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PAPER • OPEN ACCESSElectron paramagnetic resonance study of silicon28 single crystal for realization of the kilogram
To cite this article: Shigeki Mizushima and Takahide Umeda 2022 Metrologia 59 025005

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Metrologia

Metrologia 59 (2022) 025005 (13pp)

https://doi.org/10.1088/1681-7575/ac5584

Electron paramagnetic resonance study ofsilicon-28 single crystal for realization of
the kilogram
Shigeki Mizushima1,2,∗
1
2

and Takahide Umeda1

Institute of Applied Physics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan
National Metrology Institute of Japan NMIJ, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8563, Japan

E-mail: s.mizushima@aist.go.jp
Received 7 December 2021, revised 11 February 2022
Accepted for publication 15 February 2022
Published 14 March 2022
Abstract

To realize the kilogram with high accuracy using the x-ray crystal density method, mass deficit
correction to account for the number of vacancy defects in silicon-28 crystals is necessary. In
this paper, we present the results of electron paramagnetic resonance (EPR) spectroscopy
experiments performed on a silicon-28 crystal. The crystal was cut from near the radial centerof the Si28–23Pr11 crystal boule where a relatively high vacancy defect concentration was
expected within the boule, which was grown by the floating zone method. We obtained boththe in-phase and out-of-phase EPR spectra, which focused on the fast- and slow-spin
relaxation centers, respectively. Based on EPR measurements both in the dark and underillumination, the concentrations of phosphorus impurities with unpaired electrons were foundto be between 1.6 × 1012 cm−3 and 3.4 × 1012 cm−3. In contrast, EPR-active vacancy defects
in the silicon-28 crystal were not detected at the detection sensitivity of the present
measurements (around 1 × 1012 cm−3 ), although at the Si/SiO2 interface, dangling-bonddefects (Pb0 centers) were detected with an areal density between 0.4 × 1012 cm−2 and1.7 × 1012 cm−2 . The experimental detection of a larger amount of phosphorus donors underillumination indicates the presence of compensation centers (electronic levels acting as
acceptors) with a concentration of at most 1.6 × 1012 cm−3 . We determined that such a
concentration is mainly compensated by the boron acceptors, and that there is little
contribution of electrically active vacancy defects, 0.0(6) × 1012 cm−3. This value is muchsmaller than the previously reported concentrations of 1 × 1014 cm−3 to 4 × 1014 cm−3 forvacancy defects determined by positron annihilation spectroscopy.
Keywords: kilogram, x-ray crystal density method, silicon-28, vacancy defects, electron
paramagnetic resonance(Some figures may appear in colour only in the online journal)∗

Author to whom any correspondence should be addressed.

Original content from this work may be used under the termsof the Creative Commons Attribution 4.0 licence. Any further
distribution of this work must maintain attribution to the author(s) and the titleof the work, journal citation and DOI.
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Metrologia 59 (2022) 025005

S Mizushima and T Umeda

1. Introduction

from more than 15 mm from the radial center of the crystal boule. Only the in-phase spectra at the microwave powerp = 2 mW were presented in those studies.
This paper presents the latest EPR results for a silicon-28
crystal. For the EPR measurements, a sample near the radial
center, where the concentration of vacancy defects in the crystal boule had the highest value, was used. In addition, theout-of-phase spectra, and in-phase absorption spectra at a low
microwave power are presented. These spectra are effective
for the detection of defects with long relaxation times. The
principle of EPR, EPR measurements in the dark and underillumination, the saturation phenomenon, out-of-phase spectra observation, and the sample preparation procedures aredescribed in section 2. The EPR measurement results of thisstudy are described in section 3. In section 4, the concentrationof vacancy defects is estimated based on the EPR observationsin the dark and under illumination, and previously reported
PAS results and the EPR results are summarized. Finally, thepaper is concluded in section 5.

The kilogram is the unit of mass in the International System ofUnits and had been defined based on the mass of an artifact,
the International Prototype of the Kilogram (IPK), since 1889.
To ensure the long-term stability of the unit of mass, the 26th
meeting of the General Conference on Weights and Measuresapproved the kilogram to be redefined by the realization experiments by fixing the value of the Planck constant h to 6.626 070
15 × 10−34 J s in November 2018 [1]. The redefinition tookeffect from May 20, 2019.
At present, there are two independent methods officiallyrecognized as primary methods by the Consultative Committee for Mass and Related Quantities (CCM) for realizing thedefinition of the kilogram with the highest accuracy [2]. In thefirst method, an electromechanical balance, such as a Kibble
balance or a joule balance, is used. The second method is the
x-ray crystal density (XRCD) method, in which the kilogramis realized by counting the number of atoms in a crystal.
To date, two international comparisons on the realization of
the kilogram by national metrology institutes and the Bureau
International des Poids et Mesures have been conducted. The
first is the CCM.R-kg-P1 pilot study conducted before theredefinition in which the consistency and continuity of the unitof the mass after the redefinition was confirmed [3]. The relative standard uncertainty of the weighted mean of the fiverealization experiments was 1.0 × 10−8. The second comparison is the CCM.M-K8.2019 key comparison carried out after
the redefinition in which the key comparison reference value(KCRV) was determined as the weighted mean of seven realization experiments [4]. The relative standard uncertainty of
the KCRV was 7.5 × 10−9. Along with the data directly traceable to the IPK, the results of these two international comparisons provided inputs for the calculation of the consensus value
for the international coordination of the dissemination of thekilogram [5].
The XRCD method, which uses silicon-28 crystals, hasbeen under development by the International Avogadro Coordination Project since 2004 [6]. To realize the kilogram using
the XRCD method, the number of atoms contained in a singlecrystal silicon-28 sphere with a diameter of approximately
93.7 mm is counted by measuring its volume and lattice constant. If the crystal contains vacancy defects, the estimatednumber of atoms counted from the volume and lattice constant will be larger than the actual number of atoms. Therefore,a mass deficit correction must be applied to account for thenumber of vacancy defects in the crystal in the XRCD method.
To date, the vacancy defect concentration in silicon crystals has been measured for the purpose of determining themass deficit correction with microgram-level uncertainty using
positron annihilation spectroscopy (PAS) [7, 8]. At present, theresult of PAS is adopted for the mass deficit correction in the
XRCD method.
To verify the reliability of this mass deficit correction,
vacancy defect concentration has been measured by anotherindependent method, electron paramagnetic resonance (EPR)[9, 10]. The samples used in the EPR studies were extracted

