appears to be limited by the depletion current rather than the diffusion current, making the comparison difficult. However, noise and dark current values give a Tobin estimation of $\alpha_T = 3 \cdot 10^{-5}$. This value may, however, be underestimated due to the fact that the DUT is not guarded properly.

In a more recent paper from CEA, Soria¹² showed values in the $10^{-6}$ for extrinsic n/p MW devices with a confined 15-µm pitch diode, which is the best-reported Tobin coefficient to our knowledge.

Finally, to conclude on this non-exhaustive II-VI Tobin benchmark for MCT photodiodes, note that in the 2019 II-VI workshop communication, Teledyne showed that “good” diodes (18-µm pitch) exhibit an $\alpha_T$ value in the $10^{-5}$ to $2 \cdot 10^{-4}$ range.¹⁵

Compared to previous examples on II-VI FPAs, there are fewer examples of papers on III-V detectors, among which the work of Ramos et al.²⁰ is worth mentioning. They report on a 15-µm pixel pitch 15 × 15 array, MW type II-super lattices (T2SL) detector. In this case, the pixels are fully reticulated so that no lateral diffusion is collected; the guard ring is therefore pointless in this particular configuration. Our understanding of their data is that the Tobin factor of a single 15-µm pitch pixel should be $\alpha_T = 2 \cdot 10^{-4}$. This value is higher than that the state-of-the-art II-VI detectors but could be explained by the lower maturity of the new technology, and in particular by the use of deep mesa etch that could induce fluctuating sources onto the mesa walls.

Ciura et al.¹⁹ recently compared II-VI and III-V technologies. Unfortunately, there is no mention of diode surface, pixel pitch, or use of a confinement ring. In InAsSb-based devices, the dark current seems to be limited by sources other than diffusion, making the Tobin coefficient irrelevant. For MCT, a Tobin coefficient in agreement with a value of $\alpha_T = 8 \cdot 10^{-5}$ is reported, but, as already noticed, the comparison with state-of-the-art values is subjected to caution because we do not know either the test conditions or the pixel pitch of the test devices.

In Wan et al.,³⁰ 1/f noise has been studied on extended SWIR InGaAs photodiodes. The noise was measured on 20-µm pitch electrically isolated diodes. From their data, a value of $\alpha_T = 10^{-4}$ may be estimated for different samples.

## Conclusions

Ultimately, the image quality of IR imaging systems is limited by LF noise. The dimensionless Tobin factor, defined as the ratio between the 1/f current noise at 1 Hz and the dark current, may be considered as an additional FOM for IR imaging technology comparison (in addition to QE and dark current). Indeed, it appears as a measurable parameter
reflecting the average LF features of the device. It might also be a very efficient tool for the optimization and design of a given technology. Indeed, when correctly measured, this parameters appears to be well correlated with image quality at operating temperatures (noise distribution tails, RTS rates, etc.). Typical values observed on state-of-the-art MCT diodes are between 10⁻⁵ and 2 · 10⁻⁴ for a 15-µm pixel pitch.

However, care must be taken when using this parameter.

First, the measurement has to be carried out making sure that the measured devices are properly limited by 1/f noise and diffusion dark current. By increasing the temperature, both the dark current and the 1/f noise increase (an increasing part of the diodes exhibit LF noise). There even comes a point where all the diodes are dominated by this 1/f noise, which is referred to as systemic 1/f noise. Reliable Tobin measurements must usually be carried out above this temperature. If relevant for the device, when dealing with MW detectors, a very convenient way to respect those two conditions is to measure the device at room temperature. This offers a great advantage in terms of measurement simplicity, with no need for cryogenic cooling and no issue with parasitic photonic current (dark currents are easily dominating).

Then, the value of the Tobin factor depends on the detector surface. Thus, in order to be fully understandable, the surface must be specified together with the Tobin values. Also of importance, in planar configuration (as opposed to mesa configuration), the tested diodes must be preferably guarded by collecting diodes. These guard diodes must be biased such that they collect the carriers, thus limiting the area of the central tested diode to its real operating surface. Also, when the test pattern consists of a cluster of diodes, the number of diodes must be specified to enable comparison with a single pixel value.

