PHYSICAL REVIEW B 97, 245303 (2018)

Positronium formation at Si surfaces
A. Kawasuso,* M. Maekawa, A. Miyashita, and K. Wada
National Institutes for Quantum and Radiological Science and Technology, 1233 Watanuki, Takasaki, Gunma 370-1292, Japan

T. Kaiwa and Y. Nagashima
Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan(Received 2 April 2018; published 5 June 2018)
Positronium formation at Si(111) and Si(001) surfaces has been investigated by changing the doping levelsystematically over the range 300–1000 K. The temperature dependence of the positronium fraction varied withthe doping condition, and there were practically no differences between the two surface orientations. In heavily
doped n-type Si (n  1018 cm−3 ), the positronium fraction (IPs ) increased above 700 K and reached more than 95%
at 1000 K. In undoped and lightly doped Si (n, p  1015 cm−3 ), IPs decreased from 300 to 500 K and increased
above 700 K. In heavily doped p-type Si (p  1018 cm−3 ), IPs increased in two steps: one at 500–600 K and one
above 700 K. Overall, the positronium fraction increased with the amount of n-type doping. These phenomena
were found to be dominated by two kinds of positronium with energies of 0.6–1.5 eV and 0.1–0.2 eV, whichwere attributed to the work-function mechanism and the surface-positron-mediated process, respectively, with
contributions from conduction electrons. The positron work function was estimated to be positive. This agrees
with first-principles calculation. The positive positron work function implies that the formation of excitonic
electron-positron bound states begins in the bulk subsurface region and transits to the final positronium state in
the vacuum.
DOI: 10.1103/PhysRevB.97.245303

I. INTRODUCTION

Positronium, which was predicted by Mohorovičić [1]and discovered by Deutsch [2], is the purely leptonic boundstate composed of an electron and a positron. It is a very
simple system, but it is still extensively investigated suchas in the verification of the standard model of elementary
particle physics [3] and quantum electrodynamics [4], therealization of high-TC Bose-Einstein condensation (BEC) [5],and atomic physics [6]. In these studies, the generation ofhigh-density positronium (e.g., 1018 cm−3 for positronium
BEC) is an important issue. Solid surface is a medium togenerate positronium. The positronium formation is sensitive
to the surface condition such as adsorption. The angularand energy distributions of surface positronium reflect the
momenta and energy levels of electrons in the surface layersof the solid. Hence, positronium may also be a unique probe
to study the electronic state of the first surface layers [7,8]. It
is, therefore, important to understand the generation processesof positronium at solid surfaces in detail.
At metal surfaces, positronium is generated via (i) the
work-function mechanism, (ii) the surface positron state, and(iii) energetic positron scattering. Here, we focus on (i) and (ii).
The positronium work function (Ps ) is given as the summationof electron and positron work functions (φ− , φ+ ) and theenergy gain due to positronium formation (−6.8 eV): Ps =φ− + φ+ − 6.8 eV. If Ps is negative, spontaneous emission of
positronium occurs with a maximum kinetic energy of |Ps |.

*kawasuso.atsuo@qst.go.jp

2469-9950/2018/97(24)/245303(8)Positrons are also trapped at the mirror potential formed at thesurface-vacuum interface (surface positron state). In this case
we define ES = φ− + EB − 6.8 eV, where EB is the binding
energy of the surface positron. Similar to (i), if ES is negative,
spontaneous positronium emission occurs with a maximum
energy of |ES |. Even in the case of positive ES , positronium
emission occurs as a thermal activation process with an energyof dozens of meV (thermal positronium).
For semiconductors, though surface positronium may beformed in ways similar to metals [9,10]. However, because of
the band gap and the suppressed screening effect, dissimilar
processes may also be expected.
In 2011, the group at the University of California, Riverside(UCR) found that at a Si(001) surface the positronium fraction
increases with increasing temperature, while its kinetic energy
exceeds the thermal region and hence it is not classified asconventional thermal positronium [11]. They also found enhanced positronium formation as the result of optical excitation[12]. Consequently, it was proposed that positronium is formed
through an excitonic bound state between surface positrons andelectrons (PsX state) [13,14]. Since only lightly doped Si(001)samples were used in the above work, further investigationthat involves changing the doping level more systematicallyand using different surface orientations is required in order toexplore the PsX hypothesis.
In this study, we measured the positronium fractions at
Si(111) and Si(001) surfaces with different doping levels as
a function of temperature. We also determined the positronium energy from the time-of-flight measurements. Throughdescribing the positronium formation processes based on
the experimental findings and first-principles calculation, we