2. Measurement principle and sample preparation
In this section, the principle of EPR, EPR measurements inthe dark and under illumination, the saturation phenomenon,
out-of-phase spectra observation, and the sample preparation
procedures are described.
2.1. Principle of EPR

EPR is a spectroscopic tool for detecting unpaired electronspresent in a substance. When an external magnetic field isapplied to a sample that includes unpaired electrons, Zeeman splitting occurs in the spin eigenlevels of these electrons. Denoting the direction of the external magnetic fieldas the z direction, the magnetic interaction between themagnetic moment of an electron and the external magnetic
field is represented by the Hamiltonian H = gμB BSz . Hereg is a dimensionless constant called the gyromagnetic factor(g factor), which takes the value of ge = 2.002 319 . . . fora free electron, μB is the Bohr magneton, B is the magnitude of the external magnetic field, and Sz is the z componentof the electron spin operator. There are two allowed orientations for the electron spin, namely, parallel or antiparallel tothe external magnetic field. The orientation is specified by theelectron spin quantum number ms , which takes the value ofeither −1/2 or +1/2.
A magnetic-dipole transition between the Zeeman-split
eigenlevels occurs when the unpaired electrons are irradiatedby microwave radiation with an oscillating magnetic field perpendicular to the external magnetic field under the resonantcondition(1)hν = gμB B,
where h is the Planck constant and ν is the frequency ofthe microwave. This transition causes the absorption of the
microwave and is called EPR.
Continuous EPR absorption is observed, when the population of the lower energy eigenlevel (ms = −1/2) is larger than2

Metrologia 59 (2022) 025005

S Mizushima and T Umeda

ν was stabilized at 9.445 GHz, and the external magnetic field
B was swept from 333 mT to 342 mT. Copper(II) sulfate pentahydrate (CuSO4 · 5H2 O) with a mass of 12.53 mg was usedas the reference sample for quantitative determination.
To improve the detection sensitivity, a lock-in amplifier anda magnetic field modulation technique were employed in the
EPR spectrometer. In the magnetic field modulation technique,the differential change in the EPR absorption synchronizedwith the magnetic field modulation is measured to provide
the first derivative of the absorption spectra. In this study, themodulation frequency and amplitude were set to 100 kHz and
0.1 mT, respectively. The EPR absorption peak can be obtainedby integrating the measured EPR signal. The lock-in amplifier
typically monitors the in-phase EPR signals that are synchronized exactly with the modulation of the magnetic field. In theabsence of saturation phenomenon (section 2.3), the in-phase
EPR signal offers the maximum detection sensitivity. In addition to conventional in-phase measurements, we also adopt an
out-of-phase EPR detection for EPR signals under saturation,
as explained in the next subsection.
The EPR measurements were carried out at 20 K, unlessotherwise stated. Performing the measurements at lower
temperatures drastically increases the population difference
between the two spin levels and enhances the EPR absorption
intensities. Furthermore, the dielectric loss of the microwave
excitation is drastically reduced at lower temperatures, whichfurther improves the detection sensitivity. However, such temperatures can also increase the spin relaxation time of unpaired
electrons, which leads to the saturation phenomenon. The saturation phenomenon, which is discussed in the next subsection,needs to be considered in the measurements.
We performed EPR measurements both in the dark
and under band-gap illumination (100 W halogen whitelight). Under illumination, EPR-inactive defects (defects
without unpaired electrons) can be converted into EPRactive defects through the capture of photo-excited carriers.
However, because the photoinduced conversion rate foreach type of defect is unknown, the EPR absorption underillumination provides only the lower limit for the defectconcentration.
A quartz glass rod used for introducing the sample into
the microwave resonator was also used as a light guide forthe halogen light. The intensity of the halogen light was
adjusted to maximize the intensity of the EPR signals from thedefects.
Anisotropic and high-intensity cyclotron resonance signals[11] were obtained in the EPR measurements performed ataround 4 K in magnetic fields ranging from 52 mT to 185 mT.
This indicates that the illumination generated a large numberof electron–hole pairs in the silicon crystal in this study.

that of the higher energy eigenlevel (ms = +1/2). Such a population imbalance is maintained via spin–lattice relaxation, in
which the absorbed energy of the spin system is dissipated via
lattice vibrations (i.e., phonons).
In thermal equilibrium, the population ratio of the two
spin eigenlevels
 (ms = −1/2 and ms = +1/2) approaches
exp gμB B/kT due to the Boltzmann distribution, where kis the Boltzmann constant and T is the absolute temperature. This ratio is approximately given by (1 + gμB B/kT)
when the magnitude of the Zeeman splitting is sufficiently
small compared to the thermal energy (gμB B  kT). Thiscondition is satisfied in this study. Thus, if the total number of spins in a sample
 of spins in the is N, the number
two eigenstates are N/2 1 ± gμB B/2kT for ms = ∓1/2,
respectively.
The magnetic moment of each unpaired electron takes thevalue of ±gμB /2 for ms = ∓1/2. Therefore, the net magnetization of a sample with
 N unpaired  electronsin thermal equi
librium
is
given
by
+gμ
/2
N/2
1
+
gμB B/2kT +
B





