Journal of the Physical Society of Japan 83, 034702 (2014)
http://dx.doi.org/10.7566/JPSJ.83.034702

Elastic Softening of Surface Acoustic Wave
Caused by Vacancy Orbital in Silicon Wafer
Keisuke Mitsumoto1, Mitsuhiro Akatsu1, Shotaro Baba1, Rie Takasu1, Yuichi Nemoto1, Terutaka Goto1+,
Hiroshi Yamada-Kaneta2, Yuji Furumura3, Hiroyuki Saito4, Kazuhiko Kashima4, and Yoshihiko Saito5
2

1
Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan
Department of Electrical Engineering and Electronics, Kyushu Institute of Technology, Kitakyushyu 804-8550, Japan
3
Philtech Inc., Bunkyo, Tokyo 113-0033, Japan
4
GlobalWafers Japan Co., Ltd., Seiro, Niigata 957-0197, Japan
5
Toshiba Corporation, Yokohama 225-8522, Japan(Received September 13, 2013; accepted December 6, 2013; published online February 7, 2014)

We have performed surface acoustic wave (SAW) measurements to examine vacancies in a surface layer of a borondoped silicon wafer currently used in semiconductor industry. A SAW with a frequency of fs = 517 MHz was optimallygenerated by an interdigital transducer with a comb gap of w ¼ 2:5 µm on a piezoelectric ZnO ﬁlm deposited on the(001) silicon surface. The SAW propagating along the [100] axis with a velocity of vs ¼ 4:967 km/s is in agreementwith the Rayleigh wave, which shows an ellipsoidal trajectory motion in the displacement components ux and uz within apenetration depth of ­p = 3.5 µm. The elastic constant Cs of the SAW revealed the softening of ¦Cs/Cs = 1.9 © 10¹4
below 2 K down to 23 mK. Applied magnetic ﬁelds of up to 2 T completely suppress the softening. The quadrupole
susceptibilities based on the coupling between the electric quadrupoles Ou, Ov, and Ozx of the vacancy orbital consistingof ¥8–¥7 states and the symmetry strains ¾u, ¾v, and ¾zx associated with the SAW account for the softening and its ﬁeld
dependence on Cs. We deduced a low vacancy concentration N = 3.1 © 1012/cm3 in the surface layer within­p = 3.5 µm of the silicon wafer. This result promises an innovative technology for vacancy evaluation in the fabricationof high-density semiconductor devices in industry.

1.

Introduction

Acoustic waves propagating on a free surface of substratematter were investigated for the ﬁrst time by Rayleigh.1) One
hundred years later, electronic technology has widely adopted
a surface acoustic wave (SAW) in miniature electronic
modules of ladder-type delay lines, resonators, and band-passﬁlters using an interdigital transducer (IDT) with high-Q
resonance and low insertion loss.2–5) The SAW is regarded asa diagnostic tool for investigating 4f-electronic states of rareearth compounds,6–8) quantized Hall effects,9,10) topologicalinsulators,11) and graphene.12) The observation of vacancies
in the surface layer of substrate silicon wafers currently
used in semiconductor industry is a prominent issue in the
diagnostic application of the SAW.
Silicon wafers contain two different types of intrinsic point
defects, which consist of a vacancy denoting an unoccupied
lattice site and a silicon interstitial corresponding to a silicon
atom at an irregular lattice site. The low-temperaturesoftening of the elastic constants of boron-doped siliconobserved by bulk ultrasonic waves was successfully described in terms of the quadrupole susceptibilities of a vacancy
orbital with a 8 quartet ground state and a 7 doublet
excited state at 1 K.13–16) Since the fabrication of semiconductor devices for information technology is carried out
using the active region beneath the surface of the substratesilicon wafer, the evaluation of vacancy concentration in thesurface layer of the silicon wafer is keenly required. No
research on vacancy evaluation in the surface layer of siliconwafers has been reported so far to the best of our knowledge.
In the present study, we show low-temperature measurements of the elastic constant of the SAW propagating on aboron-doped silicon wafer. The excitation of the SAW using
an IDT and low-temperature SAW measurement are described in Sect. 2. The low-temperature elastic softening of the

SAW and its magnetic ﬁeld dependence are presented in
Sect. 3. The equations of motion for specifying the SAW on
the silicon surface and the quadrupole susceptibilities forthe vacancy orbital for describing the softening are shownin Sect. 4. We give our discussion in Sect. 5 and ourconclusions in Sect. 6.
2.

Experimental Procedure

We fabricated an IDT on a boron-doped silicon wafer of#9018, which was sliced from a Czochralski (CZ) siliconingot with a diameter of 300 mm. The wafer shows aresistivity range of 21–23 ³0cm at room temperature
indicating a boron concentration of about 0:7  1015 /cm3.
An infrared spectrometer detected an absorption peak at
1107 cm¹1 of interstitial atomic oxygen with a concentrationof 1:0  1018 /cm3 in the wafer. Using a laser cutter, we
reduce the diameter of the wafer concerned from 300 to
200 mm. Because we placed an order for the optimized
treatment of 200 mm wafers with inland semiconductor
facilities, all steps for the IDT fabrication for the SAW
measurement are needed to process the wafers with the same
diameter of 200 mm.
A piezoelectric ZnO ﬁlm was deposited on the (001)surface of the wafer using an AC-sputtering method. The
hexagonal c-axis of the ZnO ﬁlm is aligned perpendicular
to the surface of the wafer. Aluminum (Al) comb-shaped
electrodes were fabricated on the ZnO ﬁlm. There were two
cut-in patterns of 5-tooth-comb ﬁngers with a separation gapof w ¼ 2:5 µm and a ﬁnger length of 2.025 mm as the design
rule 800w. Ordinary semiconductor processing by photoresist lithography and reactive ion etching was used to pattern
the IDT.
The sample of #9018-8 with a rectangular shape of
10  20  0:775 mm3 was diced out from the center of thewafer of #9018. A schematic view of the IDT is shown in

034702-1©2014 The Physical Society of Japan

J. Phys. Soc. Jpn. 83, 034702 (2014)

Interdigital transducer for SAW
Frequency fs = 517 MHzx
Comb gapz λ = 9.37 μm
w = 2.5 μm ~ λ/420
Ampl (mV)

Input ysignalK. Mitsumoto et al.

OutputsignalAl comb
Propagation
Piezoelectric
ZnO film

Penetration depth
λp = 3.5 μm

0-10

tS=1.510 μs

-20
0.0

0.5

1.0

1.5

2.0

2.5

3.0

Time (μs)

Fig. 1. (Color online) Schematic view of the IDT for the SAW measurements on the silicon wafer.

(a)

10

Fig. 3. (Color online) Pulsed shape SAW traveled from the sender IDT to
the receiver IDT, which is d ¼ 7:5 mm away from the sender, on the (001)surface of the silicon wafer. A time of ﬂight ts ¼ 1:510 µs gives a groupvelocity vs ¼ d=ts ¼ 4:967 km/s of the SAW.

20 mm
d = 15.0 mm
d = 7.5 mm

10 mm

2.025 mm

Magnetic fields
Propagation

[010]
[001]
[100]

t = 0.775 mm(b)

4w = 10 μm ~ λPiezoelectric
ZnO-film
Al comb
10 μm

Al film

Fig. 2. (Color online) (a) Top view of the IDT fabricated on a siliconsample with the (001) surface diced out to 10  20  0:775 mm3. A pair of
IDTs with a distance of d ¼ 7:5 mm fabricated on the lower side was
employed in the present SAW measurements. The magnetic ﬁelds wereapplied along the [100] axis parallel to the propagation direction of the SAW.
(b) Expanded view of the Al comb electrode deposited on the piezoelectric
ZnO ﬁlm photographed by a laser microscope.

