We report on a model for the bipolar amplification effect (BAE), which enables defect density measurements utilizing BAE in metal–oxide–semiconductor field-effect transistors. BAE is an electrically detected magnetic resonance (EDMR) technique, which has recently been utilized for defect identification because of the improved EDMR sensitivity and selectivity to interface defects. In previous work, BAE was utilized exclusively in EDMR measurements. Although BAE EDMR improves the sensitivity of EDMR in studies of semiconductor/oxide interface defects, an understanding of BAE in both electrical measurements and EDMR has not yet been investigated. In this work, we develop a BAE theory based on a modified Fitzgerald–Grove surface recombination methodology, which, in theory, may be utilized to fine-tune conditions for EDMR measurements. BAE may also now be utilized as an analysis tool in purely “electronic” measurements. The model presented here may ultimately prove useful in the development of resonance-based theories of BAE EDMR.

## I. INTRODUCTION

Understanding spin-dependent transport from a theoretical standpoint dates back to work by Lepine,^{1} who investigated spin-dependence of photo-generated carriers on silicon (Si) surfaces. Kaplan, Solomon, and Mott (KSM)^{2} further showed that the spin-dependence involves an intermediate state formation between conduction band/valance band carriers and trapped carriers. This new approach was appealing as it explained the large response of spin-dependent recombination (SDR) relative to electron paramagnetic resonance (EPR) ($107$ $\xd7$ improvement)^{3} and the field-and-frequency independent response of spin-dependent transport measurements.^{2} Refinements have been made to the original work of KSM and Lepine.^{4,5} However, recent investigations have introduced the stochastic quantum Liouville equation (SLE) and density matrix formalism to investigate spin-dependent transport.^{6–8}

Harmon *et al.* developed a model of spin-dependent recombination (SDR) involving Si/silicon dioxide (SiO$2$) interface defects utilizing solutions to the SLE based on the Fitzgerald–Grove surface recombination theory.^{9} In this work, they model direct current–current-voltage (dc-IV) SDR experimental measurements, in which the SDR response was modeled as a function of the forward voltage on the source/drain terminals. It is shown that the forward voltage applied to the source/drain terminals alters the rates at which conduction electron and defect electron pairs dissociate or recombine. This is particularly important in near-zero field magnetoresistance (NZFMR),^{10,11} where these rates greatly influence the line shape of the NZFMR spectrum.^{7,8}

The theoretical framework for the dc-IV method for MOSFET defect density measurements was developed by Fitzgerald and Grove.^{9} Reddi *et al.*^{12} also utilized a similar theoretical framework to model recombination peaks observed in oxidized bipolar-junction transistors.

In this work, we develop a quantitative treatment of an electrically detected magnetic resonance (EDMR) technique, the bipolar amplification effect (BAE)^{13} to measure interface electron–hole recombination at interface defects in MOS devices and present proof-of-concept results on Si MOSFETs. The goal of this work is to provide a surface recombination model of BAE, which may be utilized as a resource for performing EDMR and purely electronic BAE measurements and for future modeling of the SDR response of BAE EDMR measurements. BAE EDMR is widely utilized as it greatly improves the sensitivity and selectivity of EDMR measurements.^{13–19} In systems that have a large contribution from bulk semiconductor defects, such as SiC MOSFETs, dc-IV is ineffective since large contributions from bulk defects would obscure the surface recombination in EDMR measurements. BAE EDMR overcomes the limitations of dc-IV-based EDMR.^{13} Therefore, this model may be adapted to solutions of the SLE in the same manner as demonstrated by Harmon *et al.*^{7,8} We utilize a modified model based on theory developed by Fitzgerald and Grove^{9} and Reddi,^{12} in which we account for the variation in carrier distribution throughout the channel region.