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TABLE I. Sample characteristics.
DopantConduction type

Carrier density(cm−3 )Si(111)

Sb
P
Undoped
B
B

n
n
n
p
p

1 × 1018
3 × 1015
1 × 1012
1 × 1015
4 × 1018

Si(001)P
Sb
UndopedBnnnp

1 × 1019
1 × 1018
1 × 1011
4 × 1018

concluded that the excitonic bound state exists universally
between positrons not only at the surface state but also in thebulk and conduction electrons.

FIG. 1. Annihilation γ -ray spectra at 511 keV obtained forGe(111) at 1000 and 300 K with positron energies of E+ = 200 eV
and E+ = 30 keV, respectively. In the latter condition, positronium
formation is negligible.

II. EXPERIMENT
A. Procedure

The samples used in this study were Czochralsky-grown
Si(111) and Si(001) wafers doped with phosphorus, antimony, or boron, and floating-zone-grown undoped Si(111) andSi(001) wafers. The sample thickness was 525 μm in all cases.
The doping levels are listed in Table I. The samples were dipped
into 50% hydrofluoric acid for 5 min and subsequently cleaned
with ultrapure water without exposing them to air. From thehydrophobicity, the removal of the natural oxide layer andsurface hydrogen termination were confirmed.
The samples were transferred into the vacuum chamber(base pressure: 8 × 10−7 Pa) equipped with a slow positron
beam apparatus during 20 min in air. The beam apparatus
is composed of a 22 Na source (1.1 GBq), a tungsten meshmoderator, and a magnetic transport system. The samples
were injected with positrons with the energy of E+ = 200 eV
and annihilation γ -ray energy spectra were measured using ahigh-purity Ge detector as a function of temperature at 300 ∼1000 K (27 ∼ 727 ◦ C). In one spectrum, 5 × 105 counts wereaccumulated. Two-γ events created a sharp peak at 511 keV(= m0 c2 , where m0 is the electron rest mass and c is the light
speed). Three-γ events gave rise to a continuous spectrum
below 511 keV. The positronium fraction was determined using
the 511 keV peak intensity denoted as P and the lower energy
spectrum intensity denoted as R:


P100% R100% − R
IPs = 1 +P0% R − R0%

−1
(1)

,

TABLE II. Calculated positron affinity (A+ ), positronium work
function (Ps ), electron and positron work functions (φ− , φ+ ), surface
dipole barrier (), and surface positron binding energy (EB ) in eV.
Bulk calculation

Slab calculation

A+

Ps

Surface

φ−

φ+



EB

−6.38

−0.42

(111)
(001)