Fig. 1. The SAW propagates along the x-direction on the z
surface of the silicon wafer. The IDT consisting of Al
electrodes with a comb gap of w ¼ 2:5 µm fabricated on thepiezoelectric ZnO ﬁlm was designed to optimally generate
the SAW with a wavelength of   4w. The opposite IDT
receives the traveled SAW.
Figure 2(a) shows a top view of the sample on which twopairs of IDTs were placed. One IDT consists of a pair ofsender and receiver electrodes. They are separately placedfacing each other with distances of d ¼ 15:0 mm in the upper
line and d ¼ 7:5 mm in the lower line. The present SAW
measurements were carried out on the lower line withd ¼ 7:5 mm. Figure 2(b) shows an expanded view of the
Al comb electrode of the IDT photographed by a laser
microscope.
The low-temperature elastic softening of the SAW was
measured using a dilution refrigerator (Oxford Instruments
Kelvinox 400HA), which was equipped with coaxial linesof CuNi tubes for efﬁcient transmission of high-frequency
electronic signals of the SAW.17) The sample of #9018-8 for
the SAW measurements was placed in thermal contact with a
Ag plate with 1.5 mm in thickness. The Ag plate was attached
to a Cu rod of an extended tail from a mixing chamber forthermal grounding using a Ag wire with a diameter of
1.0 mm. The adoption of the Ag plate has the advantage ofminimizing low-temperature nuclear speciﬁc heat enhancement in applying magnetic ﬁelds.18,19) The temperature of thesample was measured using a calibrated RuO2 resistance. A
superconducting magnet with a bore of 77 mm was equippedto the dilution refrigerator for the generation of magnetic
ﬁelds of up to 9 T.
The SAW with a pulsed shape of 0.3 µs width at arepetition rate of 1 kHz was generated. The velocity change
vs =vs of the SAW as a function of temperature andmagnetic ﬁeld was determined by measuring by phase delayof the pulsed shape SAW. The relatively high frequency of
fs ¼ 517 MHz in the SAW measurements has the advantage
of a high resolution better than vs =vs ¼ 106 . This
excellent resolution leads to the reliable detection of low
vacancy concentrations in the silicon wafer.
3.

Results

The IDT sends an incident pulsed shape SAW of 0.3 µs in
width with a carrier wave of frequency fs ¼ 517 MHz. The
pulsed shape SAW travels along the [100] direction on the(001) surface to the opposite IDT receiver. As is shown in
Fig. 3, a time of ﬂight of ts ¼ 1:510 µs for the signal received
gives a group velocity of vs ¼ d=ts ¼ 4:967 km/s at 4.2 K.
This corresponds to a wavelength s ¼ vs =fs ¼ 9:61 µm,
which is fairly consistent with the wavelength 4w ¼ 10 µm
expected from the IDT design. Furthermore, the experimentally determined group velocity vs ¼ 4:967 km/s of the SAW
on the silicon wafer is in agreement with the Rayleigh wave
velocity v ¼ 4:844 km/s, which is theoretically calculatedusing the equations of motion for the SAW in Sect. 4.1. The
velocity vs ¼ 4:967 km/s and density  ¼ 2:330 g/cm3 of thesilicon wafer give an elastic constant Cs ¼ v2s ¼ 5:748 
1010 J/m3 for the SAW at low temperatures.
Figure 4 shows the temperature dependence of the elastic
constant Cs of the SAW propagating along the [100] axis on
the (001) surface of the silicon wafer. Appreciable softening
with a relative change Cs =Cs ¼ 1:9  104 below 2 K
down to 23 mK was observed. As is shown in the theoreticalexpression of Eq. (39) in Sect. 4.2, the amount of softeningof Cs is proportional to the vacancy concentration N in thesurface layer of the silicon wafer. In Fig. 4, we schematically
picture the dangling bonds of 1 , 2 , 3 , and 4 at thevacancy site with a cubic site symmetry of Td in the silicon

034702-2©2014 The Physical Society of Japan

J. Phys. Soc. Jpn. 83, 034702 (2014)

K. Mitsumoto et al.

(a)Boron doped CZ Silicon
SAW k//H//[100]3dangling bondsyz[1 1 1]10[ 1 1 1]
x

2.0 T
1.0 T
0.4 T

5.7480Cs
5.74850T[1 1 1]

5.74755.7475

H (T) 0

0.0
Fig. 4. (Color online) Low-temperature softening of the elastic constant
Cs of the boron-doped silicon wafer of #9018-8. The SAW propagating along
the [100] direction with the (001) surface was used for the measurements.
The discontinuity at 1.17 K is an extrinsic phase shift due to thesuperconducting transition of the Al comb electrodes used in the IDT. The
solid line is a ﬁt in terms of the quadrupole susceptibilities. A schematic viewof the four dangling bonds i (i ¼ 1; 2; 3; 4) of a vacancy is also shown.

lattice, which form the vacancy orbital and bring about thesoftening of Cs .
There exists a distinct discontinuity in Cs at 1.17 K of
Fig. 4. This is caused by a superconducting transition of theAl ﬁlm used in the comb electrodes of the IDT. The slight
modiﬁcation in the effective impedance of the Al comb
electrodes brings about an extrinsic phase shift at 1.17 K
in superposing on the phase delay of the traveled SAW.
Nevertheless, the softening of Cs is properly analyzed by thequadrupole susceptibility of the vacancy orbital in Sect. 4.2
in merely offsetting the extrinsic phase shift at 1.17 K. The
solid line in Fig. 4 is a ﬁt using a theoretical expression
Eq. (39), which is described in terms of the quadrupole
susceptibility of Eq. (34) in Sect. 4.2.
In Fig. 5(a), we show the low-temperature dependence of
the Cs of the SAW below 0.8 K down to 23 mK at appliedmagnetic ﬁelds of up to 2.0 T along the [100] axis parallel tothe propagation direction. The appreciable softening wasobserved at zero magnetic ﬁeld, while the softening wasconsiderably reduced at applied magnetic ﬁelds of 0.4 and
1.0 T. The higher magnetic ﬁeld of 2.0 T fully suppresses the
low-temperature softening of Cs .
The inset of Fig. 5(a) shows the magnetic ﬁeld dependenceof Cs of the SAW up to 6 T at low temperatures. The
appreciable increase in the amount of Cs =Cs ¼ 1:9  104
at a base temperature of 23 mK with increasing magnetic
ﬁelds from zero to 6 T in the inset of Fig. 5(a) is wellconsistent with the amount of softening of Cs =Cs ¼ 1:9 
104 below 2 K down to 23 mK at zero magnetic ﬁeld shownin Fig. 4(a). The increased temperature of 0.3 K reduces theﬁeld dependence to Cs =Cs ¼ 0:7  104 . Further increasein temperatures to 0.7, 1.5, and 4.0 K completely smears out
the ﬁeld dependence of Cs . We show the results of thetheoretical calculation by the solid and dashed lines inFig. 5(b). Details of the calculation of Cs at the magnetic
ﬁelds in terms of the quadrupole susceptibilities are shown in
Sect. 4.2.
Phase shifts associated with the critical ﬁeld Hc  0:02 Tof the superconducting phase in the Al electrodes were

0.2

0.4
T (K)

2

4

0.6

-4

4.0 K
1.5 K
0.7 K
0.3 K
23 mK

5.7480[1 1 1]

ΔCs/Cs = 1.9x10

Cs (10 J/m )5.7485

6

0.8

(b)

Fig. 5. (Color online) (a) Temperature dependence of the elastic constant
Cs of the SAW on the boron-doped silicon wafer of #9018-8 below 0.8 K
down to 23 mK at applied magnetic ﬁelds along the [100] axis. The inset
shows the magnetic ﬁeld dependence of Cs at low temperatures. Anomalies
associated with the superconducting critical ﬁeld of the Al comb electrodes
were observed at 0.02 T. (b) Theoretical calculation of temperaturedependence of Cs in terms of the quadrupole susceptibilities. The inset
shows the magnetic ﬁeld dependence of the calculation of Cs at low
temperatures.

observed in the inset of Fig. 5(a). The alternative adoption of
Cu electrodes of a normal metal instead of Al electrodes canbe favorable for avoiding the extrinsic phase shift due to thesuperconducting transition in Al electrodes.
4.