### A. Direct-current-current–voltage scheme

The dc-IV measurement technique involves the application of a slight forward bias to the source/drain contacts to inject minority carriers into the channel while holding the gate voltage such that the surface is in depletion. A diagram of this biasing scheme is illustrated in Fig. 1. Under the conditions where the surface electron concentration $ns$ is equal to the surface hole concentration $ps$, surface recombination will be maximized. This occurs when^{9}

Here, $q=1.602\xd710\u221219$ C is electronic charge, $VF$ is the source/drain-to-body junction forward bias, $k$ is Boltzmann’s constant, and $T$ is absolute temperature. Under these conditions, there will be a peak in the recombination current and, subsequently, the body current $IB$. According to Fitzgerald and Grove,^{9} the amplitude of the current peak can be approximated as

Here, $AG$ is the gate area of the MOSFET and $s0$ is the surface recombination velocity given by

Here, $\sigma s$ = $\sigma e$ = $\sigma h$ is the electron and hole capture cross section, which are assumed to be equivalent. This assumption results in the same expressions for recombination rate as the case when the electron and hole capture cross sections are treated separately.^{9} $vth$ is the thermal velocity for both electrons and holes and $Dit$ is the density of interface states in cm$\u22122$ eV$\u22121$. Our discussion of the derivation of this expression follows that of Fitzgerald and Grove.^{9} We consider an n-channel MOSFET subjected to a gate bias and forward bias applied to the source and drain terminals. Our analysis is for an n-channel device. However, mirror image expressions would also be applicable to p-channel devices. First, we assume that centers have a constant distribution throughout the bandgap. Since recombination is dominated by carriers near midgap, according to Shockley–Read–Hall statistics,^{20} this assumption will introduce only modest error, assuming that the density of states does not vary greatly in the near midgap region.^{9} The recombination rate per unit area is then given by

Here, $Ev$ and $Ec$ are the valence and conduction band minimum and maximum, $Est$ is the surface trap energy, and $Ei$ is the intrinsic Fermi energy. Evaluation of (4) leads to the following expression for recombination rate (we only consider the case for $ps+ns\u22652ni$):^{9}

If it is further assumed that the condition of quasi-equilibrium is maintained throughout the surface space charge region, the concentration of surface electrons and holes will be functions of the surface potential $\varphi s$ and $VF$,

Here, $nA$ is the acceptor doping concentration of the body. By substituting (6) into (5), one may arrive at the model equation for the recombination rate as a function of $\varphi s$ and $VF$. One may relate the surface potential $\varphi s$ to a gate voltage $VG$ via^{21}

Here, $VFB$ is the flatband voltage, $QS(\varphi s)$ is the induced semiconductor surface charge, and $Cox$ is the oxide capacitance. What falls out of these substitutions is an expression $US=US(\varphi s,VF)$, which may conveniently be considered in three parts. Consider in (5) the first part of the expression on the right, before the fraction, which is characteristic of a velocity involving surface recombination. Next, the fraction represents the effect of surface potential on carrier concentrations. The third part represents the drive of $US$ toward equilibrium.^{9} This rate will be largest when (1) is satisfied.

The last approximate equation for $USm$ is valid for $VF\u226bkT/q$. Finally, one may arrive at (2) by utilizing

More generally, the recombination rate $US(\varphi s,VF)$ may be utilized to model the recombination current as a function of gate voltage by utilizing (5)–(7) and

This equation, plotted with gate voltage, yields a peak in the current, in which the amplitude may be computed via (2) and peak position occurs at precisely $\varphi s=\varphi fp\u221212VF$, where $\varphi fp=kTqln\u2061(nAni)$. This condition is illustrated in Fig. 2. The quantitative BAE method relies on similar principles as those first utilized by Fitzgerald and Grove.^{9}

### B. Bipolar amplification effect

The BAE^{13} scheme is similar to that of dc-IV. However, instead of applying a forward voltage $VF$ to both source and drain, the source is disconnected from the drain and the body terminal is grounded. The current is measured through the drain, which is held at virtual ground, and the source bias is held at $IS$ with a gate bias $VG$ held in depletion. The source and drain may be flipped here. A diagram of this biasing scheme is illustrated in Fig. 3. This biasing scheme has been extremely effective in EDMR measurements as the sensitivity of the EDMR measurement using BAE is an order of magnitude greater than the dc-IV EDMR approach.^{13}