+4.56
+4.54

+1.82
+1.84

+10.86+10.84

+2.70
+2.41

where the subscript denotes the 100(0)% positronium intensity
determined from measurements of Ge(111) at 1000 K andmica at 300 K [15–17]. As shown in Fig. 1, a very sharp
511 keV peak is seen for the Ge sample at 1000 K and
E+ = 200 eV as compared to 300 K and E+ = 30 keV, where
positronium formation is negligible. The full width at half
maximum is 1.5 keV. This is comparable to the detector
resolution, suggesting the peak is arising from nearly 100 %(para) positronium.
The samples were also subjected to positronium time-offlight measurements at the Slow Positron Facility of the HighEnergy Acceleration Research Organization in Japan [18].
Pulsed positrons generated by an electron linac were injected
into the samples at E+ = 200 eV. Positronium annihilation
events were detected using scintillation-photomultiplier detectors located 40 and 120 mm upstream from the sample position.
Positronium time-of-flight spectrum was acquired using thepulse signals from the linac and the annihilation γ rays.
In the preliminary measurements, after heating the samples
to 1000 K, the positronium formation was observable. Aftercooling the samples to 300 K, the positronium fraction was
found to decrease exponentially to approximately half inhalf a day. Whereas, when the samples were kept above∼348 K (75 ◦ C), the degradation of the positronium fraction
was suppressed. After keeping the samples at 300 K for along time, the temperature dependence obtained in the firstrun was no longer reproduced. These results show that the
sample surfaces were contaminated during the transportationin air and even in the vacuum chamber with residual gases.
During heating to 1000 K, surface contaminants and hydrogenatoms were considered to be desorbed together. Thus, in all
the experiments, the positronium fraction was measured from1000 to 300 K, and then again to 1000 K to confirm the absenceof thermal hysteresis. After one cooling and heating run, thesamples were replaced by new ones.
B. Results

Figure 2 shows the temperature dependence of the positronium fraction (IPs ) obtained for all the samples. In the cooling

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FIG. 3. Positronium fraction obtained for the doped and undopedSi(111) samples at 373 K (100 ◦ C) as a function of incident positronenergy.

FIG. 2. Temperature dependences of positronium fraction obtained for the Si(111) and Si(001) samples with different doping
conditions. Filled and open circles are obtained in the cooling andheating runs, respectively. Solid lines are the fitting curves using
Eqs. (5)–(7).

and heating runs, no thermal hysteresis is seen. The overall
positronium fraction increases in the order of heavily doped
p-type → undoped and lightly doped → heavily doped n-type.
The increase of IPs from n ∼1018 to 1019 cm−3 is only small,as seen for the Si(001) sample. Several thermal processes areseen, which can be described as follows. In the heavily dopedn-type samples, IPs increases above 700 K and reaches almost
100% at 1000 K. In the heavily doped p-type samples, IPs
increases in two steps (i.e., at 500–600 K and above 700 K).
In the undoped and lightly doped samples, IPs decreases from
300 to 500 K and then increases above 700 K. The increase of
IPs above 700 K seems to be common for all the samples.
Thus, the amount of positronium and its temperature dependence strongly depends on the doping condition. There areno essential differences between Si(111) and Si(001) surfaces.
Some of these features complement the results of the previous
works [11–14], which mainly used lightly doped Si(001)samples.
Figure 3 shows the positronium fraction for the Si(111)
samples at 373 K (100 ◦ C) as functions of incident positronenergy and mean positron implantation depth [19]. The results
for the Si(001) samples were nearly the same. In the heavily doped p-type and undoped samples, only small amountof positrons implanted in shallow regions participated inpositronium formation. Whereas, in the heavily doped n-typesample, implanted positrons effectively returned to the surface
giving rise to a large amount of positronium. The effective
drift diffusion of positrons in n-type semiconductors has been
known for a long time. It is explained as an effect of the electricfield induced by the band bending near the surface [20,21].
Figure 4 shows the positronium time-of-flight spectra obtained for the heavily doped n-type, undoped, and heavily
doped p-type Si(111) samples at different temperatures after
subtracting the ortho-positronium lifetime backgrounds [∝exp(−t/142) ns]. There are two characteristic components.
One has a peak at EPs ∼ 0.6 eV and threshold at EPs ∼ 1.5 eV.
The other has a peak at EPs = 0.1–0.2 eV. The energies of two
kinds of positronium agree with those obtained by the UCR
group for Si(001) and Si(111) surfaces [11–14]. Here, thesecomponents are termed type A (EPs = 0.6–1.5 eV) and type B(EPs = 0.1–0.2 eV). Apart from these components, no thermal
positronium (EPs  90 meV) is seen.
In the heavily doped n-type and undoped samples, type A
positronium exists in the whole temperature range, while type
B positronium appears at high temperatures. In the heavily
doped p-type sample, type A positronium is nearly absent at
348 K and appears above 600 K. Type B positronium exists inthe whole temperature range and increases above 700 K.
III. THEORETICAL CALCULATION