−gμB /2 N/2 1 − gμB B/2kT = Ng2 μB 2 B/ (4kT).
The net magnetization of the sample is proportional to N andinversely proportional to T.
The number of EPR-active defects (i.e., defects with
unpaired electrons) in a sample can be quantitatively determined by comparing its EPR signal intensity with that ofa reference sample with a known number of unpaired electrons. In general, the EPR absorption intensity for each defect(area of EPR absorption peak) is proportional to the numberof unpaired electrons in the defect when the EPR measurements are carried out in the absence of saturation phenomenon(section 2.3). The number of a given type of defect X in thetest sample, nX , is given by  AX
TX
nX = f
(2)
nR ,
AR
TR
where nR is the number
 of unpaired electrons in the reference sample, AX /AR is the ratio of the observed EPR
absorption intensities in the test and reference samples,
TX /TR is the ratio of the absolute temperatures of the test
and reference samples, and f is a correction coefficient toaccount for the displacement of the sample position (or sample size) from the pin-point central position of the microwave
resonator used in the study. The defect concentration is determined in cm−3 when the above number is divided by the
sample volume.
In addition to the defect concentration, the types of defects
present are reflected in the measured g factors. The g factor of adefect in a semiconductor is generally dependent on the direction of the external magnetic field. The majority of defects insilicon exhibit anisotropic g factors that can be used to identify
the type of each individual defect.

2.3. Saturation phenomenon and out-of-phase EPR
observation

2.2. EPR measurements in the dark and under illumination

To observe EPR absorption continuously, spin relaxation processes are required to dissipate the Zeeman energy of theunpaired electrons as lattice vibrations. When the relaxation
processes are not sufficiently activated, the continuous EPR
absorption is decreased with increasing microwave power,

EPR measurements were performed using a Bruker E500 Xband spectrometer equipped with a super-sensitive microwave
ER 4122SHQ resonator (resonator Q value ∼ 12 000) and an
Oxford ESR900 He-flow cryostat. The microwave frequency
3

Metrologia 59 (2022) 025005

S Mizushima and T Umeda

and eventually approaches zero. This is called the saturation
phenomenon.
When saturation occurs, the line shape of the EPR absorption peak Y (B) is given by [12]Y (B) =

sY0,
1 + s(B − B0 )2 γ 2 T2 2

(3)

where s is the saturation coefficient, Y 0 is the amplitude of
the EPR absorption at the center of the absorption peak without saturation, B0 is the magnetic field at the center of theabsorption peak, γ is the electron magnetic rotation ratio(γ = ge μB /h̄), and T 2 is the spin–spin relaxation time. The
saturation coefficient s is given bys=

1,
1 + γ 2 B1 2 T1 T2

Figure 1. Crystal orientation of the sample and the direction of theapplied external magnetic field B. The dotted line shows the outlineof the sample. The external  magnetic field B was applied
perpendicular to the 011 crystal orientation. The angle θ is theangle between the external magnetic field B and the [100] crystal
orientation.

(4)

where B1 is the amplitude of the microwave magnetic field for
exciting unpaired electrons, and T 1 is the spin–lattice relaxation time, which is larger than T 2 . The saturation phenomenon
occurs when γ 2 B1 2 T1 T2  1 and s approaches zero.
Equation (3) implies that the amplitude of the EPR absorption peak is proportional to s, and its line width is proportionalto s−1/2. As the extent of saturation increases, the line shapeof the EPR absorption peak becomes increasingly broad, andthe peak amplitude increasingly weak until EPR absorption
finally disappears. Such situations may occur more frequently
at lower temperatures because the spin–lattice relaxation timeT 1 of every defect (and hence also T 2 ) becomes longer atlower temperatures, owing to the lack of phonons. Therefore,
at lower temperatures, some of the EPR defect signals aresuppressed because of the saturation phenomenon.
To detect such EPR signals with long relaxation times,
an out-of-phase detection technique is conventionally used inwhich the EPR spectra are measured with the lock-in-detection
phase shifted from 0◦ (in-phase) to 90◦ (out-of-phase) [13, 14].
The saturated signals are not synchronized with the magneticfield modulation and thus appear as delayed signals. This setup
is very useful for maximizing the delayed signals in the lockin-detection system. The simultaneous application of excess
microwave power and/or a rapid field modulation/sweep canfacilitate out-of-phase detection because the saturation phenomenon is enhanced in such setups. Accordingly, in thisEPR study, we adopted the so-called rapid passage method,in which a combination of out-of-phase (90◦ phase) detection, excess microwave power, and twice the field modulation
rate (100 kHz → 200 kHz) was used. This measurement technique can selectively focus on EPR signals with long relaxation times. When the EPR spectra are observed in this way,each EPR signal shows a lineshape similar to an absorption
peak.