In our discussion of BAE, we consider a constant current bias applied to the source junction $IS$ instead of the forward bias $VF$ ($VF$ is maintained at a value slightly greater than the junction built-in voltage). The drain current, by Kirchoff’s law, obeys the following:

Here, $IB$ is the body current (we assume negligible leakage currents through the gate). If we assume that $IS$ is a constant and $IB$ $\u226b$ $ID$, then the drain current $ID$ represents only a small fraction of the total source current $IS$. We also assume ohmic contacts for the drain/source and body contacts. Under this approximation, the drain current will consist entirely of electrons and the body will consist entirely of holes since the drain is highly doped ($n+$) with ohmic contact for electrons, producing a barrier for holes, and the body is doped p-type with ohmic contact for holes, producing a barrier for electrons. Thus, under EDMR, the change in the drain current $\Delta ID$ is, to a reasonable approximation, equal and opposite to the EDMR-induced change in body current $\Delta IB$. Since $IB$ $\u226b$ $ID$, the sensitivity, defined as the relative EDMR change in current $\Delta IDID$, is orders of magnitude greater at the drain.^{13}

Aichinger and Lenahan^{13} also note that the EDMR BAE technique is exclusively sensitive to interface defects. Thus, BAE EDMR has been utilized extensively in studies of MOS interface defects.^{14–18,22–24} BAE EDMR is quite powerful as the EDMR scheme can be implemented in a wide variety of MOSFET geometries due to the ability to control the range at which minority carriers may diffuse across the channel with a gate voltage $VG$. The BAE EDMR scheme is not as limited with regard to channel length, whereas spin-dependent charge pumping (SDCP)^{25} is because in BAE EDMR, the gate voltage is utilized to control the length over which minority carriers may travel at the surface. Thus, a small user controlled diffusion current, strongly influenced by surface recombination, may be detected in BAE EDMR.

Two limitations of BAE EDMR are that (1) it cannot provide quantitative information regarding defects and (2) optimal biasing currently relies on trial and error since no predictive models of BAE have been realized. Another weakness is that a model based on BAE EDMR is currently not feasible since spin-dependent recombination and dissociation rates depend upon the forward bias voltage $VF$. We hope to address these weaknesses with this model.

### C. Electron paramagnetic resonance and electrically detected magnetic resonance

In order to understand some of the results of this paper, a brief discussion of electron paramagnetic resonance (EPR) and EDMR is necessary. The following will briefly discuss the fundamentals of EPR. Consider the simple case of an unpaired electron spin residing at a defect, in which the electron is unperturbed by its local environment. Suppose that a sample with these paramagnetic defects is placed in a microwave cavity and that this cavity is between the pole faces of an electromagnet. The cavity is critically coupled with a microwave source, which generates microwave radiation with photon energy $E=h\nu $, where $h$ is Planck’s constant and $\nu $ is the microwave frequency. The electromagnet provides a large linearly swept field $B0$. The magnetic field splits the energy of the defect electron’s $+1/2$ and $\u22121/2$ spin states. This energy difference is given by $\Delta E=ge\mu BB0$. Here, $ge\u22482.0023\u2026\u2033$ is the Landé $g$ factor and $\mu B$ is the Bohr magneton. When the photon energy is equal to the energy difference between the $+1/2$ and $\u22121/2$ spin states, resonance occurs and the electron’s spin “flips.” This resonance condition is^{26,27}

Perturbations to the simple case caused by the defect electron’s environment provide key information for defect identification. Two of the most important perturbations are spin–orbit coupling and electron–nuclear hyperfine interactions. Spin–orbit coupling alters the Landé $g$ factor $ge$ into an orientation dependent number $g$, which is often expressed as a second-rank tensor $g$. Electron–nuclear hyperfine interactions are caused by nearby magnetic nuclei, which interact with the defect electron and cause additional splitting of the electron energy levels. These interactions can be described via a spin Hamiltonian of the form^{26,27}

Here, $B$ is the applied magnetic field vector, $g$ is a second rank tensor whose parameters depend upon the spin–orbit coupling interactions, $S$ is the electron spin angular momentum operator, $Ii$ is the nuclear spin angular momentum operator for the $i$th nucleus, and $Ai$ is the hyperfine coupling tensor for the $i$th nucleus. Other perturbations exist, such as dipolar and exchange interactions. However, these additional interactions need not be considered in this work.