To interpret the experimental data quantitatively, twocomponent density functional theory (TC-DFT) calculations(conventional scheme) were carried out by using the ABINIT
code [22] with the projector-augmented-wave method [23]within the generalized gradient approximation (GGA) [24].
The electron-positron correlation energy functional was basedon the GGA method [25]. The valence electron configuration of the Si atom was assumed to be 2s 2 2p6 3s 2 3p2 . The
bulk calculation was carried out using a primitive cell composed of two Si atoms with full structural optimization. The
k-point sampling was 15 × 15 × 15. The optimized lattice
constant was a = 5.47 Å. The positronium work function was

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The positron work function and surface dipole barrier were
determined, respectively, by
φ+ = −A+ + φ− and  = −A+ + μ− .

(3)

For the calculations of electron work function and surface
positron binding energy, Si(111) and Si(001) slab crystals
composed, respectively, of 12 and 13 monolayers oriented in
the surface normal direction with the primitive cell oriented in
the surface parallel direction were constructed. The vacuum
layer was initially assumed to be 20 Å. For the electron
calculation, the k-point sampling was 9 × 9 × 1. The (111) and(001) surfaces were assumed to be bulk truncated and dimer
structures, respectively. To determine the positron surface state
precisely, a corrugated mirror potential was implemented as
the surface potential for positrons [26–28]. The contact pointof image potential and electron-positron correlation potential(effective image plane) was determined in a way similar
to the previous study [29]. Traditionally, the local-density
approximation (LDA) electron-positron correlation potential
is connected to the corrugated mirror potential. Here, theGGA electron-positron correlation potential was used since
it gives better results in the evaluations of surface positronbinding energy and annihilation lifetime as compared to the
LDA potential [28,30]. For the positron calculation, only the point was considered. Full structural optimization was also
carried out.
Table II shows the results of the calculation. The positronium work function is in good agreement with that obtained by
Kuriplach and Barbiellini [31]. The positron work function ispositive for both (111) and (001) surfaces. The surface positronbinding energies are in good agreement with those obtained by
Fazleev et al. [29]. These results imply that the spontaneousemission of positronium occurs, there is no positron emission,and the surface positron state exists.

IV. DISCUSSION

FIG. 4. Positronium time-of-flight (Ps-TOF) spectra obtained for
the Si(111) samples for different doping conditions at the highest(1000 K), medium (573 and 673 K) and lower (373 and 348 K)
temperatures. Vertical broken lines denote the threshold of the TOF
spectra (∼1.5 eV). Arrows denote the approximate peak positions of
two types of Ps distinguished by their energies.

determined asPs = −A+ − 6.8 eV = −(μ− + μ− ) − 6.8 eV,(2)where A+ is the positron affinity and μ∓ is the electron(positron) chemical potential measured from the crystal zero.

Figure 5(a) shows the schematic representation of the temperature dependence of the positronium fraction for the three
typical doping conditions shown in Fig. 2. As denoted by thearrow, the overall positronium fraction increases from heavily
doped p-type to heavily doped n-type Si. There are threekinds of temperature dependence: (I) the process decreasing
with temperature, which is seen only for undoped and lightly
doped Si; (II) the process increasing with temperature, which
is seen only for heavily doped p-type Si; and (III) the processincreasing with temperature, which is seen for all the doping
conditions.
Figure 5(b) shows the temperature ranges of type A andtype B positronium. In undoped and lightly doped Si, andheavily doped n-type Si, type A positronium exists in theentire temperature range, while type B positronium appears
only above 700 K. In heavily doped p-type Si, type A positronium appears only above 600 K, while type B positronium
exists in the whole temperature range and increases above
700 K. The problem here is how to explain these experimental
facts.