Theory4.1 Equations of motion for SAW
We shall treat equations of motion for the SAW
propagating along the x-direction on the z surface of thesubstrate silicon wafer.20,21) We set a free space for z < 0 anda solid region for z > 0. This boundary condition gives
constraint for stress components of iz (i ¼ x; y; z) acting on
the free z surface as@ux @uzþxz ¼ zx ¼ C44 "zx ¼ C44
¼ 0;
@z
@x


@uz @uyþ¼ 0;
yz ¼ C44 "yz ¼ C44
@y
@z
ð1Þ
zz ¼ C12 ð"xx þ "yy Þ þ C11 "zz@ux @uy
@uz
þ
¼ 0:
¼ C12
þ C11
@x
@y
@z

034702-3©2014 The Physical Society of Japan

J. Phys. Soc. Jpn. 83, 034702 (2014)

K. Mitsumoto et al.

Here, ui (i ¼ x; y; z) denote the displacement vector components, and C11 , C12 , and C44 denote the bulk elastic constantsof the silicon lattice with a cubic symmetry. The elasticstrains "ij (i, j ¼ x; y; z) are deﬁned as22)
@ux
;
"xx ¼
@x
@uy
"yy ¼
;
@y
@uz
"zz ¼
;
@z
ð2Þ
@uz @uy
þ
;
"yz ¼
@y
@z
@ux @uz
"zx ¼
þ
;
@z
@x
@uy @ux
"xy ¼
þ
:
@x
@y
The SAW propagating along the x-direction on the z
surface leads to a constraint that the displacement vector
components ux and uz are relevant and uy is irrelevant. The
SAW propagating along the x-direction keeps the angles forthe x–y and y–z axes orthogonal. The equations of motion for
the SAW consisting of the displacement components ux anduz in each inﬁnitesimal volume are described as@ 2 ux@ 2 ux@ 2 ux@2 uz;¼ C11 2 þ C44 2 þ ðC12 þ C44 Þ
2
@t
@x
@z
@z@x




 C11  v2  C44 q2
ðC12 þ C44 Þqii ¼ 0:
2
2
 ðC12 þ C44 Þqi
C44 þ v þ C11 qi
Equation (8) gives an algebraic equation of qi as
C11 C44 q4i  fC11 ðC11  v2 Þ þ C44 ðC44  v2 Þð8Þ ðC12 þ C44 Þ2 gq2i þ ðC11  v2 ÞðC44  v2 Þ ¼ 0: ð9Þ
Now we shall go back to the boundary conditions of thestress tensor iz ¼ 0 acting on the free z surface at z ¼ 0 of
Eq. (1). Substituting Eq. (4) into Eq. (1), we obtain simultaneous equations for U1 and U2 as
U1 ð1  q1 Þ exp½kq1 z
þ U2 ð2  q2 Þ exp½kq2 z ¼ 0;
U1 ðC12 þ C11 1 q1 Þ exp½kq1 z
þ U2 ðC12 þ C11 2 q2 Þ exp½kq2 z ¼ 0:

ð10Þ

Nontrivial solutions of Eq. (10) require that the determinantshould satisfy1  q 1
2  q 2
ð11Þ ¼ 0:
 C12 þ C11 1 q1 C12 þ C11 2 q2 
Using the i of Eq. (7) and the qi of Eq. (9), we obtain an
algebraic equation from the determinant of Eq. (11):


2
2
C11
C12
R 1  2  R ¼ R2 ð1  RÞ:
ð12Þ
1
C44
C11

P
Here, 2i¼1 means the sum over two solutions of penetration
factors qi (i ¼ 1; 2) for the z-axis. Ui and Wi denote
amplitude of ux and uz of the SAW, respectively.
Substituting ux and uz of Eq. (4) into Eq. (3), we obtain
simultaneous equations

Here, we introduce the ratio of the elastic constant Cs ¼ v2of the SAW to the bulk elastic constant C11 as R ¼ Cs =C11 .
Substituting the elastic constants C11 ¼ 15:681, C12 ¼
6:181, and C44 ¼ 8:063 in 1010 J/m3 units at 4.2 K15) and themass density  ¼ 2:330 g/cm3 of the silicon into Eq. (12),
we obtain three solutions for SAW velocities of v ¼ 4:844,
8.341, and 11.873 in km/s units. Among them, the lowest
velocity of v ¼ 4:844 km/s corresponds to the Rayleighwave, while other higher velocities signify the pseudoSAW.23) The group velocity vs ¼ 4:967 km/s determined by
the time of ﬂight of the pulsed shape SAW in Fig. 3 is in fairagreement with the lowest velocity v ¼ 4:844 km/s of theRayleigh wave in the calculation of Eq. (12).
Substituting the solution v ¼ 4:844 km/s and the bulk
elastic constants of the silicon crystal into Eq. (9), we obtain
the penetration factor qi of the Rayleigh wave. Because theSAW is expected to penetrate into the solid for z > 0 during
decay, we adopt the solution q with a positive value in the realpart, qRe , and a negative one in the imaginary part, qIm , asðC11  v2  C44 q2i ÞUi þ ðC12 þ C44 Þqi ðiWi Þ ¼ 0;

q ¼ qRe þ iqIm ¼ 0:4311  0:5216i:

ð3Þ
@ 2 uz@ 2 uz@ 2 uz@2 ux
 2 ¼ C11 2 þ C44 2 þ ðC12 þ C44 Þ
:
@t
@z
@x
@x@z
The SAW with the displacement components ux and uz
propagates along the x-axis with a phase velocity v and awavenumber k ¼ 2f=v for a frequency f. Furthermore, theSAW penetrates into the silicon surface along the z-axis forz > 0. Therefore, ux and uz are given as
ux ¼

2
X

Ui exp½kqi z þ ikðx  vtÞ;

i¼1

uz ¼

2
X

ð4Þ
Wi exp½kqi z þ ikðx  vtÞ:i¼1

2

 C11 q2i ÞðiWi Þ ¼ 0:

ð5Þ

ðC12 þ C44 Þqi Ui  ðC44  v
The second line of Eq. (5) gives a relation between theamplitude of Ui and Wi asiWi ¼

ðC12 þ C44 Þqi
Ui ¼ i Ui :C44  v2  C11 q2i

ð6ÞHere, i is a ratio denoting the difference in the amplitude andphase between Ui and Wi :
i ¼

ðC12 þ C44 Þqi:
C44  v2  C11 q2i

ð7Þ

Nontrivial solutions of Eq. (5) require an equation for thedeterminant:ð13ÞThis solution gives the ratio £ as
 ¼ Re þ iIm ¼ 0:4311  1:1120i:

ð14Þ

The corresponding ux and uz of the SAW of Eq. (4) are givenasux ¼ U exp½kqz þ ikðx  vtÞ;uz ¼ iux ¼ 1:19 exp½iU exp½kqz þ ikðx  vtÞ:

ð15Þ

Here, we have an angle of  ¼ 158:81° for the phase shift of
i. The displacement components of ux and uz vary at thepositions x and z and the time t with a phase velocity
v ¼ 4:844 km/s along the x-direction. The real parts of ux
and uz are written as

034702-4©2014 The Physical Society of Japan

J. Phys. Soc. Jpn. 83, 034702 (2014)

K. Mitsumoto et al.

Displaced surface
Damped motion
Particle trajectory

consists of the four symmetry strains "B , "u , "v , and "zx . The
real parts of Eq. (18) are written as