crystal orientations, respectively. Figure 1 shows the arrangement of the crystal orientation of the sample and the directionof the external magnetic field B applied to the sample.
This sample was cut from crystal boule Si28–23Pr11,
which is a silicon-28 single crystal produced by PhysikalischTechnische Bundesanstalt (PTB) in 2015. Two 1 kg spheres,Si28kg01a and Si28kg01b, were fabricated from this crystal
and used for the determination of the Avogadro constant andthe realization experiments of the kilogram [15].
The sample used in this study was located 115 mm awayfrom the seed and 2.5 mm from the radial center of crystal boule Si28–23Pr11. The sample was located near theradial center of the crystal boule where the defect concentration would be higher than that in the surrounding areas if
vacancy defects exist in the boule [16]. Therefore, the sample
used in this study is advantageous for the detection of the small
number of vacancy defects in the crystal boule. Table 1 summarizes the locations of the silicon-28 crystal samples measured using EPR so far. The samples measured before thisstudy were located more than 15 mm away from the radial
center.
The sample surfaces were first mirror polished, and a layer
with a thickness of about 2 μm was removed by etchingwith tetramethylammonium hydroxide (TMAH) solution. Thiseliminated the signals from mechanical damage defects on thesurface (Si3 ≡ Si•, where the lines denote chemical bonds andthe dot denotes an unpaired electron) which would otherwise
overwhelm the signals from vacancy defects. After etching, thesample was placed in air at room temperature, and a naturaloxide film was formed on the sample surface.
The mass of the sample is 139.2413(10) mg, and the volume of the sample calculated based on the crystal density[15] is 0.060 016(4) cm3 . The geometric surface area of thesample calculated from its dimensions is 1.16(2) cm2 , which
consists of the surface area of 0.36(1) cm2 from the Si(100)surfaces and the surface area of 0.80(1) cm2 from the Si(110)
surfaces. To introduce the sample into the cavity resonatorof the EPR instrument, the sample was mounted on one endof a quartz glass rod with a length of 250 mm and diameterof 4 mm.

2.4. Sample preparation

The shape of the silicon-28 single-crystal sample used in thisstudy is almost the same as those used in previous studies[9, 10]. The sample is a solid cuboid with the dimensions of
1.8 mm × 3.4 mm × 10.0 mm. The 1.8 mm-long, 3.4 mm-long,and 10.0 mm-long edges are along the [011], [100], and 011
4

Metrologia 59 (2022) 025005

S Mizushima and T UmedaaTable 1. The location of the samples in the crystal boules measured by EPR so far .

Crystal name

Sample region

Axial distance from seed/mm

Radial distance from center/mm

Reference

Si28–10Pr11
Si28–10Pr11
Si28–23Pr11
Si28–23Pr11

9.R1
9.R1
M.R2
M.R2

∼420
∼420
∼115
∼115

35.0
15.3
18.8
2.5

[9]
[10]
[10]
This studya
The Si28–10Pr11 crystal is also called the AVO28 crystal. Two 1 kg spheres, AVO28-S5c and AVO28-S8c, were produced from this crystal [17]. Two 1 kg
spheres, Si28kg01a and Si28kg01b, were produced from the Si28–23Pr11 crystal [15].

3. Experimental results

value when the dangling bond orbital is parallel to (perpendicular to) the external magnetic field. Figure 3 shows theangular dependence of the Pb0 center. The points indicate thepositions obtained from the out-of-phase spectra under illumination. These positions are in good agreement with the lines
simulated from the above principal g factors.
The areal density of dangling bonds with unpaired electrons (Pb0 centers) under illumination was evaluated to be
0.9(1) × 1012 cm−2 based on the absorption intensity in thein-phase spectra. Because the Pb0 centers exhibited more saturation behavior than the phosphorus donors, they appearedstrongly in the out-of-phase spectra.

EPR spectra were obtained under various conditions to detectvacancy defects in the silicon-28 crystal sample. We measured(i) the in-phase and out-of-phase spectra under illumination,(ii) the in-phase and out-of-phase spectra in the dark, and (iii)
the in-phase spectra at low microwave power, as described in
the following three subsections.
3.1. In-phase and out-of-phase EPR spectra underillumination

To search for vacancy defects in the sample, the in-phase andout-of-phase EPR spectra were obtained under illumination,
as shown in figure 2. Measurements were carried out at the
microwave frequency of ν = 9.445 GHz with the microwave
power of p = 2 mW. The out-of-phase EPR spectra can reveal
defects that are difficult to detect from the in-phase spectra
because of the saturation phenomenon.
To identify the type of defects, the EPR measurements
were performed at the four different angles of θ = 0◦ , 30◦ ,60◦ , and 90◦. Each spectrum was obtained as the average of
20 scans performed over 30 min. The rotation of the sample
resulted in slight fluctuations in the Q factor of the microwave
resonator and therefore, fluctuations in the signal intensities,
as shown in figure 2. The highest Q factor was obtained at θ
= 0◦ , at which the highest detection sensitivity was obtained.
Therefore, we quantified the impurity and defect concentrations at θ = 0◦ .
Two isotropic signals with the same intensity were observedin the first-derivative in-phase EPR spectra. Their mean position corresponds to the g factor of 1.9989, and their splitting
width is 4.2 mT. The signals are consistent with a doublet signal of the phosphorus donors (Si4 == P•, where ‘•’ denotes
an unpaired electron) in silicon [18]. The signal splitting is
attributed to the hyperfine interaction between an unpaired
electron and a 31 P nuclear spin (I = 1/2). Based on the absorption intensity, the concentration of phosphorus impurities with
unpaired electrons was determined to be 3.4(5) × 1012 cm−3
under illumination.
In addition, anisotropic signals with g factors ranging from2.0021 to 2.0095 were observed. These angular-dependent
signals arose from dangling bonds at the Si/SiO2 interface(Si3 ≡ Si•, the so-called Pb0 centers). This identification isbased on the similarity of the observed g factors to those of
the Pb0 centers that predominantly form at the Si/SiO2 interface at low oxidation temperatures (principal g factors of
gZ = g = 2.0018 and gX = gY = g⊥ = 2.0082 [19, 20]). The
observed g factor of the Pb0 center corresponds to the g (g⊥ )3.2. In-phase and out-of-phase EPR spectra in the dark

