PHYSICAL REVIEW B 102, 054427 (2020)

Gadolinium-implanted GaN studied by spin-polarized positron annihilation spectroscopy
M. Maekawa , A. Miyashita , S. Sakai , and A. Kawasuso*

National Institutes for Quantum and Radiological Science and Technology, 1233, Watanuki, Takasaki, Gunma 370-1292, Japan(Received 25 March 2020; revised 23 July 2020; accepted 6 August 2020; published 19 August 2020)
In this study, Gd ion implantation and annealing were performed at 900◦ C for nominally undoped wurtzite
GaN grown by metal-organic chemical vapor deposition. Spin-polarized positron annihilation measurements
showed that vacancy clusters including at least 12 vacancies per cluster were the major positron-trapping centers
and that the electrons in the vacancy clusters were spin-polarized. These observations could be explained byfirst-principles calculations. The previous speculation about the defect-assisted ferromagnetism of Gd-implanted
GaN may be supported if vacancy clusters are considered.
DOI: 10.1103/PhysRevB.102.054427

I. INTRODUCTION

The synthesis of dilute magnetic semiconductors is a key
technology toward semiconductor spintronics. A successful
dilute magnetic semiconductor is Mn-doped p-type GaAs and
InAs with a Curie temperature of 180 K [1–3]. Furthermore,
it was predicted theoretically that doping Mn into GaN andZnO will result in a high Curie temperature (i.e., above 300 K)
due to a high hole concentration of 1020 cm−3 [4]. However,Mn-doping into GaN seems to be very problematic due tothe formation of secondary phases, and carrier-mediated ferromagnetism by this method remains controversial.
Meanwhile, in 2002, Teraguchi et al. [5] demonstrated theroom-temperature ferromagnetism of Gd-doped GaN grownby molecular beam epitaxy, showing that the secondary phaseformation is suppressed well. Thereafter, several groups reported not only room-temperature ferromagnetism, but also
colossal ferromagnetism in Gd-doped GaN during crystal
growth [6–8] and Gd-implanted GaN [9–15]. Here, “colossal”
ferromagnetism means that the magnetization per an introduced Gd atom far exceeds 7 μB , which is the intrinsic magnetization of a Gd atom. From comparisons among implanted
elements, the presence of Gd atoms is thought to be crucial
for the appearance of ferromagnetism [14].
In the above studies, hysteresis loops observed by superconducting quantum interference device (SQUID) magnetometry were considered to be the evidence of ferromagnetism.
However, the SQUID magnetometry creates serious artifacts
via the pinning of magnetic flux [16,17]. To prove the existence of ferromagnetism, some methods with elemental specificity were required. The hard x-ray absorption spectroscopy
studies showed that Gd atoms are only paramagnetic and thespin polarization of Ga atoms is almost null [18–20]. Therefore, some defect species and oxygen impurities are thought togive rise to the magnetic order. Then, it has been proposed thatsingle Ga vacancies and Ga vacancy-oxygen complexes are
the source of ferromagnetism [21–23]: an electrically neutral*

Corresponding author: kawasuso.atsuo@qst.go.jp

2469-9950/2020/102(5)/054427(8)Ga vacancy has a magnetic moment of 3 μB , and multiple
moments can be coupled ferromagnetically with those of Gd
atoms [24,25]. It is thought that only positron annihilation
spectroscopy (PAS) can confirm such a hypothesis. However,a previous study in 2011 implied that there was only a poorcorrelation between the ferromagnetism of Gd-doped GaN
and single Ga vacancies [26], but since then there have been
no further investigations in that direction.
To confirm the existence of vacancy-assisted magnetism,
it is important to detect the spin polarization associated withvacancies. Using spin-polarized PAS (SP-PAS), we showed
previously that spin-polarized cation vacancies may assist theferromagnetism of oxygen-irradiated ZnO [27]. In this study,
we used SP-PAS to investigate Gd-implanted GaN. We foundthat instead of single Ga vacancies, vacancy clusters including
at least 12 vacancies per cluster form during post-implantation
annealing, and the electrons therein are spin-polarized.
II. EXPERIMENTThe samples were cut from a nominally undoped
wurtzite GaN(0001) film (2-μm thick) that was grownby metal-organic chemical vapor deposition (MOCVD) on
Al2 O3 (0001) and purchased from POWDEC K.K. These were
implanted with 370-keV Gd ions to the maximum dose (φ)
of 1016 Gd+ /cm2 . Using the SRIM code [28], the Gd concentration at the mean range (∼70 nm) was simulated to beapproximately φ × 1.5 × 105 cm−1 (i.e., 1.5 × 1020 Gd/cm3
for φ = 1015 Gd+ /cm2 ). After implantation, annealing was
performed in a N2 atmosphere for 30 min at 900◦ C, which has
been reported to be the optimum annealing temperature [10].
X-ray diffraction (XRD) structural analysis was performed inthe θ -2θ orientation using a RIGAKU SmartLab diffractometer without Ge(220) monochromators. Magnetization measurements were performed using a Quantum Design MPMS
SQUID apparatus from 10 to 300 K. In each M–H curve, thediamagnetic linear background was subtracted. M–T curves
were obtained in both zero-field and field-cooling conditions
at 200 Oe. Conventional PAS based on the Doppler broadening of annihilation radiation (DBAR) method was performed