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POSITRONIUM FORMATION AT Si SURFACESPHYSICAL REVIEW B 97, 245303 (2018)
(a)

(c)

(b)

(d)

(a)

(b)

FIG. 5. Schematic representations of (a) temperature dependenceof positronium fraction for the three typical doping conditions shownin Fig. 1, and (b) temperature ranges of type A and type B positronium
distinguished by their energies.
A. Origins of the observed features: Type A positronium

In undoped and lightly doped Si and heavily doped n-type
Si, type A positronium (EPs = 0.6–1.5 eV) exists in the wholetemperature range, and hence it is likely to be formed via thework-function mechanism. The work function may be givenby the threshold energy in the time-of-flight spectrum (i.e.,
exptPs = −1.5 eV) rather than the peak energy of 0.6 eV. Thisis somewhat larger than the theoretical work function shown in
Table II (calc
Ps = −0.42 eV). Ordinarily, valance electrons arethought to take part in positronium formation. The theoretical
positronium work function is indeed obtained using valence
electrons. Assuming that conduction electrons are the sourceof the positronium, due to an additional energy gain of theband gap of Si, 1.1 eV, the theoretical work function wouldbe calc
Ps = −0.42 − 1.1 ≈ −1.5 eV which agrees with theexperimental value.
Actually, heavily doped n-type Si contains enough conduction electrons. In addition, as discussed for Fig. 3, the electricfield due to the band bending effect promotes positrons to thesurface as shown in Fig. 6(a). The high positronium yield canalso be explained from this aspect. Since near the surface ofheavily doped n-type Si, the depletion layer will be formed,
the combination of positrons and conduction electrons may besuppressed there. Probably, conduction electrons outside thedepletion region participate in type A positronium formation.

FIG. 6. Schematic representations of (a) band bending effecton positrons, (b) electron-hole plasma created by positron impact
and resultant positronium formation, (c) positronium formation via
work function mechanism, and (d) positronium formation via surface
positron state.

The suppressed type A positronium formation in heavily
doped p-type Si is explained as being due to the lack of conduction electrons. The repulsive surface potential to positrons
may also reduce the type A positronium formation.
Since undoped and lightly doped Si are also deficient in
conduction electrons, positronium formation from positronsand conduction electrons seems to be unlikely at first glance. As
for this, the excitation of electron-hole (e-h) pairs by the impact
of positrons should be considered, as shown in Fig. 6(b) [11].
In the present experiments (E+ = 200 eV), approximately
50 e-h pairs will be created per positron in a sphere with aradius of about 10 Å. That is, the local e-h plasma density
is 5 × 1021 cm−3 . Positronium formation may be possible
between positrons and conduction electrons eluded from thee-h recombination.
The e-h plasma process seems to be more important in
undoped and lightly doped Si, since the recombination lifetimeof e-h pairs is very long (typically in the milliseconds) when thedoping level is low. Furthermore, the carrier lifetime becomesshorter at elevated temperatures since the carrier capture crosssection by gap states normally obeys the Arrhenius law. This
may explain the decrease in the positronium fraction from 300
to 500 K in undoped and lightly doped Si [process I in Fig. 5(a)].
In heavily doped n-type Si, the conduction electrons excitedby positron impact will decay rapidly through the capture
process via impurity levels and also the drift into the deeper
region by the band bending effect. Hence, the e-h excitation
creates no significant effects in the temperature dependence. In

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heavily doped p-type Si, the excited conduction electrons arealso relaxed rapidly due to impurity levels and also via the drift
to the surface by the band bending. Conversely, positrons will
be repelled from the surface. Hence, positronium formationfrom positrons and conduction electrons will be rare at least
at low temperatures. The small increase in the positronium
fraction at 500–600 K [process II in Fig. 5(a)] may be partly
explained as the thermal activation of positrons beyond thesurface potential barrier and the combination with the excitedconduction electrons [Fig. 6(a)].
Thus, type A positronium may be formed between
positrons and conduction electrons via the work-function
mechanism. Using the experimental positronium work funcexpt
tion (Ps = −1.5 eV), the positron work function is estimated to be φ+ = Ps − φ− + 6.8 = −1.5 − (5.2 − 1.1) +6.8 = +1.2 eV. This estimate is a bit smaller than the present
calculation listed in Table II (φ−calc = +1.8 eV). Anyway, thepositron work function should be positive in Si. This may havean important meaning in the detailed positronium formation
process, as discussed in Sec. IV D below.