If the guard ring is not possible/available in planar configuration, it is then very useful to consider different collector geometries. To go into more detail, the ratio between the fluctuating surface and the total surface appears in the expression of the Tobin parameter with the factor $\sqrt{A_T/A}$. Therefore, a good understanding of the 1/f noise usually requires a careful analysis of the noise and dark current evolution with different geometries (junction size, mesa size, and/or pitch) in order to identify the surfaces contributing to 1/f fluctuations in the resulting dark current. In the case of state of the art MCT photodiodes from LETI and Lynred, for instance, it is very instructive to be able to differentiate the fluctuations coming from the lateral diffusion from the effect coming from the near proximity of the collector. The *pn* junction area appears generally degraded compared to the lateral diffusion area, away from the junction. In other words, the implanted junction area tends to induce more LF fluctuation sources than the passivated interface away from the *pn* junction.

Further measurements with other material systems are now to be considered in order to strengthen this analysis, in

particular III-V detectors. Bulk or superlattices are currently being studied at LETI in order to extract reliable and comparable Tobin values.

**Acknowledgments** This work was performed through DEFIR (Common Lab Lynred/CEA-LETI), with DGA (French MoD) funding.

**Funding** Open access funding provided by Commissariat à l'Énergie Atomique et aux Énergies Alternatives.

**Conflict of interest** The authors declare that they have no conflict of interest.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

## References

1. W.E. Tennant, D. Lee, M. Zandian, E. Piquette, and M. Carmody, MBE HgCdTe technology: a very general solution to IR detection, described by "rule 07", a very convenient heuristic. *J. Electron. Mater.* 37(9), 1406–1410 (2008). https://doi.org/10.1007/s11664-008-0426-3.

2. S.P. Tobin, Noise in (Hg, Cd)Te photodiodes. *IEEE Trans. Electron Devices* 29(1), 43–48 (1980). https://doi.org/10.1109/T-ED.1980.19817.

3. C.T. Elliott, N.T. Gordon, R.S. Hall et al., 1/f noise studies in uncooled narrow gap Hg₁-xCdxTe non-equilibrium diodes. *J. Electron. Mater.* 26, 643–648 (1997). https://doi.org/10.1007/s11664-997-0209-2.

4. R.L. Strong, M. Kinch, and J.M. Armstrong, Performance of 12-µm- to 15-µm-pitch MWIR and LWIR HgCdTe FPAs at elevated temperatures. *J. Electron. Mater.* 42(11), 3103–3107 (2013). https://doi.org/10.1007/s11664-013-2658-0.

5. M. Kinch, R. Strong, and C. Schaake, 1/f noise in HgCdTe focal-plane arrays. *J. Electron. Mater.* 42, 3243–3251 (2013). https://doi.org/10.1007/s11664-013-2636-6.

6. D. Pogany, S. Ababou, G. Guillot, X. Hugon, B. Vilotitch, and C. Lenoble, Study of RTS noise and excess currents in lattice-mismatched InP/InGaAs/InP photodetector arrays. *Solid-State Electron.* 38(1), 37–49 (1995). https://doi.org/10.1016/0038-1101(94)E0053-H.

7. P. Guinedor, A. Brunner, L. Rubaldo, D. Bauza, G. Reimbold, and D. Billon-Lanfrey, Low-frequency noises and DLTS studies in HgCdTe MWIR photodiodes. *J. Electron. Mater.* 48(10), 6113–6117 (2019). https://doi.org/10.1007/s11664-019-07213-7.

8. S. Dinand, E. Deborniol, N. Baier, F. Rochette, S. Rizzolo, V. Goiffon, and O. Gravrand, *Study of damages caused by proton irradiation on MCT n/p focal plane cooled arrays*, vol. 12534 (SPIE, 2023), p. 149–162 https://doi.org/10.1117/12.2663763

9. R. L. Strong, M. Kinch, and J. M. Armstrong, *Temperature dependence of 1/f noise, defects, and dark current in small pitch*
MWIR and LWIR HDVIP® HgCdTe FPAs, vol. 8704 (SPIE, 2013), p. 87042O https://doi.org/10.1117/12.2015816

10. C.T. Elliott, N.T. Gordon, R.S. Hall, T.J. Phillips, C.L. Jones, and A. Best, 1/f noise studies in uncooled narrow gap HgI_xCdTe non-equilibrium diodes. *J. Electron. Mater.* 26(6), 643 (1997).

11. W. Hassis, O. Gravrand, J. Rothman, and S. Benahmed, Low-frequency noise characteristics of HgCdTe infrared photodiodes operating at high temperatures. *J. Electron. Mater.* 42(11), 3288–3296 (2013). https://doi.org/10.1007/s11664-013-2728-3.