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unimpl.

unimpl.

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V6 [3 (VGaVN)]

Bulk

15

as-impl. 1x10+

Gd /cm2

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V8 [4(VGaVN)]Ga

V12 [6

S parameter

0.5(VGaVN)]

N

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0.55

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as-impl. 1x10

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0.55

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+
2
as-impl. 1x10 Gd /cm

900oC ann.

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900oC ann.

0.5

FIG. 1. Schematic representations of vacancy clusters considered in the calculation. The filled circles represent Ga and N atoms.
The open circles represent vacancies.

0.45
0.55

as-impl.

0.5

with incident positron energy (E+ ) of 2–15 keV at 300 K. As
an index of the presence of vacancy defects, the line-shape S
parameter, which is defined as the peak area intensity of the
DBAR spectrum (i.e., 511 ± 1 keV) was determined. Also,
coincidence-mode DBAR spectra were obtained for selected
samples. Magnetic DBAR (MDB) measurements were performed using a longitudinally spin-polarized positron beamwith a polarization of 27% in an out-of-plane magnetic fieldof 0.91 T at 20 and 300 K. Also obtained were the differential
spectra N+ (p) − N− (p), where the subscripts + and − denotepositive and negative magnetic fields, respectively, with respect to the positron spin polarization. The MDB intensity
is defined as the total area of the differential spectrum. The
details are described elsewhere [27].
III. THEORETICAL CALCULATIONS

To interpret the experimental results, calculations basedon density functional theory (DFT) were performed within
the generalized gradient approximation [29]. The valence
electron wave functions were calculated using the projector
augmented-wave method [30] with the ABINIT8.10.3 code[31]. The valence electron configurations were 3d 10 4s2 4p1 ,2s2 2p3 , and 4 f 7 5s2 5p6 5d 1 6s2 for Ga, N, and Gd atoms,
respectively. For Gd, the Hubbard U correction (DFT + U )was considered with U = 6.7 eV and J = 0.7 eV [24,25].
Supercells including 64 and 128 atoms that correspond to2 × 2 × 2 and 4 × 2 × 2 conventional wurtzite unit cells,
respectively, were constructed. These systems with one Gd
atom correspond to Gd contents of approximately 3 and 2 at.
%, respectively. In addition to a perfect lattice, single N and
Ga vacancies (VN , VGa ) and divacancy (VGaVN ) with 2 × 2 × 2
cell, and 6-vacancy (V6 ), 8-vacancy (V8 ), and 12-vacancy (V12 )clusters with 4 × 2 × 2 cell were examined. The configurations are shown schematically in Fig. 1. Note that the vacancy“cluster” does not mean the group of isolated Ga vacancies[32], but corresponds to the void geometry including both

0.45
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10
15 0
5
10
Incident positron energy, E+ (keV)

15

FIG. 2. Positron annihilation S parameter versus incident
positron energy (S−E ) plots measured at 300 K for the Gd-implanted
samples before and after annealing at 900◦ C for 30 min (filled
circles) and for the unimplanted sample (open circles). The upperhorizontal axes show the mean positron implantation depth. Dashedlines denote the mean Gd range.