Propagation
direction¼ AB kU exp½kqRe z cos½kðx  vtÞ  kqIm z þ B ;
"ðReÞ
B

λ

x

¼ Au kU exp½kqRe z cos½kðx  vtÞ  kqIm z þ u ;"ðReÞu
¼ Av kU exp½kqRe z cos½kðx  vtÞ  kqIm z þ v ;
"ðReÞ
v

z

"ðReÞ
zx ¼ Azx kU exp½kqRe z cos½kðx  vtÞ  kqIm z þ zx :ð19ÞThe numerical values q of Eq. (13) and £ of Eq. (14) give thesharing parameters in the amplitude A (phase shift  ) asAB ¼ 0:346 ( B ¼ 42:61°), Au ¼ 1:491 ( u ¼ 281:37°), Av ¼
1:000 ( v ¼ 90:00°), and Azx ¼ 1:045 ( zx ¼ 214:41°).
Introducing a function of f ðz; x; t;  Þ depending on x, z,
and t with the phase shift  as

λpf ðz; x; t;  Þ ¼ exp½kqRe z cos½kðx  vtÞ  kqIm z þ  ;
Fig. 6. (Color online) Instantaneous displacement components ux and uzof the SAW with a wavelength  ¼ 9:37 µm propagating along the x-axis on
the z surface of the silicon wafer. The dot-dashed line denotes the displaced
surface of the SAW. The dotted line denotes the ellipsoidal particle trajectoryof uz and ux when penetrating into the z-axis on the silicon wafer with apenetration depth p ¼ 3:5 µm. The damped motion of ux and uz is shown by
the dashed line.

uðReÞ¼ U exp½kqRe z cos½kðx  vtÞ  kqIm z;xuðReÞ
¼ 1:19U exp½kqRe z cos½kðx  vtÞz kqIm z þ 158:81 :
ð16Þ
The ellipsoidal trajectory of the instantaneous amplitude
and directions of ux and uz in Eq. (16) is shown by arrows in
Fig. 6. The dot-dashed line in Fig. 6 shows the displaced
surface of ux and uz along the x-direction oscillating witha wavelength  ¼ 9:37 µm in the present SAW of fs ¼
517 MHz. As shown in Fig. 6, the amplitude of ux and uz
decreases to 1=e when penetrating in the z-direction at a depthp ¼ =ð2qRe Þ ¼ 3:5 µm accompanied by a oscillation witha wavelength of z ¼ 2=ðkjqIm jÞ ¼ =jqIm j ¼ 18:0 µm.
In order to analyze the softening of the SAW in terms of
the quadrupole-strain interaction in Sect. 4.2, we shall treat
the symmetry strains "B , "u , and "v deﬁned as@ux @uy @uzþþ;"B ¼ "xx þ "yy þ "zz ¼
@x
@y
@z


2"zz  "xx  "yy1@uz @ux @uy
pﬃﬃﬃ
"u ¼
¼ pﬃﬃﬃ 2; ð17Þ
@z@x@y3
3@ux @uy"v ¼ "xx  "yy ¼

:
@x
@y
Substituting Eq. (15) into Eqs. (2) and (17), we obtain thesymmetry strains"B ¼ ikðq þ 1Þux ;ik"u ¼ pﬃﬃﬃ ð2q  1Þux ;3"v ¼ ikux ;ð18Þ

"yz ¼ 0;ð20Þ
we can write the symmetry strains "B , "u , "v , and "zx with theamplitude A and phase shifts of  of Eq. (19) as"ðReÞ ¼ A f ðz; x; t;  Þ :ð21Þ

Here, an external perturbation parameter ¼ kU ¼ 2U=,proportional to the injection amplitude of the pulsed shape
SAW, is introduced.
Figure 7(a) shows the instantaneous amplitude of thesymmetry strains "B , "u , "v , and "zx of Eq. (19) along x-axis
at z ¼ 0 and t ¼ 0. The symmetry strains "u with the phaseshift u ¼ 281:37° and "v with v ¼ 90:00° in Fig. 7(a)oscillate in keeping the almost antiphase relationj u  v j ¼ 191:37°. Furthermore, the symmetry strains "Bwith B ¼ 42:61° and "zx with zx ¼ 214:41° also sharealmost the antiphase relation j B  zx j ¼ 171:80°. This iscaused by the fact that the SAW travels along the x-direction
like a plane wave but considerably decays along the zdirection. The SAW located within the surface layer of p isreminiscent of a stationary wave conﬁned in an end-closed
waveguide.
The instantaneous amplitude of the symmetry strains "B ,"u , "v , and "zx along the penetrating z-direction for z > 0 isshown in Fig. 7(b) for kx ¼ 1:79 and t ¼ 0. The amplitude of
an individual strain in Fig. 7(b) considerably decays along
the z-direction as deep as kp ¼ 2p = ¼ 1=qRe ¼ 2:3. The
almost antiphase sharings between the symmetry strains "uand "v and between the strains "B and "zx are again observedin Fig. 7(b).
The total elastic energy Utotal ðz; x; t; Þ depending on x, z,
and t associated with the SAW is written by the sum of theelastic energies UB , Uu , Uv , and Uzx , which are caused by thesymmetry strains "B , "u , "v , and "zx , respectively. Taking
account of the symmetry strain " with the individual
amplitude A and the phase shift  , we obtain the totalelastic energy
X1
2C ð"ðReÞUtotal ðz; x; t; Þ ¼
 Þ
2


"zx ¼ kð  qÞux ;¼"xy ¼ 0:Consequently, the Rayleigh wave of Eq. (15) propagating
along the x-direction on the z surface of the silicon wafer
034702-5

¼

X1
2


C fA f ðz; x; t;  Þg2 2

1
CB ðAB fB ðz; x; t; B ÞÞ2 2
2©2014 The Physical Society of Japan

J. Phys. Soc. Jpn. 83, 034702 (2014)

K. Mitsumoto et al.

(a)

(a)

(b)

(b)

Fig. 7. (Color online) (a) Instantaneous amplitude for the symmetry
strains "B , "u , "v , and "zx induced by the SAW at the surface z ¼ 0 andtime t ¼ 0. The strains "B and "zx , and "u and "v oscillate in an almost
antiphase manner relative to each other. This implies the stationary wave
character of the SAW during decay along the z-axis with the penetration
depth p . (b) The instantaneous amplitude for the symmetry strains "B , "u , "v ,and "zx induced by the SAW along the z > 0 direction for kx ¼ 1:79 andt ¼ 0.

1
CT ðAu fu ðz; x; t; u ÞÞ2 2
21
þ CT ðAv fv ðz; x; t; v ÞÞ2 2
21
þ C44 ðAzx fzx ðz; x; t; zx ÞÞ2 2 :ð22Þ
2Here, weP abbreviate the elastic constant as CT ¼ ðC11 
C12 Þ=2.  denotes the sum over the symmetry strains "B ,"u , "v , and "zx . In Fig. 8(a), the instantaneous feature of thetotal elastic energy Utotal ðz; x; t; Þ along the x-direction at
z ¼ 0 and t ¼ 0 for the perturbation parameter ¼ kU ¼ 1 isshown together with the instantaneous energies caused by thesymmetry strains. The total elastic energy Utotal ðz; x; t; Þ of the
SAW at t ¼ 0 in Fig. 8(a) shows its maximum for kx ¼ 1:79.
Employing the power-reduction formula and linear
combination of trigonometric functions, we can reduce thetotal energy Utotal ðz; x; t; Þ in Eq. (22) to the time-dependentand time-independent parts of the elastic energies as

Fig. 8. (Color online) (a) Instantaneous elastic energy Utotal of the SAW of
Eq. (22) for ¼ kU ¼ 1 along the x-axis at the surface of z ¼ 0 and for time
t ¼ 0 together with the elastic energies UB , Uu , Uv , and Uzx . The
Utotal ðx; z; t; Þ of the SAW is described by the sum of the time-dependentand time-independent parts. (b) The instantaneous elastic energy Utotal of the
SAW penetrating in the z-direction for kx ¼ 1:79 and t ¼ 0 is shown together
with the elastic energies UB , Uu , Uv , and Uzx for ¼ kU ¼ 1. The elastic
energy shows a rapid decay with a penetration depth of kp ¼ 2:3.