12. M. Soria et al., High operating temperature n-on-p extrinsic MWIR HgCdTe photodiodes, vol. 117411C (SPIE, 2021), https://doi.org/10.1117/12.2587706

13. M. Guénin, S. Derelle, M. Caes, L. Rubaldo, and I. Ribet-Mohamed, RTS noise detection and voltage effect on RTS in HgCdTe focal-plane arrays. *J. Electron. Mater.* 49(11), 6963–6970 (2020). https://doi.org/10.1007/s11664-020-08271-y.

14. A. Cathignol et al., RMS noise modeling and detection for high-reliability HgCdTe infrared focal plane arrays development, vol. 9451 (SPIE, 2015), p. 945126 https://doi.org/10.1117/12.2178034

15. R.L. Strong and M. Kinch, Quantification and modeling of RMS noise distributions in HDVIP® infrared focal plane arrays. *J. Electron. Mater.* 43(8), 2824–2830 (2014). https://doi.org/10.1007/s11664-014-3099-0.

16. F. Hooge, T.G. Kleinpenning, and L.K. Vandamme, Experimental studies on 1/f noise. *Rep. Prog. Phys.* 44, 31 (1981).

17. T.G.M. Kleipenning, 1/f noise in p-n junction diodes. *J. Vac. Sci. Technol. A* 3(1), 176 (1985).

18. F.N. Hooge, T.G.M. Kleinpenning, and L.K.J. Vandamme, *Rep. Prog. Phys.* 44, 479 (1981).

19. L. Ciura et al., *Sens. Actuators A* 305, 111908 (2020).

20. D. Ramos, M. Delmas, R. Ivanov, L. Höglund, E. Costard, P. Hellström, and G. Malm, 1/f noise and dark current correlation in midwave InAs/GaSb type-II superlattice IR detectors. *Phys. Status Solidi A* 218, 2000557 (2021). https://doi.org/10.1002/pssa.202000557.

21. Zhu et al., *Opt. Express* 28(16), 23660–23669 (2020). https://doi.org/10.1364/oe.399565.

22. S.H. Bae, S.J. Lee, Y.H. Kim, H.C. Lee, and C.K. Kim, Analysis of 1/f noise in LWIR HgCdTe photodiodes. *J. Electron. Mater.* 29(6), 877–882 (2000). https://doi.org/10.1007/s11664-000-0242-x.

23. D. Lee et al., High-operating temperature HgCdTe: a vision for the near future. *J. Electron. Mater.* 45(9), 4587–4595 (2016). https://doi.org/10.1007/s11664-016-4566-6.

24. R.A. Schiebel, D. Blanks, D. Bartholomew, and M. Kinch, Evidence for 1/f noise in diffusion current due to insulator trapping and surface recombination velocity fluctuations. *J. Electron. Mater.* 22(8), 1081–1085 (1993). https://doi.org/10.1007/BF02817528.

25. M. Kinch, C.-F. Wan, H. Schaake, and D. Chandra, Universal 1/f noise model for reverse biased diodes. *Appl. Phys. Lett.* 94(19), 193508 (2009). https://doi.org/10.1063/1.3133982.

26. A. Kerlain et al., Mid-wave HgCdTe FPA based on P on N technology: HOT recent developments. NETD: dark current and 1/f noise considerations. *J. Electron. Mater.* 45(9), 4557–4562 (2016). https://doi.org/10.1007/s11664-016-4506-5.

27. M. Kinch, C. Wan, and J.D. Beck, 1/f noise in HgCdTe photodiodes. *J. Electron. Mater.* 34(6), 6–10 (2005). https://doi.org/10.1007/s11664-005-0044-2.

28. O. Gravrand, J.C. Desplanches, C. Delbegue, G. Mathieu, and J. Rothman, Study of the spatial response of reduced pitch HgCdTe dual-band detector arrays. *J. Electron. Mater.* 35(6), 1159–1165 (2006). https://doi.org/10.1007/s11664-006-0236-4.

29. D. Lee et al, II–VI workshop communication, not published (2019).

30. L. Wan et al., Dark current and 1/f noise characteristics of In<sub>0.74</sub>Ga<sub>0.26</sub>As photodiode passivated by SiN<sub>x</sub>/Al<sub>2</sub>O<sub>3</sub> bilayer. *Infrared Phys. Technol.* 109, 103389 (2020). https://doi.org/10.1016/j.infrared.2020.103389.

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