Conventional EPR has a sensitivity of about $1011$ defects per mT linewidth.^{28} EPR sensitivity is also field and frequency-dependent and the detected response is influenced by every paramagnetic defect within the sample. These limitations are detrimental in studies of micro- and nano-scale electronics devices. EDMR overcomes these limitations. EDMR is about $107$ times more sensitive than classical EPR^{3} and has a sensitivity that is independent of the EDMR field and frequency. An additional advantage of EDMR in nano-and micro-electronic device studies is that EDMR is only sensitive to electrically active defects.

### D. Spin-dependent recombination

As previously mentioned, the method of EDMR detection we will be discussing is SDR, which can be can be understood through the seminal work of KSM^{2} and subsequent work refining their ideas.^{4–6} The following provides a simplified discussion of SDR. Consider a conduction electron that encounters an unpaired electron spin residing at a deep level defect. The conduction electron and trapped electron will couple with form an intermediate spin state. These spin states can either be singlet states ($S=0$) with basis state $S0=(|\u2191\u2193\u27e9\u2212|\u2193\u2191\u27e9)/2$ with $mS=0$ or triplet states ($S=1$) with basis states $T+=|\u2191\u2191\u27e9$, $T0=(|\u2191\u2193\u27e9+|\u2193\u2191\u27e9)/2$, and $T\u2212=|\u2193\u2193\u27e9$ with $mS=+1,0,$ and $\u22121$ respectively. For the triplet case, the conduction band to deep level transition will be forbidden by the Pauli exclusion principle, since both electrons will have the same spin quantum number. These triplet states will tend to dissociate and there will be no subsequent electron–hole recombination. However, at resonance, spin flipping events will transform triplet states into singlet states. The resonance-induced singlet state condition renders the transition allowed and permits subsequent electron–hole recombination. In SDR, the flipping of the trap center spins is generally observed.

## II. MODIFIED FITZGERALD-GROVE MODEL

In our modified model, a forward bias $VF$ is applied to the source and the current at the drain $ID$ is monitored. Thus, the drain is at a potential of $0$ $V$, while the source is at a potential of $VF$. Therefore, the approximation of quasi-equilibrium throughout the space charge region of (6) must be modified to account for the surface concentrations of electrons and holes along the length of the channel. In order to develop an accurate model for $ns$ and $ps$, we must account for the variation in carrier concentrations along the channel. Assume that the $x$- direction is along the length of the channel, whose length is defined as $LC$.

The MOSFET operation in BAE is similar to that of a gated bipolar-junction transistor in the forward active mode. We begin by writing down the transport equation. Here, we assume that the electron concentration varies only in the $x$-direction because the high electron concentration layer thickness is negligible compared to $LC$. This is a reasonable approximation and is used elsewhere for similar systems,^{21,29}

Here, $Dn$ is the diffusion coefficient for electrons at the surface, $\delta ns$ is the excess surface electron concentration, and $\tau $ is the lifetime of electrons. The solution here has the form

Here, $Ln$ is the electron diffusion length, defined as

The values of $\tau $ utilized to calculate $Ln$ is $30$ $\mu s$, which is consistent with lifetime measurements in Si.^{30}

The coefficients of (15) can be solved with the boundary conditions

The solution to this yields^{21}

Thus, the surface electron concentration is $ns(x)$ $=$ $\delta ns(x)+ns$, where $ns$ $=$ $ni2nAeq\varphi skT$ is the surface concentration far away from the source. With the boundary conditions of (17), (a) is the case at the drain terminal and (b) is the case at the source terminal. We may next utilize (7) to relate $\varphi s$ to $VG$. Note that when $LC\u226bLn$, $\delta ns(x)$ can be approximated by $\u221d$ $e\u2212xLn$. If $LC$ $\u226a$ $Ln$, a linear approximation may be utilized in place of (18). The linear approximation is detailed elsewhere.^{21} A plot of $ns(x)$ for both extremes, where $Ln$ $\u226a$ $LC$ and $Ln$ $\u226b$ $LC$, is shown in Figs. 4(a) and 4(b), respectively.