Ga and N vacancies. In all the calculations, the experimental
lattice constants were used and the atomic positions were optimized in accordance with the molecular dynamics simulation
installed in the code. The cutoff energy of the plane-wave
basis set was 544 eV. The core electron wave functions wererepresented by the Slater function parameterized by Clementi
and Roetti [33]. A self-consistent positron wave function was
calculated based on the two-component DFT to minimize theenergy functional [34]. The Borónski-Nieminen enhancement
factor was adopted. The DBAR spectra were obtained byconvoluting the one-dimensional angular correlation of theannihilation radiation spectra obtained from the momentum
density with the Gaussian resolution function having a half
width of 3.92 × 10−3 m0 c.
IV. RESULTS
A. Defect characterization by PAS and XRD

Figure 2 shows plots of the S parameter versus the incident
positron energy (S–E plots) for the unimplanted sample andall the implanted samples before and after annealing at 900◦ C.
The mean positron implantation depth is shown as the top horizontal axes. In the as-implanted states, the S parameter keeps
low values at φ  1013 Gd+ /cm2 , but starts to increase from

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PHYSICAL REVIEW B 102, 054427 (2020)

As-implanted

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Ratio, N(p) / Nref(p)Experiment1016 Gd+/cm2 E+=3keV
1012 Gd+/cm2 E+=5keV

1

Calculation
VN
VGa
V6
V8
V12

0.5

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900oC annealed

1.5

Experiment1015 Gd+/cm2 E+=5keV

Ratio, N(p) / Nref(p)φ = 1014 Gd+ /cm2 . For φ  1015 Gd+ /cm2 , the increase in
the S parameter is more pronounced at E+  3 keV. These
indicate that small vacancies are introduced in the low dose
conditions. The average S parameter for φ = 1012 Gd+ /cm2
relative to the bulk value is ∼1.043. This is compatible withthe theoretical value for single Ga vacancy (1.046). Although
single N vacancies may also be created, the S parameter istheoretically very close to the bulk value and these are hardly
detected. In the high dose conditions, larger vacancy clusters
seem to be created. The pronounced vacancy production at
E+  3 keV is consistent with the SRIM simulation predicting
a vacancy profile in a region that is a little shallower than themean Gd range (∼70 nm). Figure 2 also shows that vacancy
defects are created even in the deeper region up to 400 nm,
which is much deeper than the simulated maximum Gd range(∼150 nm). Possible reasons for this might be the diffusionof created vacancy defects and/or the cascade damage due torecoiled atoms with channeling.
After annealing at 900◦ C, the pronounced increase in the
S parameter at E+  3 keV observed for φ  1015 Gd+ /cm2
vanishes. For φ = 1015 Gd+ /cm2 , a bump appears near the
Gd range (E+ ∼ 5 keV), while such a bump is not appreciable for φ = 1016 Gd+ /cm2 . The S parameter for φ 1014 Gd+ /cm2 also increases upon annealing. These results
indicate that upon annealing, the initial vacancy defects
created by implantation are transformed into large vacancy
clusters. The formation of vacancy clusters is enhanced withincreasing dose up to φ = 1015 Gd+ /cm2 , but any further
increase of dose is not efficient.
Figure 3 shows the so-called ratio DBAR spectraN(p)/Nref (p) for some selected samples before and after
annealing at 900◦ C. Here, Nref (p) denotes the reference
DBAR spectrum given by the unimplanted sample. In the
as-implanted state, the spectrum for φ = 1012 Gd+ /cm2 isin good agreement with the theoretical curve for single Ga
vacancy (VGa ). While, the spectrum for φ = 1016 Gd+ /cm2 issignificantly changed, i.e., the intensity in the low-momentum
region greatly increases. The spectrum shape is compatible
to the theoretical curve for 12-vacancy cluster (V12 ). Afterannealing, the spectrum for φ = 1015 Gd+ /cm2 is also ingood agreement with the theoretical one for V12 . For φ =1012 Gd+ /cm2 , the intensity in the low-momentum regionis suppressed compared to that for φ = 1015 Gd+ /cm2 , but
the curve shape itself does not change appreciably. To reproduce the experimental spectrum with the theoretical curvefor V12 , the positron trapping fraction of 40% (described as
the amplitude adjustment in Fig. 3) is required. Similarly, theexperimental spectrum for φ = 1012 Gd+ /cm2 is reproducedby V6 with 50% trapping fraction and by V8 with 60% trapping
fraction, respectively. As discussed later, V12 with the positron
trapping fraction of 40% can explain the dose dependenceof effective magnetization. These again confirm that small
vacancies such as single Ga vacancies are predominant for the
as-implanted low-dose conditions. Also, these results leave
no doubt about the formation of large vacancy clusters in
the high-dose conditions and/or upon annealing. The average
open volume of the detected vacancy clusters is comparable
to that of V12 .
The calculations were also performed with Gd decorationof vacancy defects. It was found that replacing one of the1012 Gd+/cm2 E+=5keV