fs ðz; x; t; s ÞþUtotal ðz; x; t; Þ ¼¼ exp½kqRe z cos½kðx  vtÞ þ kqIm z þ s ;and the amplitude

The phase shift

ð25Þ

s of the SAW is represented bygc
cos½2 s  ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ;2
gc þ g2s

gs
sin½2 s  ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ;2
gc þ g2s
1X
C A2 ;
g¼
4 

1 0
1
C ðjAs jfs ðz; x; t; s ÞÞ2 2 þ K0 ðzÞ 2 : ð23Þ2 s
2The former time-dependent energy means the part propagating along the x-direction, which is described in terms of thefunctionsﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
4 g2c þ g2s:
jAs j ¼
Cs0

ð24Þ

gc ¼

1X
C A2 cos 2  ;
4 

gs ¼

1X
C A2 sin 2  :
4 

ð26Þ

The latter time-independent energy denotes the non-propagating part along the penetrating z-direction. This is
described by the function of

034702-6©2014 The Physical Society of Japan

J. Phys. Soc. Jpn. 83, 034702 (2014)

K0 ðzÞ ¼ 2ðg 

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
g2c þ g2s Þ exp½2kqRe z:K. Mitsumoto et al.

Γ7

ð27Þ

The time-dependent elastic energy for the propagating partof the SAW is described by the square of As fs ðz; x; t; s Þ forthe amplitude jAs j ¼ 1:505 and the phase s ¼ 77:47°. The
time-independent elastic energy for the stationary wave iswritten as ð1=2ÞK0 ðzÞ 2 for K0 ðzÞ ¼ 6:428  1010 J/m3 at
z ¼ 0. The time-dependent and time-independent parts of theinstantaneous energy of Eq. (23) for ¼ kU ¼ 1 at z ¼ 0
and t ¼ 0 are also pictured in Fig. 8(a). The Rayleigh wave
shares out the amplitudes and phase shifts among thesymmetry strains of Eq. (19) to minimize the time-dependent
elastic energy of ð1=2ÞCs fAs fs ðz; x; t; s Þg2 2 .
Figure 8(b) shows the instantaneous feature of the totalenergy of Utotal ðz; x; tÞ and its symmetry parts of UB , Uu , Uv ,and Uzx in penetrating into the z-direction for z > 0. Here, we
adopt the position of kx ¼ 1:79 and t ¼ 0 for ¼ kU ¼ 1.
The instantaneous energy of the SAW rapidly decays within
the surface layer of the penetration depth of kp ¼ 2:3. The
diagnostic application of the SAW guarantees for the vacancy
evaluation within a surface layer as deep as p ¼ 3:5 µmof the silicon wafer. This prominent feature of the SAW
measurement is deﬁnitely distinguished from the bulk
ultrasonic measurement to evaluate the vacancy concentration of the inside of the wafer.
4.2 Quadrupole susceptibility
The vacancy orbital of a charge state V þ in the borondoped silicon wafer accommodates three electrons owing tocharge neutrality, where one electron is released from thevacancy orbital to an acceptor state of the boron dopant. Foursp3 dangling bonds of i (i ¼ 1; 2; 3; 4) shown in Fig. 4 at a
cubic site symmetry of Td are lifted to a singlet a1 state withan energy of 3 and a triplet t2 state with £ owing to thetransfer energy of  ¼ hj jH0 ji i ( > 0).24) The energy
gap of 4  1:7 eV  2:0  104 K between the singlet a1
state and the triplet t2 state was theoretically estimated.25,26)
The former singlet a1 state located beneath the top of thevalence band accommodates two electrons with an antiparallel spin orientation, while the latter triplet t2 state locatednear the chemical potential possesses an electron with spin
S ¼ 1=2. The softening of the elastic constants of the siliconwafer measured by the bulk ultrasonic waves was successfully described in terms of the quadrupole susceptibility ofthe vacancy orbital.13,27,28)
The magnetic ﬁeld dependence of the elastic softening
is responsible for the magnetism of the vacancy orbital
accommodating three (odd) number electrons. The magnetism of the vacancy orbital is caused by the spin–orbit
interaction of HSO ¼ L  S ( < 0) of the electron withorbital L ¼ 1 and spin S ¼ 1=2 in the triplet t2 state. This
HSO gives rise to the 8 quartet ground state with the energyof =2 and the 7 excited state with the energy of as29)
8
rﬃﬃﬃﬃ



>
2 
1
1  1
>
>
p
ﬃﬃ
ﬃ

1;

0;
ji
¼
>
<
3
2
3 2
j7 i
ﬃﬃﬃ
ﬃ
r



;
>


>
2
1
1
1
>


>
 pﬃﬃﬃ 0; 1;:j i ¼
3
2
2
3

Charge stateV+
L=1
S = 1/2

α, β

|ξ|
|ξ|/2

Spin-orbit interaction
HSO = ξL • S(ξ < 0)

Γ8
ν, κ

λ, μ

Fig. 9. The triplet t2 state with L ¼ 1 and S ¼ 1=2 of the charge state V þof the vacancy orbital in boron-doped silicon forms the 8 quartet groundstate (¯, ¬, ­, and ®) and 7 doublet excited state (¡ and ¢) due to the spin–
orbit interaction HSO ¼ L  S ( < 0). The 8 –7 states ¯ and ¬, ­ and ®,and ¡ and ¢ on the right-hand side have polar, donut, and spherical charge
distribution, respectively.

8 rﬃﬃﬃﬃ
>1
1
2  1
>

>
p
ﬃﬃ
ﬃ
þ
j
i
¼
1;

0;
>
>
>
2
3 2
3
>
>
ﬃﬃﬃ
ﬃ
r
>



>
>
1 
1
2 
1
>
>
þ
0; 
< ji ¼ pﬃﬃﬃ 1;
2
3
2 :
3
j8 i


>
>
> ji ¼ 1; 1
>
>
 2
>
>
>

>

>

>
1
>

>
: ji ¼ 1; 
2ð28ÞHere, jLz ; Sz i means the state with angular momentum
components over Lz ¼ 1; 0 for L ¼ 1 and spin components
over Sz ¼ 1=2 for S ¼ 1=2. The quadrupole susceptibilities
for the vacancy orbital with 8 (0 K) and 7 (1 K) for the
spin–orbit coupling of ¼ 2=3 K well described thesoftening of the elastic constants measured by the bulk
ultrasonic waves.14–16)
The spatial charge distribution distinguishes among three
Kramers doublets of the vacancy orbital of Eq. (28). As
shown in Fig. 9, the Kramers doublet of ¯ and ¬ in the 8quartet shows a polar charge distribution elongating along thez-axis, while the Kramers doublet of ­ and ® bears a donuttype distribution spreading in the x–y plane. The 7 Kramers
doublet of ¡ and ¢ has a spherical charge distribution. The
characteristic charge distribution in the 8 –7 states of thevacancy orbital is responsible for the coupling between theelectric quadrupoles and the symmetry strains associated with
the SAW.
It is known that the 8 quartet has three magnetic dipoles,
ﬁve electric quadrupoles, and seven magnetic octupoles.30)
Among them, the electric quadrupoles couple to the elasticstrains associated with ultrasonic waves. Actually, theconsiderable elastic softening with decreasing temperature
was observed in CeB631,32) and Ce3Pd20Ge633) with the 8quartet ground state. The coupling between the electric
quadrupoles of the 8 quartet ground state of the vacancy
orbital and the elastic strains of the SAW is relevant for thedescription of the softening of Cs .
The applied magnetic ﬁelds of H lift the Kramers doubletsof the vacancy orbital. This is caused by the Zeeman energy as
HZeeman ¼ B ðL þ 2SÞ  H:

ð29Þ

Here, B ¼ eh =ð2mcÞ  0:67 K/T is the Bohr magneton.
Among the two Kramers doublets of the 8 quartet ground034702-7©2014 The Physical Society of Japan

J. Phys. Soc. Jpn. 83, 034702 (2014)

Irrep
Γ

Γ1

Electric
MultipoleO ΓγElastic
StrainElastic
Constant
C Γγ

Γ3

Γ5

Ou
H0

ε Γγ

K. Mitsumoto et al.

εB= ε xx + ε yy + ε zz
CB =C11 + 2C12
3=

2L2z − L2x − L2yOyz = Ly Lz + Lz Ly Ozx = Lz Lx + Lx Lz Oxy = Lx Ly + Ly Lx

3

εu
=

Ov = L2x − L2y

2ε zz − ε xx − ε yy

ε v = ε xx − ε yy

ε yz

ε xy

ε zx

3

CT =

C11 − C122

C44

Fig. 10. (Color online) Electric multipoles, counterpart symmetry strains, and corresponding elastic constants. The quadrupole-strain interaction for thevacancy orbital brings about the softening of the elastic constant Cs . H0 means the electric hexadecapole described by the 1 -irreducible representation (irrep).

state, the ¯ and ¬ states with magnetic moments of ð2=3ÞB
and ­ and ® states with 2B are distinct from each other atmagnetic ﬁelds along the z-axis. The 7 Kramers doublet
state ¡ and ¢ have different magnetic moments of ð1=3ÞB .
In the present experiments, we applied the magnetic ﬁelds
along the x-axis parallel to the propagating direction of theSAW. The corresponding Zeeman energy HZeeman ¼ B ðLx þ
2Sx ÞHx is responsible for the magnetic ﬁeld effects of theelastic constant Cs of the present SAW measurements on thesilicon wafer.
The coupling between the electric monopole and thevolume strain "B with the full symmetry 1 is relevant for thevalence instability system like SmB6.34) Because the charge
state V þ of the boron-doped silicon is stable, the coupling
between the electric monopole and "B seems to be irrelevant.
The coupling between the higher order electric hexadecapole
H0 ¼ 35L4z  f30LðL þ 1Þ  25gL2z þ 3L2 ðL þ 1Þ2  6LðL þ1Þ þ ð5=2ÞðL4þ þ L4 Þ and the volume strain "B is principally
possible in the highly degenerate 8 –7 states. The present
analysis, however, excludes it because of its negligible
contribution to the softening. We restrict ourselves to thecoupling between the symmetry breaking strains associatedwith the SAW and the appropriate electric quadrupoles of thevacancy orbital.
The softening of Cs of the SAW down to the basetemperature of 23 mK in Fig. 4 is accounted for in terms of
the quadrupole susceptibility for the vacancy orbital, which islocated in the surface layer within the penetration depth of
p . The scalar potential energy V of the vacancy orbital ismodulatedby the elastic strains of the SAW as V ¼ V0 þ
PðReÞ ðReÞ ð@V=@" Þ" . Here, V0 denotes an unperturbed
potential energy. The derivative of @V=@"ðReÞ is represented
by the electric quadrupole O , which is written by thesymmetry square of the Cartesian coordinates x, y, and z, andthe quadrupole-strain coupling constant g . Therefore, theperturbation energy due to the elastic strains of the SAW is
written by the quadrupole-strain interaction HQS as35,36)
ðReÞ
HQS ¼ g3 ðOu "ðReÞþ Ov "ðReÞuv Þ þ g5 Ozx "zx :ð30Þ

The eigenfunctions of the 8 –7 states of the vacancy orbitalof Eq. (28) are represented by the quantum numbers of Lz
and Sz . Taking account of the Wigner–Eckert theorem,37) we
can write the electric quadrupoles using the symmetry squareof the orbital angular momentum operators of Lx , Ly , and Lz .
We p
adopt
the electric quadrupoles of Ou ¼ ð2L2z  L2x 
ﬃﬃﬃ
2Ly Þ= 3 and Ov ¼ L2x  L2y with the 3 -irreducible representation and Ozx ¼ Lz Lx þ Lx Lz with the 5 -irreducible representation. In Fig. 10, we list the electric quadrupoles(multipoles) and the counterpart symmetry strains in thequadupole-strain interaction of the cubic system.
The perturbation energy Ei ð"ðReÞ Þ for the ith state (i ¼; ; ; ; ; ) of the vacancy orbital consisting of the8 –7 states is calculated using the perturbation Hamiltonian
for the appropriate symmetry strain of
HQS ¼ g O "ðReÞ

"ðReÞ
 as
ðReÞ
0
Ei ð"ðReÞ Þ ¼ Ei þ g hijO jii"þ g2

X jhijO jjij2
jð6¼iÞ

Ei0  Ej02
ð"ðReÞ Þ :

ð31Þ

Here, Ei0 denotes the unperturbed energy of the ith state of thevacancy orbital of Eq. (28).
The total free energy of the system consisting of thevacancy orbital with the vacancy concentration N in thesurface layer of the silicon wafer is written as
1 0 ðReÞ 2
C ð" Þ
Fð"ðReÞ Þ ¼
2   ""##X
Ei ð"ðReÞ Þ: ð32Þ
 NkB T ln
exp 
kB T
i

034702-8©2014 The Physical Society of Japan

J. Phys. Soc. Jpn. 83, 034702 (2014)

K. Mitsumoto et al.

ﬁrst term on the right-hand side of Eq. (34) denotes thevan Vleck term associated with off-diagonal processes andthe second term is the Curie term associated with the
diagonal processes. The former van Vleck term leads to thetemperature-independent softening, while the latter Curie
term brings about the steep softening proportional to thereciprocal temperature. The quadrupole interaction among thevacancy orbitals at different sites is excluded in the present
analysis, because the strong quadrupole-strain interaction
dominates the low-temperature softening of the system andthe intersite quadrupole interaction scarcely contributes to thesoftening.15)
In order to explain the appreciable softening of Cs causedby the Curie term of Eq. (34), we present the matrix elements
hijO jji of the electric quadrupoles Ou , Ov , and Ozx for the8 –7 states of the vacancy orbital in Eq. (28) at zeromagnetic ﬁeld as

The second derivative of the free energy with respect tothe symmetry strain "ðReÞ leads to the elastic constant C
as
C ¼

@2 F
¼ C0   Ng2 ðO Þ:@"ðReÞ2ð33Þ

Here, the quadrupole susceptibility ðO Þ for the electric
quadrupole O is deﬁned as7,8,35)
X nj  niðO Þ ¼
jhijO jjij2
0
0
E

E
i
j
i; j
Ei0 6¼Ej0

2
þ

X

X

1 4
jhijO jiij2 ni 
kB T
i

!2 3
jhijO jiijni 5: ð34ÞiHere, ni ¼ Z 1 exp½Ei0 =kB T is the Boltzmann
P occupation
probability with the partition function Z ¼ i exp½Ei0 =kB T. On the analogy of the magnetic susceptibility, the
0 
B 0
0
B
B 0
0
B pﬃﬃﬃ
B 2 0
1
hijOu jji ¼ pﬃﬃﬃ B
pﬃﬃﬃ
3 B
B 0
2
B
B
@ 0
0
0


0





1

0

0

0
pﬃﬃﬃ
2

0

1

0

0

0

1

0

0

0

1

0C
C
0C
C
0C
C;
C
0C
C
C
0A

0

0

0

1

pﬃﬃﬃ
2

0


B
B 0
B
B
B
B
B 0
B
B
B
B
B 0
1
hijOzx jji ¼ pﬃﬃﬃ B
3 B
B rﬃﬃﬃﬃ
B
3
B
B
2
B
B 1
B
B  pﬃﬃﬃ
B
2
B
@