In order to account for the varying level of surface recombination across the channel caused by the difference in potential between the source and drain, we may write the surface recombination rate $US$ $=$ $US(\varphi s,VF,x)$ utilizing the boundary conditions of (6) modified with spatially dependent surface electron concentration of (18). In order to account for the change in recombination rate over the channel, (10) can be modified by utilization of^{31}

We utilize the recombination rate from (5) with modified boundary conditions from (6) and (18) and numerically solve for the drain current utilizing (19).

In Fig. 5, we plot $US(\varphi s,x)$ for a simulated $VF=0.6$ V. The color bar here represents the magnitude of $US(\varphi s,x)$ in units of cm$\u22122$ s$\u22121$. First note that recombination takes place closer to the source of the device (where $VF$ is applied) and at low enough band bending such that the average electron and hole carriers are about equal. It should be noted that at 0 along the $x$-direction (at the source), the rate is maximum at a specific $\varphi s$. If one were to multiply this rate by $qAG$, one would arrive at the dc-IV peak amplitude. This is so because at position of 0 along $x$ (at the source), the physics is the exact same as that derived by Fitzgerald–Grove.^{9} However, in BAE, we must account for the tapering-off of $US$ across the channel. A good approximation is that there is an average $US(\varphi s,VF)$ over channel position. Thus, we utilize integration to establish a mean value over the channel length such that we may find a $US$ related to only $\varphi s$ and $VF$. $\varphi s$ may then be related to $VG$ via (7).

## III. EXPERIMENTAL

In our study, we utilized Si MOSFETs with 7.5 nm thick SiO$2$ gate oxides. The samples were doped $nA\u22485\xd71016$ cm$\u22123$, in which $nA$ was extracted from CV measurements. The samples had gate areas $1\xd715$ $\mu $m$2$ with 126 transistors chained together. The channel length was 1 $\mu $m. The channel mobility utilized is about $300$ cm$2$ V$\u22121$ s$\u22121$, which was chosen as a limiting case for Si n-channel MOSFETs.^{32} The Si MOSFETs were stressed with high field stressing with $VG=\u22129.5$ V for $20$ min. The stressing created a high density of Si/SiO$2$ interface traps.

Our electrical measurements utilized a semiconductor parameter analyzer. The EDMR data was collected utilizing a modified commercial X-band system (9.6 GHz) with an electromagnet and power supply with a temperature-compensated Gaussmeter and Hall probe, a X-band microwave bridge, and a TE$102$ cavity. EDMR device currents were measured with a low-noise current preamplifier. Since EDMR current changes are on the order of $pA$ and our device currents are in the $nA$ range, lock-in detection and magnetic field modulation is utilized. A virtual lock-in amplifier was utilized and magnetic field modulation frequencies in the range of 1 KHz was utilized. Modulation amplitudes were about 0.1 mT.

For the EDMR data, our current measurements have an uncertainty of less than 0.1$%$, unless otherwise noted. The uncertainty is limited by signal to noise. For the X-band measurements, the uncertainty in g-value calculations is less than 0.0003. The absolute magnetic field was calibrated with a “strong pitch” spin standard. For the Si EDMR data, a backward and forward moving average filter was utilized. The signal to noise before filtering was about 40:1. The uncertainty in peak heights of the electrical curves is no greater than 3$%$.

## IV. RESULTS

Figure 6 shows IDVD measurements from the Si MOSFETs. It should be noted again that these samples were stressed with high field stressing, which is discussed in the Experimental section. The stressing induces a high density of Si/SiO$2$ interface defects.