Following the previous section, the in-phase and out-of-phase
EPR spectra were obtained in the dark. The purpose of thesemeasurements is to quantitatively compare the EPR spectra
obtained under illumination and in the dark to reveal the presence of defects in the sample that are otherwise invisible. In theprevious section, we described the observation of EPR signals
from phosphorus donors. If the sample includes compensation
defects that can receive electrons from donors, the donor signal
should be reduced in the dark because of the loss of unpaired
electrons from the donors. The differential change in the donor
signal indicates the amount of compensation defects (deeplevel defects and acceptors). Because vacancy-type defects
generally have deep energy levels, they act as compensation
defects.
Figure 4 illustrates the electronic band structures in thedark (i.e., in thermal equilibrium) at cryogenic temperatures.
The sample in this study includes phosphorus impurities
that act as donors, as well as boron impurities and vacancy
defects that act as acceptors. Because the phosphorus-donor
level is very close to the conduction band edge (0.046 eV
below the edge), nearly all the defect levels in the band gap act
as acceptors when there is a sufficient amount of phosphorus
donors.
In figure 4(a), where nD > nA , phosphorus donors that havelost an electron (indicated by an open circle) could not be
observed by EPR because of the lack of an unpaired electron.
Only (nD − nA ) of the phosphorus donors were observable. In
contrast, in figure 4(b), where nD  nA , the phosphorus donors
were not observed by EPR because all the phosphorus donors
have lost an electron.
The measurements were carried out at the microwave frequency of ν = 9.445 GHz with the microwave power of
p = 2 mW. To identify the type of defects, measurements were5

Metrologia 59 (2022) 025005

S Mizushima and T Umeda

Figure 2. In-phase and out-of-phase EPR spectra under illumination. θ is the angle between the external magnetic field and the [100]crystalline axis. For each out-of-phase spectrum, a background spectrum from the quartz glass components of the EPR instrument was
subtracted.

taken at the four different angles of θ = 0◦ , 30◦ , 60◦ , and 90◦ .
The measurement results are shown in figure 5. Each spectrum
is the average of 20 scans performed over 30 min.
In the in-phase spectra, two isotropic signals from thephosphorus donors split by the hyperfine interaction wereobserved, similar to the measurements under illumination.
Based on the intensity of these signals, the concentration ofphosphorus impurities with unpaired electrons in the dark
was determined to be 2.3(3) × 1012 cm−3 . This result clearly
demonstrates that the present sample satisfied the case shownin figure 4(a); namely, the concentration of phosphorus impurities acting as donors was larger than the sum of the concentrations of boron acceptors and vacancy defects acting as
acceptors.

In addition to the phosphorus donors, weak EPR signals
from the Pb0 centers were observed, similar to the measurements under illumination. Based on the signal intensity, thedangling-bond areal density at the Si/SiO2 interface in the dark
was evaluated to be 0.4(1) × 1012 cm−2.
3.3. In-phase EPR spectra with low microwave power

In-phase EPR spectra were also obtained at the microwave
frequency of ν = 9.445 GHz with the low microwave powerof p = 0.002 mW under illumination and in the dark toinvestigate the defects in the sample. Under this conditionof a small microwave field amplitude B1 , the magnitude ofγ 2 B1 2 T1 T2 was sufficiently small to avoid the saturation
phenomenon, even when the relaxation times of the6Metrologia 59 (2022) 025005

S Mizushima and T Umeda

Figure 3. EPR signal positions of the dangling bonds at the Si/SiO2 interface (Pb0 centers). The points denote the positions obtained fromthe out-of-phase spectra under illumination. The lines are simulations with the principal g factors of gZ = g = 2.0024 andgX = gY = g⊥ = 2.0094. The angle θ on the vertical axis is the angle between the external magnetic field B and the [100] crystal orientation.
To convert the g factor to the magnitude of the external magnetic field B, the microwave frequency of ν = 9.445 GHz was used. Owing to therotational symmetry of the silicon crystal, up to 12 absorption peaks may appear depending on the direction of the external magnetic field.

Figure 4. Electronic band structures of silicon in the dark at cryogenic temperatures with (a) nD > nA and (b) nD  nA , where nD is thedonor concentration and nA is the acceptor concentration. The gray circles indicate electrons. The electrons occupy the energy levels fromthe lowest level to the upper levels in sequence in the dark at cryogenic temperatures. Because the centers observable by EPR must have
unpaired electrons, phosphorus donors are partially EPR-active in case (a), and all of them are EPR-inactive in case (b) in the dark at
cryogenic temperatures.