1

Calculation
V12
V12 x0.4
V6 x0.6
V8 x0.5

0.5

0

0

5
10
15
20
Electron momentum, p (10-3 m0c)

25

FIG. 3. The Doppler broadening of annihilation radiation (ratio)
spectra at 300 K for the samples with doses of 1016 Gd+ /cm2 (opencircles) and 1012 Gd+ /cm2 (open squares) in the as-implanted states(upper panel) and with doses of 1015 Gd+ /cm2 (open circles) and
1012 Gd+ /cm2 (open squares) after annealing at 900◦ C for 30 min(lower panel). The solid curves denote the theoretical calculations
assuming VN (black), VGa (green), V6 (blue), V8 (sky blue) and V12(red). The broken lines are for V6 (blue), V8 (sky blue), and V12 (red)with the amplitude adjustments of 0.6, 0.5, and 0.4, respectively.

nearest-neighbor Ga atoms around the V12 cluster with a Gd
atom causes no distinct changes in the spectra without Gddecoration. Therefore, it is difficult to judge if the vacancy
clusters are decorated with Gd atoms or not.
Figure 4 shows the XRD spectra obtained for the unimplanted sample and the implanted samples with φ = 1012 and1016 Gd+ /cm2 after annealing at 900◦ C. Only the reflections related to GaN and sapphire substrate are seen and thesecondary phases such as metal Gd (32.4◦ ) and GdN (30.9◦ )are absent. However, the formation of GaGdN phase (34.1◦ )is also not seen. The full width of half maximum (FWHM)
of GaN(0002) reflection is 0.058◦ for the unimplanted sample and the implanted sample with φ = 1012 Gd+ /cm2 .
This is even better than that of Si(111) reflection (0.065◦ )measured as a reference, suggesting that the crystallinity is
maintained in the low-dose condition. For the sample with
φ = 1016 Gd+ /cm2 , the FWHM is 0.083◦ , suggesting that
defective structures are developed with increasing Gd dose.

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Magnetization [emu/cm3]

Sapphire(0006) CuK

Sapphire(0006) WK

Sapphire(0006) CuK

GaN(0002) CuK

GaN(0002) WL

GaN(0002) CuK

Log intensity (arb. units)

MAEKAWA, MIYASHITA, SAKAI, AND KAWASUSO

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H [kOe]unimplanted
35

40

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2 (degree)
FIG. 4. The XRD spectra for the unimplanted sample and theimplanted samples with the doses of φ = 1012 and 1016 Gd+ /cm2
after annealing at 900◦ C.

(f) 1

(e)
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Tm
o

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B. Magnetic property by SQUID

Figures 5(a) to 5(d) show the M–H curves obtained at 10
and 300 K for the unimplanted sample and all the implanted
samples before and after annealing at 900◦ C. In the unimplanted state, hysteresis is almost null or very weak. Thatis, this is the level of artificial response of SQUID apparatus
and/or magnetic contamination during sample handling. After
implantation, the saturation magnetization clearly increases
to nearly the same level except for φ = 1016 Gd+ /cm2 , anddecreases slightly from 10 to 300 K. After annealing at
900◦ C, the saturation magnetization starts to increase fromφ = 1012 Gd+ /cm2 and saturates at the level higher than thatfor the as-implanted states at φ  1013 Gd+ /cm2 .
The dose dependence of M–H curve after annealing seems
to be systematic. However, one may feel that the dose dependence in the as-implanted state is peculiar. That is, the saturation magnetization only for φ = 1016 Gd+ /cm2 is lost andthat for φ = 1012 Gd+ /cm2 is higher even without annealing.
For φ = 1016 Gd+ /cm2 , it is speculated that heavy irradiation
damage and/or extremely high Gd concentration (supersaturation) prohibit the appearance of magnetization in the asimplanted state, while damage recovery and dispersion of Gd
atoms upon annealing result in a finite magnetization. For φ =1012 Gd+ /cm2 , it is speculated that the defect concentration
is appropriate to assist the magnetization in the as-implanted
state, while upon annealing the defect concentration decreases
and the magnetization is lost.
As shown in Fig. 5(e), the annealed samples exhibit hysteresis loops. (The curve for φ = 1012 Gd+ /cm2 is not plottedsince the saturation magnetization is small.) The coercivity is
estimated to be approximately 150 Oe. In the as-implanted
states, such hysteresis loops were not clearly seen.