In order to confirm defect density counts in the Si MOSFETs, charge pumping (CP)^{33–35} measurements have been performed (not shown). We extract a defect density of $1.9\xd71012$ cm$\u22122$ eV$\u22121$. The range of the bandgap explored with our charge pumping measurements is about 49%. This percentage is in agreement with the literature for charge pumping measurements in Si MOS at room temperature.^{36}

We also performed dc-IV measurements on the Si MOSFETs. The results are shown in Fig. 7. Here, the source/drain-to-body junctions were forward biased at $VF$ $=$ to $0.48$ V. We assumed a capture cross section of $4\xd710\u221216$ cm$2$.^{37} From the peak of the dc-IV curve, utilizing (2), we extract a defect density of $1.7\xd71012$ cm$\u22122$ eV$\u22121$.

We performed BAE electrical measurements by forward biasing the source junction and measuring current at the source/drain. We utilize $VF=0.48$ to $0.60$ V in increments of $0.04$ V. The results are shown in Fig. 8. Here, the solid lines represent the experimental BAE current, extracted from the difference between the magnitude of the source and drain current (this was required because the drain current rises steeply as threshold voltage is approached). The dashed line shows the model BAE current calculated from (19). Note the close correspondence between experimental and theory curve peaks and positions. As Fitzgerald and Grove explains in their original surface recombination theory, the difference in peak width can be explained by microscopic nonuniformities of the surface potential caused by a random distribution of surface charges.^{9} From the fit of the theory curves, a defect density of $1\xd71012$ cm$\u22122$ eV$\u22121$ was extracted. It should be noted that a window of $\u224812qVF$ of the middle of the bandgap is explored with both dc-IV and BAE.

We next compare the original Fitzgerald–Grove model to the modified model taking into account different quasi- Fermi levels on each side of the channel, resulting in a gradient of electron concentration across the channel. The results are plotted in Fig. 9.

It is clear from this plot that the modified model peaks are smaller than those predicted by Fitzgerald–Grove. This is because of the difference in quasi-Fermi levels at the source and drain. Assuming a linear carrier distribution along the channel (for $Ln$ $\u226b$ $LC$), which is usually the case for $Si$ MOSFETs, and assuming $qVF$ $\u226b$ $kT$, a good approximation for the peak amplitude of the BAE is given by

Here, $\Delta IB$ is the same as in (2) and is the body current measured during dc-IV utilizing the same biasing conditions. Through numerical integration of the simulated BAE peaks and comparing amplitudes to simulated dc-IV peak amplitudes, one arrives at a ratio of $\u22480.62$, which is fairly consistent with $VF$. Thus, the BAE peaks are $\u224823$ the size of the dc-IV peaks at the same conditions. Again, this is assuming that $qVF$ $\u226b$ $kT$ and, again, $Ln$ $\u226b$ $LC$. If the surface electron concentration gradient is not quite linear, the actual ratio will be smaller than $23$. Thus, $23$ is an approximate upper limit to $\Delta IBAE\Delta IB$.

Thus, the peak amplitudes depend upon the free carrier concentration gradient at the interface and this gradient alters both the effect of the surface potential on carrier concentrations and on the driving force toward equilibrium. It should be noted that in reference to Fig. 2, the point at which electron and hole densities are equal (1) in BAE is $\varphi s\u2248\varphi fp\u221212VF$, approximately the same as dc-IV.

For the case of Si, this is observed in comparing the two schemes at the same forward bias $VF$, as shown in Fig. 10. Note the close correspondence between the $trend$ of the data with bias between Figs. 9 and 10. A similar trend is observed for even larger biases. The dc-IV peaks are larger than $32$ the BAE peaks. The actual measured ratios are consistently between $0.5$ to $0.6$. Again, the discrepancy between the theoretically predicted ratios and experimentally observed is within the limits of our approximations. If the interface surface electron gradient strays from linear, the form of (18) would have to be utilized for the surface electron concentration and the actual ratio would be smaller than $23$.