as shown in table 2(a). This is because these EPR centers werepartially compensated (lost unpaired electrons) in the darkboth in the bulk and at the surfaces. In contrast, the EPR centers were converted into EPR-active states under illumination
when they received unpaired electrons via the photoexcitationof a large number of electron–hole pairs in the silicon crystal.
As a result, the phosphorus donors underwent the transition
Si4 == P + • → Si4 == P• (‘•’ denotes an unpaired electron)in the bulk, and the Pb0 centers underwent the transition Si3≡ Si + • → Si3 ≡ Si• at the Si/SiO2 interfaces on the sample
surface.
A clear signature of the saturation behavior was found for
the Pb0 centers. Their concentration was reduced to approximately half under p = 2 mW, as shown in table 2(b). This
indicates that the deep-level defects, such as Pb0 centers, exhibited relatively long relaxation times. By analogy with the Pb0

unpaired electrons localized at defects became longer at
low temperatures.
The measurement results are shown in figure 6. The measurements were performed at θ = 0◦ . Each spectrum is theaverage of 500 scans performed over a 12 h period. The measurements at low microwave power required a larger numberof scans to improve the signal-to-noise ratio. Similar to themeasurements under p = 2 mW shown in the previous subsections, only phosphorus donors and Pb0 centers were detected(figure 6).
Tables 2(a) and (b) summarize the measurement results ofthe phosphorus donor concentration and dangling bond areal
density at the Si/SiO2 interface based on the in-phase EPR
spectra shown in subsections 3.1–3.3.
The phosphorus donor concentration and the dangling bondareal density were higher under illumination than in the dark,
7

Metrologia 59 (2022) 025005

S Mizushima and T Umeda

Figure 5. In-phase and out-of-phase EPR spectra in the dark. θ is the angle between the external magnetic field and the [100] crystalline
axis. For each out-of-phase spectrum, the background spectrum from the quartz glass components of the EPR instrument was subtracted. To
quantify the concentrations of impurities and defects, the in-phase spectrum measured at θ = 0◦ was used.

centers, the vacancy defects may also have long relaxation
times. However, the EPR measurements that focused on suchslow-relaxation centers (the out-of-phase spectra shown infigures 2 and 5, and the low-power EPR spectra shown infigure 6) did not find any EPR signals from the nine typesof EPR-active vacancy defects, namely, V + , V − , V 2 + , V 2 − ,V 3 − , V 4 − , V 5 − , (VO)− , and (VP)0 [21]. Therefore, we concluded that the concentration of each of these defects was
below the detection sensitivity, which was estimated to beapproximately 1 × 1012 cm−3 , similar to previous EPR studies[9, 10].

4. Discussion
4.1. Concentration of electrically-active vacancy defects

The EPR spectra obtained under various conditions, namely,
the (i) in-phase and out-of-phase spectra under illumination,(ii) in-phase and out-of-phase spectra in the dark, and (iii)
in-phase spectra at a low microwave power, were describedin the previous section. In every condition, only EPR signals
from the phosphorous donors in the bulk and the Pb0 centersat the Si/SiO2 interface were detected. No other signals above
the detection sensitivity of this study (around 1 × 1012 cm−3 ;8Metrologia 59 (2022) 025005

S Mizushima and T Umeda

Figure 6. In-phase EPR spectra under illumination and in the dark obtained at low microwave power. The angle between the external

magnetic field and the [100] crystalline axis was set to θ = 0◦ . A reference EPR signal with an electron spin concentration of
1 × 1012 cm−3 and a half width at half maximum of 0.1 mT is shown.
Table 2. (a) Concentration of observed EPR-active phosphorus donors. (b) Areal density ofobserved EPR-active dangling bonds (Pb0 centers).

(a)Subsection

Phosphorus donor
concentration/(1012 cm−3 )Observation conditions

3.1
3.2
3.3

p = 2 mW, under illumination, in-phasep = 2 mW, in the dark, in-phasep = 0.002 mW, under illumination, in-phase

3.3

p = 0.002 mW, in the dark, in-phase
3.4(5)
2.3(3)
3.2(4)
1.6(2)

Difference
between underilluminationand in thedark/(1012 cm−3 )1.1(6)
1.6(5)(b)
Subsection
3.1
3.2
3.3
3.3

Observation conditions

Dangling bond areal density/(1012 cm−2 )p = 2 mW, under illumination, in-phasep = 2 mW, in the dark, in-phasep = 0.002 mW, under illumination, in-phasep = 0.002 mW, in the dark, in-phase0.9(1)
0.4(1)
1.7(2)
0.8(1)

see reference EPR signal with a spin concentration of
1 × 1012 cm−3 in figure 6) were detected. We confirmedthat there was no detectable vacancy defect signals in thesample, including those from the nine types of EPR-active
vacancy defects [V + , V − , V 2 + , V 2 − , V 3 − , V 4 − , V 5 − , (VO)− ,and (VP)0 ]. These defects exhibit anisotropic EPR signals

with principal g factors ranging from 1.9989 to 2.0151, asestablished in previous EPR studies [21].
Here, we focus on the bulk region. The concentration of phosphorus donors observed under illumination was
higher than that in the dark. The former was found to be
between 3.2(4) × 1012 cm−3 and 3.4(5) × 1012 cm−3 , as9

Metrologia 59 (2022) 025005

S Mizushima and T Umeda

Figure 7. Summary of concentrations of phosphorus donors, boron acceptors, and all other electronic levels including vacancy defects in thesample. The gray circles indicate electrons. The values of the bandgap and the phosphorus donor and boron acceptor energies were takenfrom reference [22]. FTIR values were taken from reference [15].

Table 3. Summary of the measured vacancy defect concentrations.