-3-11016 Gd+/cm2
oMagnetization [emu/cm3]30

Magnetization [emu/cm3]

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100 150 200 250 300

Temperature [K]FIG. 5. (a)–(d) M−H curves measured at 10 and 300 K for all the
Gd-implanted samples in the as-implanted states and after annealing
at 900◦ C for 30 min (solid circles) and the unimplanted sample (greycircles). (e) Magnification of M−H curves for the implanted samples
after 900◦ C for 30 min measured at 300 K. (f) M–T curves for the
implanted samples after 900◦ C for 30 min in the field cooling (filled
circles) and zero-field cooling (open circles) conditions at 200 Oe.

Figure 5(f) shows the M–T curves obtained for the implanted samples after annealing at 900◦ C. Overall, the magnetization level increases from the unimplanted state to thehighest dose systematically. Also, the magnetization increases
slightly at low temperature implying the existence of paramagnetic component. But, as expected from the comparable
magnetizations at 10 and 300 K in M–H curves, the magnetization is almost independent of temperature. This implies
that the magnetic phase transitions occur above 300 K. The
difference between the field-cooling and zero-field coolingconditions is not seen, suggesting the absence of superparamagnetic component. The temperature dependencies arisingfrom secondary GdN and metal Gd phases are not seen either.
This is consistent with the XRD observation in Fig. 4.
Figure 6 shows the effective magnetizations per Gd
atom obtained before and after annealing at 900◦ C. Thesevary from ∼3000 μB /Gd for φ = 1012 Gd+ /cm2 to below
1 μB /Gd for φ = 1016 Gd+ /cm2 .
The above results may allow us to conclude that ferromagnetism is induced in the Gd-implanted GaN samples and054427-4

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As-implanted
900oC annealed
Estimated

100
10-1

1012

1013
1014
1015
Gd dose (Gd+/cm2)

1016

FIG. 6. The effective magnetization per implanted Gd atom obtained in the as-implanted state (solid circles) and after annealing
at 900◦ C for 30 min (solid squares). The estimated values withthe concentration of vacancy clusters after annealing and assumed
magnetization per vacancy cluster are plotted as open squares.

C. Vacancy spin by SP-PAS

Figure 7 shows plots of the MDB intensity versus E+
at 300 K for all the samples. Nearly the same features
were obtained at 20 K, indicating no temperature dependence. For the unimplanted sample, no finite MDB intensities were found, suggesting that electrons participating in
the positron annihilation are not spin-polarized. In the asimplanted states, finite MDB intensities are seen only for φ =1015 and 1016 Gd+ /cm2 at E+  5 keV. Considering the factthat vacancy clusters are formed there as shown in Fig. 2, suchvacancy clusters are probably spin-polarized. After annealing,
the MDB intensities at E+  5 keV are reduced considerably.
Instead, the MDB intensities at around E+ = 5 keV (aroundthe Gd range) increase with increasing dose from φ = 1013 to1015 Gd+ /cm2 . The MDB intensity for φ = 1016 Gd+ /cm2is still nonzero but is suppressed considerably. The MDB
intensity for φ = 1012 Gd+ /cm2 is under the detection limit.
Again, the increase (decrease) of the MDB intensity seems
to coincide with the appearance (disappearance) of vacancy
clusters as discussed in the S–E plots in Fig. 2.
Figure 8 shows the MDB differential spectrum N+ (p) −N− (p) at 300 K for the implanted sample with φ =1015 Gd+ /cm2 after annealing at 900◦ . The data for E+ =4−6 keV are integrated for better statistics. The spectrum has

as-impl.

1x1015 Gd+/cm2

900oC ann.

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it is colossal. However, it is known that, apart from simple
magnetic contamination of sample, the SQUID magnetometry
creates some serious artifacts, such as spurious magnetizationand ferromagnetic loop mainly due to the pinning of magnetic
flux [16,17]. Unfortunately, there are no definitive ways tojudge if the observed features arise from genuine ferromagnetism or just fakes. It is said that in early studies, these risks
involving the SQUID magnetometry were not well known andthe so-called colossal magnetism of GaN:Gd was claimed.
Therefore, at this moment, we can only suspend the argument
for the existence of ferromagnetism.