B. Origins of the observed features: Type B positronium

Type B positronium (EPs = 0.1–0.2 eV) is seen for all thedoping conditions and is independent of surface orientation.
The increase of positronium fraction above 700 K [process III
in Fig. 5(a)] is caused by type B positronium. Above 700 K,
electrons are thermally excited from the valence band to the
conduction band. In the surface depletion layer, the Fermi
level is relatively deeper than in the bulk region even forheavily doped Si [32]. Hence, the temperature dependenceof conduction electron density may be locally and temporally similar for all doping conditions giving rise to similar
temperature dependence of process III. Thus, “athermal” type
B positronium may be formed from surface positrons andthermally excited conduction electrons. Indeed, the surface
positron binding energy estimated as EB = ES − φ− + 6.8 =−(0.1 ∼ 0.2) − (5.2 − 1.1) + 6.8 = +2.5 ∼ 2.6 eV is comparable to the calculation listed in Table II.
In heavily doped p-type Si, type B positronium is also seen
at low temperatures. Probably, the conduction electrons excitedby positron impact drift to the surface region due to the band
bending effect and combine with residual surface positrons,
though positrons tend to be repelled from the surface. The small
increase in the positronium fraction at 500–600 K [process
II in Fig. 5(a)] may partly be explained as the increase in
the surface positrons resulting from the thermal activation of
positrons beyond the surface barrier. In the other types of Si,
such a drift of conduction electrons to the surface is not strong
enough to give rise to type B positronium formation at low
temperatures. This is plausible for heavily doped n-type Si in
which the conduction electrons excited by positron impact are
repelled from the surface.

C. Temperature dependence

A monotonic temperature variation of the positronium
fraction may be described as a competition of two processes,
one of which is the thermal process. The positronium fraction

is generally given asIPs (T ) =g(T )I,r + g(T )(4)where I is the maximum change of positronium fraction in
the process, r and g(T ) represent the rates of the nonthermal
and thermal processes, respectively, and T is temperature. For
instance, in the thermal desorption of positronium observed formetal surfaces, r = λS , which is the annihilation rate of surface
positrons, and g(T ) = (4kT / h) exp [−ES /(kT )], where k is
the Boltzmann constant, h is the Planck constant, and ES =φ− + EB − 6.8eV  0 [33]. The three processes, I, II, and III,are described as follows.
Process I: The positronium formation rate is assumed to be
proportional to the density of surviving conduction electrons
at a certain time after the creation of an e-h plasma by positron
impact and trapping by gap states. The competitive process
may be other events, such as free positron annihilation. The
excess electron density may be given as n ∝ exp(−kt) andhence g(T ) ∝ exp(−kt), where t is time and k is the trapping
rate of electrons into gap states. The trapping rate is k = Nt vσt ,where Nt is the electron trap density, v = (3kT /me )1/2 is thethermal velocity of conduction electrons with effective massme , and σt is the cross section for capture of electrons by thetraps. This cross section is generally given in the form of σt ∝T −2 exp[−EI /(kT )], where EI is the energy difference
between the bottom of the conduction band and the crossing
point of the adiabatic potentials for the conduction band andthe trap [34–36]. Hence k ∝ T −3/2 exp[−EI /(kT )]. Thus,
using Eq. (4), process I is described as
IIPs(T ) =

II,−3/2
1 + AI1 exp{AI2 T
exp[−EI /(kT )]}(5)

where AI1 and AI2 are constants.
Process II: The positronium formation rate is dominatedby the thermal activation of positrons to overcome the surface potential barrier. The available electron density is assumed to be independent of temperature. Hence, g(T ) ∝exp[−EII /(kT )], where EII is surface potential barrier. The
competitive process may be the free positron annihilation rate.
Thus, process II is described as
IIIPs(T ) =