Measurement
methodPAS
PAS
EPR

Crystal

Vacancy defectconcentration/(1014 cm−3 )Corresponding mass
deficit fora 1 kg sphere/μg

Reference

Isotopically natural silicon FZ crystal
Silicon-28 FZ crystal
Silicon-28 FZ crystal

1–4, quantified as monovacancy defects
3.3(1.1), quantified as monovacancy defects
0.000(6)2–8
6.6(2.2)
0.0(1)[7][8]
This study

shown in table 2(a), which agrees well with the value of4.0(1.0) × 1012 cm−3 measured using Fourier-transform
infrared spectroscopy (FTIR) on sample M.2 close to thesample of this study [15] within their combined uncertainties.
The difference in the phosphorus donor concentration
under illumination and in the dark was estimated to be
1.1(6) × 1012 cm−3 to 1.6(5) × 1012 cm−3 (see table 2(a)).
This difference reflects the total amount of electronic levelsin the band gap that were deeper than the phosphorus donor
level (0.046 eV below the conduction band edge) and hence
acted as acceptors. Therefore, the concentration of electrically active vacancy defects with energy levels in the band
gap was determined to be at most the differential concentration(1.6 × 1012 cm−3 ) indicated above. In addition, the concentration of boron impurities is included in this upper limit.
As mentioned above, sample M.2 from crystal boule
Si28–23Pr11, which was located close to the sample in thisstudy, was measured using FTIR [15]. The boron impurity
concentration was measured to be 3.9(5) × 1012 cm−3 .
Because this concentration is larger than the upper limit of
1.6 × 1012 cm−3 obtained in our measurements, we canconclude that our result is nearly fully due to boron impurities. Consequently, we estimate the contribution of electrically
active vacancy defects is zero, i.e., 0.0 × 1012 cm−3 .
We then estimate its standard uncertainty to be
0.6 × 1012 cm−3, assuming a uniform distribution witha half-width of 1 × 1012 cm−3 , the magnitude of which isthe detection sensitivity of this study. We emphasize that
this estimation is consistent with the absence of any vacancy
defect signals, as noted earlier.
Figure 7 shows a summary of the concentrations of phosphorus donors, boron acceptors, and all other electronic levels,
including vacancy defects in the sample, based on observations
under illumination and in the dark at the microwave power of
p = 0.002 mW. The phosphorus donors were partially compensated (lost their electrons) by electronic levels deeper thanthe phosphorus donor levels. The concentration of such deeplevels was at most 1.6 × 1012 cm−3. Most of these levels
were contributed by boron acceptors (estimated to be nearly
1.6 × 1012 cm−3 ). Finally, the concentration of electrically
active vacancy defects was estimated to be smaller than
0.0(6) × 1012 cm−3 .

10Metrologia 59 (2022) 025005

S Mizushima and T Umeda

4.2. Summary of PAS and EPR results

5. Conclusion

In the realization of the kilogram by the XRCD method, thevacancy defect concentration measured by PAS is currently
employed for the mass deficit correction. To verify the reliability of this mass defect correction, this study used EPR,
an independent method different from PAS, to measure thevacancy defect concentration in silicon-28 single crystal.
Table 3 summarizes the measurement results of the vacancy
defect concentration by PAS and EPR for the realization of thekilogram by the XRCD method. In these measurements, silicon crystals grown using the floating zone (FZ) method were
investigated.
In the PAS measurements, the average positron lifetime
exceeded the bulk lifetime by only a few picoseconds, and thelifetime spectrum of the sample could not be decomposed consistently. Therefore, the type of vacancy defect has not been
identified, and its concentration has not been directly determined. Details are given in the appendix of this study. The
vacancy defect concentration, estimated from the increase in
the average positron lifetime at cryogenic temperatures, wasat the 1014 cm−3 level in monovacancy equivalent.
In contrast, the EPR measurements in this study did notdetect any vacancy defects, including the nine types of EPRactive vacancy defects. In addition, EPR signals from phosphorus impurities that acted as donors at a cryogenic temperature in the dark were observed, which indicates that theconcentration of electrically active vacancy defects in the sample was 0.0(6) × 1012 cm−3 , as described in the previous subsection. The mass deficit correction for a 1 kg silicon sphere
was evaluated to be 0.0(1) μg.
The number of atoms contained in a single-crystal silicon28 sphere is counted by measuring its volume and lattice constant to realize the kilogram in the XRCD method,where the mass deficit corresponding to the vacancy defectconcentration is subtracted. Therefore, if the EPR result of
this study is adopted instead of the PAS result, the values
reported from PTB and National Metrology Institute of Japan(NMIJ) using the XRCD method in the CCM.R-kg-P1 pilot
study and CCM.M-K8.2019 key comparison would be corrected by +6.6 μg. The magnitude of this proposed correction is small compared to the uncertainties reported from PTB
and NMIJ. In addition, their values after the correction remain
consistent with the reference values in the two international
comparisons.
One of the possible causes for the discrepancy between
the PAS and EPR results could be that the vacancy defectconcentration determined using EPR was an underestimatedvalue. The properties of vacancy aggregates up to heptavacancy V 7 in silicon crystals have been studied using molecular dynamics simulations and molecular orbital methods [23].
The hexavacancy V 6 , which has a ring configuration of vacancies, is electrically inactive because it has no deep levels inthe band gap. If such ring-hexavacancy V 6 defects exist in
silicon crystals, it is highly likely that they cannot be detectedby EPR.