900oC ann.

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015

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MDB intensity (arb. units)

Meff ( B/Gd)

52

1x1016 Gd+/cm2

5
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15 0
5
10
Incident positron energy, E+ (keV)

15

FIG. 7. Plots of magnetic Doppler broadening (MDB) intensity
versus incident positron energy at 300 K (filled circles) for the Gdimplanted samples before and after annealing at 900◦ C for 30 min(filled circles) and for the unimplanted sample (open circles). The
upper horizontal axes show the mean positron implantation depth.
Dashed lines denote the mean Gd range.

an upward peak at its center. In the context so far, the problem
is if this spectrum feature can be explained considering V12
clusters. The present calculations showed that small vacancy
defects (VGa , VN , VGaVN ) have spontaneous magnetizations(3, 1, 2 μB , respectively) irrespective of the presence of Gd
atoms. However, V12 has no magnetization in the supercell
without a Gd atom. We therefore further calculated (i) with the
Gd-added supercell including separated Gd atom and V12 , and(ii) with the clean supercell including two close V12 clusters
separated by an interatomic distance to mimic the high density
condition. The first induced no magnetization on the V 12
cluster, while in the second case spontaneous magnetizationof 2 μB /V12 appeared due to the formation of partially filledstates. As seen in Fig. 8, the calculated spectrum for VGa
has a downward peak at the centers, which is because of
the preferential annihilation of positrons with spin-polarized
N valence electrons. Such a downward-peak spectrum hardly
explains the experimental upward-peak spectrum. Contrarily,
VN and V12 give rise to upward-peak spectra because of thespin polarization of Ga valence electrons. The MDB intensity
of VN is not sufficient because of the weak localization of
the positron wave function and thus is not comparable tothe experiment. Both the intensity and curve shape of V12
are comparable to the experiment. Thus, spin-polarized V12054427-5

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MAEKAWA, MIYASHITA, SAKAI, AND KAWASUSOPHYSICAL REVIEW B 102, 054427 (2020)

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VN (1 B)
VGa (3 B)
V12 (2 B)-10

Experiment1015 Gd+/cm2
o
900 C annealed

-20
-10
0
10
20
Electron momentum, p (10-3 m0c)

FIG. 8. MDB spectrum measured at 300 K for the Gd-implanted
sample with a dose of 1015 cm−2 after annealing at 900◦ C (opencircles). The data for 4–6 keV were summed for better statistics. The
solid curves denote the theoretical calculations assuming VN (black),
VGa (blue), and V12 (red) cluster.

clusters may be the major positron trapping centers. This isconsistent with the discussion based on the ratio DBAR curvein Fig. 3.
V. DISCUSSION