0

0 
B 0
0
B
B 0
0
B
B 0
1
0
hijOv jji ¼ pﬃﬃﬃ B
3 B
B 0
0
B
pﬃﬃﬃ
B
@ 0
2
pﬃﬃﬃ

2 0


rﬃﬃﬃﬃ
3
2



 1

1
 pﬃﬃﬃ
2

0

0

0

0

1

0

0

0

0

0

1

0

0

1
pﬃﬃﬃ
2

0

1

0

C
0 C
C
C
C
1 C
pﬃﬃﬃ C
2C
C
C
C
0 C
C:
C
C
C
1 C
C
C
C
0 C
C
C
C
A
0

0

0

0

rﬃﬃﬃﬃ
3

2

rﬃﬃﬃﬃ
3

2

The electric quadrupoles Ou , Ov , and Ozx commonly havediagonal elements for the ¯, ¬, ­, and ® of the 8 quartet
ground state, but no elements for the ¡ and ¢ of the 7
doublet excited state. The lifting of the 8 quartet groundstate to two Kramers doublets due to the quadrupole-straininteraction in Eq. (30) brings about the softening of theelastic constant due to the Curie term in Eq. (34). The offdiagonal elements of Ou , Ov , and Ozx between the 8 quartet
ground state and 7 doublet exited state in Eq. (35) lead tothe van Vleck term in Eq. (34).
For accurate calculations of the low-temperature softeningof Cs , we explain the perturbation Hamiltonian of Eq. (30)using the time-dependent symmetry strains of Eq. (21)induced by the SAW as

HQS ¼

X


¼

X



0 1

 1
pﬃﬃﬃ
0
2C
pﬃﬃﬃ
C
2 0 C
C
0
1 C
C;
C
1
0 C
C
C
0
0 A

1 0

0

0 0
0 0
0 0
0 0



0

ð35Þ

g O "ðReÞg O A f ðz; x; t;  Þ

¼ g3 Ou Au fu ðz; x; t; u Þ þ g3 Ov Av fv ðz; x; t; v Þþ g5 Ozx Azx fzx ðz; x; t; zx Þ :ð36ÞTaking account of the quadrupole-strain interaction of
Eq. (36), the perturbation energy for the vacancy orbital iscalculated up to the second-order processes of the external
perturbation ¤ using Eq. (31). Here, the matrix elements ofthe electric quadrupoles of Eq. (35) are employed.
The total elastic energy Utotal of Eqs. (22) and (23) for theSAW is renormalized by the quadrupole-strain interaction of
Eq. (36) as

034702-9©2014 The Physical Society of Japan

J. Phys. Soc. Jpn. 83, 034702 (2014)

K. Mitsumoto et al.

(a)

y

x
z

(b)

y

H

(c)

x
H

z

y

x
z

H

Fig. 11. (Color online) Temperature dependences of the quadrupole susceptibilities ðOu Þ in (a), ðOv Þ in (b), and ðOzx Þ in (c) under applied themagnetic ﬁelds of H ¼ ðHx ; 0; 0Þ parallel to the propagation direction of the SAW. The anisotropic charge distribution of the electric quadrupoles is alsopresented. The inset shows the magnetic ﬁeld dependences of the quadrupole susceptibilities at low temperatures.

1
1
Utotal ¼ Cs ðAs fs ðz; x; t; s ÞÞ2 2 þ K0 ðzÞ 22
2
1¼ CB ðAB fB ðz; x; t; B ÞÞ2 2
21
þ ðCT  Ng23 ðOu ÞÞðAu fu ðz; x; t; u ÞÞ2 2
21
þ ðCT  Ng23 ðOv ÞÞðAv fv ðz; x; t; v ÞÞ2 2
21
þ ðC44  Ng25 ðOzx ÞÞðAzx fzx ðz; x; t; zx ÞÞ2 2 : ð37Þ2
The second derivative of the elastic energy of the SAWof Eq. (37) with respect to the external perturbation ¤ gives
the softening and its magnetic ﬁeld dependence on therenormalized elastic constant Cs of the SAW as@2 Utotal¼ Cs ðAs fs ðz; x; t; s ÞÞ2 þ K0 ðzÞ@ 2¼ CB ðAB fB ðz; x; t; B ÞÞ2þ ðCT  Ng23 ðOu ÞÞðAu fu ðz; x; t; u ÞÞ2þ ðCT  Ng23 ðOv ÞÞðAv fv ðz; x; t; v ÞÞ2
þ ðC44  Ng25 ðOzx ÞÞðAzx fzx ðz; x; t; zx ÞÞ2 :ð38Þ

The temperature dependence of Cs shown in Fig. 4 and themagnetic ﬁeld dependence of Cs shown in Fig. 5(a) are
explained in terms of the quadrupole susceptibilities ðOu Þ,ðOv Þ, and ðOzx Þ asCs ¼ Cs0 

N½g2 ðAu fu ðz; x; t; u ÞÞ2 ðOu ÞðAs fs ðz; x; t; s ÞÞ2 3

þ g23 ðAv fv ðz; x; t; v ÞÞ2 ðOv Þþ g25 ðAzx fzx ðz; x; t; zx ÞÞ2 ðOzx Þ:ð39ÞHere, the quadrupole susceptibilities of ðOu Þ, ðOv Þ, andðOzx Þ as functions of temperature and magnetic ﬁeld along
the x-direction are shown in Figs. 11(a)–11(c), respectively.
Cs0 is the background elastic constant where the quadrupolestrain interaction is ignored as

Cs0 ðAs fs ðz; x; t; s ÞÞ2¼ CB ðAB fB ðz; x; t; B ÞÞ2 þ CT ðAu fu ðz; x; t; u ÞÞ2
þ CT ðAv fv ðz; x; t; v ÞÞ2 þ C44 ðAzx fzx ðz; x; t; zx ÞÞ2
 K0 ðzÞ:ð40Þ
0
Cs is expected to faintly increase with decreasing temperature
because of anharmonic vibrations of the Rayleigh wave.
5.

Discussion

In order to ﬁt the softening and its magnetic ﬁeld
dependence on Cs by Eq. (39), we need numerical valuesof the quadrupole-strain coupling constants of g3 and g5 in
Eq. (30). Recently, the authors’ group has reported the sum
rule between the number of the vacancies consumed in thevoid formation and the number of residual vacancies
proportional to the low-temperature softening of the C44 of
the bulk transverse ultrasonic wave.16) This sum rule leads toa strong coupling constant of g5 ¼ 2:8  105 K between theelectric quadrupole of Ozx and the counterpart symmetry
strain of "zx in Eq. (30). Namely, the deformation energy
for the external strain pofﬃﬃﬃ "zx ¼ 1 corresponds tog5 h8 jOzx j8 i ¼ 2:8  105 = 3 K ¼ 1:6  105 K ¼ 14 eV.
Furthermore, the ratio of g5 =g3 ¼ 1:6 was deduced bymeasuring of the relative amount of the low-temperaturesoftening of both elastic constants C44 and ðC11  C12 Þ=2.15)
Using the coupling constants g5 ¼ 2:8  105 K and g3 ¼
1:8  105 K and the amplitude A and the phase  in thefunction f ðz; x; t;  Þ of Eq. (19) in Sect. 4.1, we havemade a ﬁt for the softening of Cs by the solid line in Fig. 4.
We have deduced the vacancy concentration N ¼ 3:1 1012 /cm3 in the surface layer within p ¼ 3:5 µm of thepresent silicon wafer. In the ﬁtting, we simply offset theextrinsic phase shift at a superconducting transition temperature of 1.17 K of the Al electrode to continue on both sides.
Furthermore, we adopted the temperature-dependent background of the elastic constant in Eq. (40) described by
Cs0 ¼ C 0  a=ðexp½b=T  1Þ with the parameters C0 ¼
5:749  1010 J/m3, a ¼ 6:30  106 J/m3, and b ¼ 7:3 K.
The dashed line in Fig. 4 shows the background.