EPR spectra were observed under various conditions on a sample cut from the Si28–23Pr11 crystal boule, which was produced for experiments to realize the kilogram. A part close to
the radial center of the crystal boule, which has a high concentration of vacancy defects if they exist, was chosen for the
sample. We obtained the (i) in-phase and out-of-phase spectra
under illumination, (ii) in-phase and out-of-phase spectra in
the dark, and (iii) in-phase spectra at low microwave power.
We draw the following conclusions from our measurement
results:(a) In the out-of-phase spectra and in-phase spectra at low
microwave power, the signals from the dangling bonds at
the Si/SiO2 interface (Pb0 centers), which showed saturation behavior at low temperatures, were observed withhigher intensities. This indicates that even defects withlong relaxation times can be detected with high sensitivity
by measuring the out-of-phase spectra or in-phase spectra
at low microwave power.
(b) EPR measurements were performed on a silicon-28 crystal. These measurements included measurements of theout-of-phase and in-phase spectra at low microwave
power, which are useful for detecting defects with longrelaxation times. However, only signals from phosphorus
impurities in the sample and dangling bonds at the Si/SiO2
interface (Pb0 centers) were detected. Anisotropic signals from vacancy defects were not observed. Therefore,
we conclude that the concentration of each of the nine
types of EPR-active vacancy defects in the sample was
below the detection sensitivity of our EPR measurement(approximately 1 × 1012 cm−3 ).
(c) EPR signals from the phosphorus donors were observedin the dark at cryogenic temperatures, indicating thatthe concentration of phosphorus donors was higher than
the total concentration of boron impurities and vacancy
defects acting as acceptors. The concentration of thecompensated phosphorus donors was estimated to beapproximately 1.6 × 1012 cm−3. Because most of thecompensation was contributed by boron acceptors, theconcentration of electrically active vacancy defects wasestimated to be 0.0(6) × 1012 cm−3 . The mass deficit
correction for a 1 kg silicon sphere was evaluated to be
0.0(1) μg.
(d) Previously reported PAS results and EPR results in thisstudy for the vacancy defect concentration were summarized. One of the possible causes for the discrepancy
between the PAS and EPR results is that the PAS measurement did not identify the type of vacancy defect anddid not directly determine the vacancy defect concentration. Another possible cause is the presence of electricallyinactive V 6 hexavacancy defects, which have a ring configuration of vacancies and are unlikely to be detected by
EPR.

11

Metrologia 59 (2022) 025005

S Mizushima and T Umeda

Acknowledgments

trapping coefficient varies with the type and charge stateof the defects. In particular, it was reported that the trapping coefficient of negatively charged defects is more thanten times larger than that of neutral defects [25].
(b) A computational study based on the superposition of thefree-atomic densities and the three-dimensional solutionof the positron Schrödinger equation showed that thepositron lifetime in vacancy defects increases as a function of the vacancy size and rapidly saturates to the limit of
500 ps. The positron lifetime does not depend linearly on
the vacancy defect size [26]. Therefore, if the PAS spectra are not decomposed and the size of the vacancy defects
cannot be determined, the correction for the mass deficit
cannot be calculated accurately in the XRCD method, inwhich the correction is given by the product of the sizeand concentration of the vacancy defects.
(c) First-principles calculations show that negative ions insemiconductors have a large positron trapping rate at low
temperatures that is comparable to the positron trapping
rate of vacancy defects [27]. In general, FZ silicon crystals contain non-negligible concentrations of boron impurities that are negatively charged at low temperatures with
an electron density that is smaller than that of silicon
atoms, that is, ρ B− /ρ (Si) = 6/14. In the Si28–23Pr11
silicon-28 crystal, the concentration of boron impurities
measured using FTIR ranged from 1.7 × 1012 cm−3 to
9.8 × 1012 cm−3 [15]. Therefore, at least a part of theincrease in the average positron lifetime τ vac with decreasing temperature can be attributed to positron trapping byboron impurities. The increase in the average positron
lifetime does not necessarily indicate the presence of
vacancy defects.

This work was supported by JSPS KAKENHI Grant No.
17K05112. Some of the sample preparation processes werecarried out in the Clean Room for Analog–Digital Superconductivity (CRAVITY) at the National Institute of Advanced
Industrial Science and Technology (AIST) and MicroNano
Open Innovation Center (MNOIC). The silicon-28 crystal cut
from crystal boule Si28–23Pr11 was provided by the PTB
based on an international cooperation between PTB and NMIJ
in realization of the kilogram.
Appendix.
The measurement and analysis of the vacancy defect concentration in FZ silicon crystals using PAS was described in detailby Gebauer et al [7]. In that study, the average positron lifetime τ av was observed because the lifetime spectra could not
be consistently decomposed in a sample in which the average positron lifetime exceeded the bulk lifetime by only afew picoseconds. The average positron lifetime τ av increased
by approximately 1.5 ps when the temperature was decreased
from 300 K to 20 K. This increase in the average positron
lifetime was also detected after the sample was annealed at
500 ◦ C for 1 h. Therefore, it was concluded that the positron
trapping could not be explained by simple point defects(V, V 2 , VO, and V 2 O) with low thermal stability. Because thecarrier concentration in the high-resistivity FZ crystal was only
on the order of 1012 cm−3 , it was deduced that the detectedvacancies with a concentration of 1014 cm−3 were electricallyneutral. The PAS results were hence attributed to electricallyneutral vacancy aggregates such as V 6 defects. In addition, thepositron lifetime in an as-grown FZ-Si crystal was comparedwith that of the electron-irradiated crystals using the estimatedmonovacancy introduction rate of Σ = 0.1 cm−1 . It was concluded that the concentration of the monovacancy defects inthe as-grown FZ-Si crystal would be between 1 × 1014 cm−3
and 4 × 1014 cm−3 .
The trapping rate of positrons into a specific type of vacancy
defect κvac , the trapping coefficient specific to these vacancy
defects μvac , and the concentration of the defects cvac are
related byκvac =

1 (τav − τbulk )= μvac cvac ,τbulk (τvac − τav )ORCID iDs
https://orcid.org/0000-0003-1276Shigeki Mizushima
1328
Takahide Umeda https://orcid.org/0000-0002-2584-4782
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