III,
1 + AII exp[EII /(kT )](6)

where AII is a constant.
Process III: The annihilation of surface positrons andpositronium formation from surface positrons and conduction electrons that are thermally excited from the valence band compete each other. Hence, r = λS and g(T ) ∝n. As discussed in Sec. IV B, because of the depletion
layers, the temperature dependence of n for undoped Si
may be a good approximation for all doping conditions.
Thus, g(T ) ∝ (Nv Nc )1/2 exp[−Egap (T )/(2kT )], where Nv andNc are the density of states of the top of the valenceband and the bottom of the conduction band, respectively:
Nv(c) = 2(2π mh(e) kT / h2 )3/2 with the effective hole (electron) mass, mh(e) , and the temperature dependent band gap
Egap (T ) = 1.166 − 4.73 × 10−4 T 2 /(T + 636) [37]. Thus,

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POSITRONIUM FORMATION AT Si SURFACESPHYSICAL REVIEW B 97, 245303 (2018)using Eq. (4),
III
=
IPs

IIII
,
−3/2
1 + AIII T
exp[Egap (T )/(2kT )](7)

where AIII is a constant.
The temperature dependences of heavily doped n-type Si,
undoped and lightly doped Si, and heavily doped p-type Si
were fitted using Eq. (7) with a constant term, Eqs. (5) and (7),and Eqs. (6) and (7) with a constant term, respectively. The
solid lines in Fig. 2 are the obtained fitting curves. The observed
temperature dependences are well described using the concepts
described above. The potential barriers were estimated to be
EI = 90–110 meV and EII = 0.28–0.54 eV. The former
value may be reasonable as a nonradiative process [38].
From the latter value, the width of the depletion layer (=[2 EII /(eNA )]1/2 , where is the dielectric constant, e is theelementary charge and NA is the acceptor concentration) is
further estimated to be a few tens of nm [32].
D. Excitonic positronium formation

As discussed so far, positronium formation in Si may occur
between positrons and conduction electrons via the workfunction mechanism (type A) and between surface positronsand conduction electrons via the athermal process (type B).
The positivity of the positron work function gives further
insight into the work-function mechanism. If the positron work
function is negative, positronium may be formed in two ways:
either a positron is emitted into the vacuum and then an electron
hops onto the positron, or a quasipositronium state formedin the low electron density region is eventually emitted as apositronium. In the case of positive positron work function, theformer process is prohibited. In Si, therefore, quasipositronium(excitonic electron-positron bound state) should be formed in
the bulk subsurface region and these then transit to positronium
in vacuum. More apparent emission of positronium generated
in the bulk to the vacuum is known to occur in insulators[39]. In Si, the excitonic bound state probably has a large Bohr

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radius and low binding energy (∼30 meV) and hence it will beinvisible as positronium [40]. Since the excitonic bound state
will be easily broken up by a phonon attack, the transition to
the vacuum positronium occurs in a sufficiently short time.
Strictly to say, in the previous arguments of the type A
positronium work function, the binding energy of the excitonic
state should be considered. But, since it is rather low, itsomission makes no seriously wrong estimations.
Athermal excitonic positronium formation at Si surfaces
was originally proposed by the UCR group. The excitonic
state was assumed to be formed between surface positronsand electrons in surface dangling-bond states by analogy withthe surface excitons observed by time-resolved photoluminescence measurements. If surface electrons participate in
the positronium formation, the temperature dependence may
be different for different surface orientations. Also, since inheavily doped n-type Si the upper surface bands should befully occupied, thus prohibiting electron excitation from thelower band, the positronium fraction may be independentof temperature. These are not consistent with the present
observations shown in Fig. 2. Therefore, it is proposed thatthe athermal positronium is formed between surface positrons
and mainly bulk conduction electrons. This may also be a formof excitonic positronium formation.
V. CONCLUSION

The temperature dependence of the positronium fraction
at Si surfaces is rich in variety and depends on the doping
condition. The diversity arises from two kinds of positroniumthat are produced via the work function and surface positronmediated mechanisms. In both cases, an excitonic state may
be formed between positrons and conduction electrons as aprecursor of the final positronium state.
ACKNOWLEDGMENT

This work was financially supported by JSPS KAKENHI
under Grant No. 17K19061.

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