In addition, as noted by Aichinger and Lenahan,^{13} BAE EDMR measurements are highly sensitive and highly selective to only the most important defects in MOS technology, namely, the semiconductor/oxide interface. In Fig. 11, we plot the BAE EDMR spectrum of the Si MOSFET. Here, the $g$ values measured are consistent with silicon dangling bonds known as $Pb0$ and $Pb1$ centers.^{38,39}

Thus, we have shown, as a proof-of-concept, that in the Si/SiO$2$ system, BAE can now be utilized as a quantitative tool to extract defect densities and provide defect structure.

## V. DISCUSSION AND CONCLUSIONS

A model has been developed for the BAE technique, in which proof-of-concept results are demonstrated on Si/SiO$2$ MOSFETs. Much like dc-IV, the densities of defects may be extracted from utilization of theory developed by Fiztgerald and Grove.^{9} However, integration of a spatially dependent recombination rate (19) extends the original Fitzgerald–Grove model to account for the difference in the quasi-Fermi levels at the source and drain.

The presented model accurately predicts the peaks for Si and differences between dc-IV and BAE peak intensities are also accounted for utilizing the modified model presented here. The extracted defect densities for Si are reasonable. The Si $Dit$ values are quite high in comparison with the literature.^{38} The high density of defects in the Si devices is the result of high field stressing. The CP measurements in Si measures about $49%$ of the bandgap and the BAE and dc-IV schemes measure $\u224812qVF$. The defect densities measured by all three schemes (dcIV, BAE, and CP) are reasonably consistent with one another.

Since the modified model presented here depends upon differences in potential from the source and drain of the MOSFET, the modified model may be utilized to develop a relatively accurate BAE EDMR/NZFMR model utilizing solutions to the SLE. The modified model should explain differences in BAE EDMR/NZFMR and dc-IV EDMR/NZFMR through predictions of differences in singlet/triplet dissociation rates observed in comparing the two measurement schemes. However, this is the subject of future experimental and modeling investigations.

Another potential utility of the quantitative BAE method is that it may also be utilized for estimating carrier lifetimes by performing both BAE and dc-IV in devices with varying channel lengths. This is because when the surface carrier concentration is linear ($Ln\u226bLC$), the ratio of the BAE peak intensity to dc-IV peak intensity $\Delta IBAE\Delta IB$ is constant ($\u224823$) across channel lengths well below $Ln$. However, when $LC$ approaches $Ln$, the ratio $\Delta IBAE\Delta IB$ begins to drop below $23$. At the point where this occurs, $Ln$ may be estimated.

We present a simple quantitative model for BAE, which may now be utilized to determine defect densities within the midgap range in MOSFETs. The quantitative BAE measurement is demonstrated in a proof-of-concept on Si MOS devices. The densities extracted are compared against more traditional schemes, such as CP and dc-IV. We find a close agreement between CP, dc-IV, and quantitative BAE. The model presented here thus extends BAE from an EDMR measurement, which is utilized to study defect structure, to a quantitative measurement. In addition, the model may potentially be utilized in BAE EDMR measurements to optimize biasing in order to maximize the SDR in MOS devices. Future work will consist of (1) developing models for BAE EDMR, in which the results from this work will be utilized in conjunction with solutions to the SLE, which have already proven useful in models of dc-IV SDR,^{7,8} and (2) performing BAE and dc-IV measurements in devices with varying channel lengths to determine $Ln$.

## ACKNOWLEDGMENTS

The authors would like to thank Fedor Sharov for assistance in technical discussions and sample work and David R. Hughart and Gaddi S. Haase of Sandia National Laboratories for providing the samples utilized in this work. This project was sponsored by the Department of Defense, Defense Threat Reduction Agency under Grant No. HDTRA1-18-0012. The content of the information does not necessarily reflect the position or the policy of the federal government, and no official endorsement should be inferred.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## REFERENCES

*2020 IEEE International Reliability Physics Symposium (IRPS)*(IEEE, 2020), pp. 1–4.