034702-10©2014 The Physical Society of Japan

J. Phys. Soc. Jpn. 83, 034702 (2014)

K. Mitsumoto et al.

The magnetic ﬁelds applied parallel to the propagating xdirection of the SAW reduce the low-temperature softening of
Cs . As shown in Figs. 5(a) and 5(b), the softening of Cs is
completely suppressed at magnetic ﬁelds of up to 2.0 T. Thisis due to the lifting of the Kramers doublets of the 8 –7states by the Zeeman energy in Eq. (29). The magnetic ﬁeld
dependences of the quadrupole susceptibilities ðOu Þ,ðOv Þ, and ðOzx Þ under the ﬁelds along the x-directionof up to 6 T for various temperatures are shown in
Figs. 11(a)–11(c), respectively. The conﬁgurations between
the electric quadrupole and the applied ﬁeld direction along
the x-direction are also shown in Fig. 11. Furthermore, theinsets of Figs. 11(a)–11(c) show the magnetic ﬁeld dependence of the quadrupole susceptibilities of ðOu Þ,ðOv Þ, and ðOzx Þ, respectively, up to 6 T. The magnetic
ﬁelds along the x- and z-directions give the same effects on
the unperturbed 8 –7 states of the vacancy orbital at the
cubic site symmetry. When we treat the quadrupole-straininteraction of Eqs. (30) and (36) for the SAW propagating
along the x-direction on the z surface, the present magnetic
ﬁeld along the x-direction affects the softening differently
from that along the z-direction. The solid and dashed lines inFig. 5(b) are ﬁts for the softening and its ﬁeld dependence on
Cs in terms of the quadrupole susceptibilities in Eq. (34). The
results of the theoretical calculations of Cs in Fig. 5(b) using
the quadrupole susceptibilities are in fair agreement withthose of the SAW experiments on Cs in Fig. 5(a).
Despite the fairly good consistency between the results of
the SAW experiments and calculations based on the vacancy
orbital with the 8 –7 states, the softening of Cs below 0.1 K
in the zero magnetic ﬁeld in Fig. 5(b) slightly deviates fromthe theory. Because of the strong quadrupole-strain coupling
constants g3 ¼ 1:8  105 K for the symmetry strains "u and"v and g5 ¼ 2:8  105 K for "zx , even a tiny strain of"  106 may lift the 8 quartet ground state to two Kramers
doublets with an order of energy of 0.1 K. Although we haveattempted to eliminate unwanted strains, tiny strains might
still remain in the silicon sample. This may give rise to thelifting of the 8 ground state, which plausibly accounts forthe slight deviation in the softening of Cs from the theory.
The precise measurement of the low-temperature softening of
Cs as a function of externally applied stress acting on siliconsamples is reserved for future work to verify the strainsensitive feature of the vacancy orbital with the 8 groundstate.
6.

Conclusions

Focusing on boron-doped silicon wafers currently available for the fabrication of semiconductor devices, we havemeasured SAWs propagating on the surface of the wafer. An
IDT with a comb gap of w ¼ 2:5 µm optimally generated a
SAW of fs ¼ 517 MHz. A pulsed shape SAW propagatesalong the x-direction with a group velocity vs ¼ 4:967 km/s
and a wavelength s ¼ vs =fs ¼ 9:61 µm ³ 4w ¼ 10 µm. The
equations of motion for the SAW on the z surface of thesilicon wafer give the Rayleigh wave with a velocity
v ¼ 4:844 km/s, which is in agreement with the observed
SAW velocity vs ¼ 4:967 km/s. The SAW with the ellipsoidal trajectory of the displacement components ux and uz
induces the time-dependent symmetry strains "B , "u , "v ,and "zx with the different amplitude and phase shifts. The

almost anti-phase relations between "B and "zx and between"u and "v indicate the stationary wave character of the SAW
during decay along the penetrating z-direction.
The remarkable softening of Cs =Cs ¼ 1:9  104 below
2 K down to a base temperature of 23 mK is caused by the
coupling between the SAW and the vacancy orbital with the8 ground state and 7 excited state located at 1 K. Magnetic
ﬁelds of up to 2 T fully suppress the softening of the SAW
due to the lifting of the 8 –7 states at the magnetic ﬁelds.
The low-temperature softening and its magnetic ﬁeld
dependence are well described in terms of the quadrupole
susceptibilities based on the coupling between the symmetry
strains "u , "v , and "zx of the SAW and the appropriate electric
quadrupoles. Taking account of the strong quadrupole-straininteractions of g5 ¼ 2:8  105 K and g3 ¼ 1:8  105 K forthe vacancy orbital, we deduced the low vacancy concentration N ¼ 3:1  1012 /cm3 in the surface layer within apenetration depth p ¼ 3:5 µm of the silicon wafer. The
present result promises a highly sensitive diagnostic toolof the SAW where the softening of Cs =Cs ¼ 1:0  104corresponds to a vacancy concentration N ¼ 1:6  1012 /cm3in the surface layer. The advantage of the SAW is clearly seen
as compared with sensitivity of the bulk transverse ultrasonicwave measurement where the softening of C44 =C44 ¼
1:0  104 corresponds to N ¼ 1:5  1013 /cm3 in the bulk
region of the wafer.16)
The complementary metal oxide semiconductor is widelyadopted in the fabrication of various large-scale integrated(LSI) devices of center processing units, dynamic random
access memories, NAND-type ﬂash memories, and charge
couple devices to name a few.38,39) The active region beneaththe surface of the silicon wafer in the depth range of 1–3µm is particularly utilized in LSI device fabrication. The
vacancies play important roles in the proper control of adiffusion of dopants, precipitation of bulk micro-defects, andgettering of metal impurities in device fabrication. Therefore,
the vacancy evaluation in the surface layer of silicon wafers
is crucially required for the fabrication of the high-density
LSI devices beyond 19 nm in size in order to obtain a highyield. The continuous pursuit of the high-quality wafers ofpolished, annealed, and epitaxial types with 450 mm as well
as 300 mm in diameter demands the ﬁne control of thevacancy concentration in the surface layer. The low-temperature measurement of the SAW demonstrated in the present
study surely suggests an innovative technology for determining the vacancy concentration in the surface layer, which isone of the ﬁgure merits of the use of silicon wafers.
Acknowledgments
We thank Akira Hasegawa and Takehiko Bando of Niigata
University, Masashi Muromachi and Syouzou Saito of
Toshiba Corporation, and Susumu Kouyama of GlobalWafers
Japan Co., Ltd., for continuously supporting basic research inpursuit of an innovative technology for vacancy evaluation.
We also thank Akira Yoshihara of Ishinomaki Senshu
University for illuminative discussions of the SAW, HaruhikoSuzuki of Kanazawa University for the low-temperature
SAW measurements in magnetic ﬁelds, and Yoshio Satohof Taiyo Yuden Co., Ltd., for valuable suggestions regarding
the IDT fabrication on the wafer. This work was supported
by a Grant-in-Aid for Specially Promoted Research034702-11©2014 The Physical Society of Japan

J. Phys. Soc. Jpn. 83, 034702 (2014)

K. Mitsumoto et al.

(No. 18002008) “Strongly correlated quantum phases associated with charge ﬂuctuations” from the Ministry of
Education, Culture, Sports, Science and Technology ofJapan. The present work was also supported by the
Innovation Research Project on Nanoelectronics Materials
and Structure from the Ministry of Economy, Trade andIndustry of Japan (NEDO). We appreciate the support in theform of a Grant-in-Aid for Scientiﬁc Research “Supporting
Program for Creating University Ventures” from the Japan
Science and Technology Agency (JST). We also appreciate
the support through the Strategic Young Researcher Overseas
Visits Program for Accelerating Brain Circulation from theJapan Society for the Promotion of Science (JSPS).

+

goto@phys.sc.niigata-u.ac.jp
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034702-12©2014 The Physical Society of Japan