The present observations and their interpretations by thecalculation are summarized in Table I.
In the as-implanted low dose conditions, single Ga vacancies were detected as the major positron trapping centers(Fig. 3). However, SP-PAS measurements showed no spinpolarization on single Ga vacancies (Fig. 7). Therefore, if theobserved magnetizations in Fig. 5 are genuine, those are notmediated by single Ga vacancies. It is possible that the othermechanisms need to be considered. In the present calculation,
single N vacancy also possesses a magnetization of 1 μB .
Some other point defects such as interstitials and antisites may
also induce magnetizations [35]. To clarify such possibilities
further study is needed.
In the as-implanted high-dose condition, vacancy clusters
at least larger than V12 were formed in the regions shallower
than the mean Gd range (Figs. 2 and 3). These vacancy
clusters were found to be spin-polarized (Fig. 7). Therefore, if
the observed magnetizations in Fig. 5 are genuine, those may
be explained considering large vacancy clusters. However, thehighest dose, i.e., φ = 1016 Gd+ /cm2 , is exceptional since nomagnetization appeared (Fig. 5).
Upon post-implantation annealing, small vacancies disappeared. Instead, vacancy clusters larger than at least V12
were formed as the predominant defect species even in thelowest dose, i.e., φ = 1012 Gd+ /cm2 (Fig. 3). Based on theargument of the number of dangling bonds, V12 may beenergetically stable [36]. This may be why such vacancy
clusters form upon post-implantation annealing. Because theexperimental DBAR spectrum for φ = 1015 Gd+ /cm2 after
annealing coincides with the nonrescaled theoretical curve for
V12 , injected most positrons are probably trapped by vacancy
clusters. Thus, the concentration of vacancy clusters shouldbe very high 1019 cm−3 . In the SRIM simulation, each Gd
ion creates approximately 4500 vacancies. Therefore, even
if many Frenkel pairs recombine immediately during irradiation, an extremely high concentration of vacancy clusters
may be formed. The spin-polarizations associated with thesevacancy clusters were detected except for the lowest dose,i.e., φ = 1012 Gd+ /cm2 (Fig. 7). Therefore, if the observed
magnetizations in Fig. 5 are genuine, those may be explained
as mediated with vacancy clusters.
One may think the effect of Gd decoration on the spinpolarization of V12 . We examined such effects and found thatby decorating V12 with a Gd atom, no spin-polarizations are
induced except for the spin-polarization of Gd atom itself. The
experimental MDB spectrum was also not reproduced considering the Gd decoration. Even if some spin polarizations are
induced with Gd decoration, the colossal ferromagnetism (if itexists) is hardly explained, since the colossal ferromagnetism
needs some magnetic species other than Gd itself that mediate
ferromagnetism.
Assuming that vacancies and Gd atoms cause a macroscopic magnetization, the effective magnetization per Gd may
be given by CV MV /NGd + 7 μB , where CV is the vacancy concentration, MV is the magnetization associated with a vacancy,and NGd is the concentration of implanted Gd. The vacancy
concentration is given by CV = f λB /[μ(1 − f )], where μ is
the positron trapping coefficient of the vacancy, λB is the
positron annihilation rate in the bulk, and f is the fractionof positrons trapped by vacancies and given by f = (S −SB )/(SV − SB ), where SB and SV are the intrinsic S parameters
for the bulk and vacancy. One may take SB = 0.463 andSV = 0.541 from Fig. 2, and the theoretical MV = 2 μB for
V12 and λB = 6 ns−1 from the literature [37]. Normally, μ is of
the order of 1014 s−1 in either the transition-limited regime or
the diffusion-limited regime. In the diffusion-limited regime,
μ is given by 4π RD+ /NA , where R is the effective positron

TABLE I. Summary of experimental observations and interpretations by calculation.
Dose(Gd+ /cm2 )
16

10
1015
1014
1013
1012

900◦ C annealed

As-implanted
Magnetization

Vacancy spin

Invisible
Exist
Exist
Exist
Exist

Exist
Exist
Weak
Invisible
Invisible

Defect species
Vacancy clusters
Vacancy clusters
Vacancy clusters, small vacancies
Small vacancies, VGa etc
Small vacancies, VGa etc

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Magnetization

Vacancy spin

Defect species

Exist
Exist
Exist
Exist
Weak

Exist
Exist
Exist
Exist
Invisible

Vacancy clusters
Vacancy clusters
Vacancy clusters
Vacancy clusters
Vacancy clusters

GADOLINIUM-IMPLANTED GAN STUDIED BY …

PHYSICAL REVIEW B 102, 054427 (2020)

trapping radius of vacancy (approximate geometrical size of
vacancy, i.e., a few Å in the present case), D+ is the positron
diffusion constant, and NA is the atomic density. In thepositron diffusion in a clean matrix with phonon scattering,D+ is approximately 1 cm2 /s. However, in the defective
matrix due to the radiation damage, the positron diffusion may
not be necessarily the same as that in clean matrix due to thescattering by defects [38]. If we take one order of magnitude
smaller diffusion constant, then the effective magnetization
per Gd was drawn in Fig. 6. It seems that the dose dependenceof effective magnetization can roughly be explained.

detection of electron spins associated with defects, the potential methods have been limited. In this study, we applied
the SP-PAS method for Gd-implanted GaN. The electrons
associated with larger vacancy clusters were found to bespin-polarized. Such spin-polarized vacancy clusters may be
the source of ferromagnetism. However, the SQUID magnetometry has a serious problem in manifesting the existence offerromagnetism. In this respect, the improvement of SQUID
magnetometry and the development of alternative methods areindispensable.

VI. CONCLUSIONACKNOWLEDGMENTS

The defect-assisted magnetism in Gd-implanted GaN is
still far from full understanding. One reason is that in theThis work was supported, in part, by JSPS KAKENHI
under Grant No. 18K04931.

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