Czichos H., Saito T., Smith L.E. (Eds.) Handbook of Metrology and Testing

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238 Part B Chemical and Microstructural Analysis

Fig. 5.45 Halo pattern of selected area electron diffraction

from an amorphous carbon ﬁlm

quantity r[ρ(r) −ρ

] is given by the inverse Fourier

transformation of the quantity on the left, which is mea-

surable in diffraction experiments.

For multicomponent solids, the similarly deﬁned

partial distribution functions are deduced from diffrac-

tion patterns if one uses techniques such as anomalous

dispersion that allow one to enhance the atomic scat-

tering factor f

of only a speciﬁc element by tuning

–

hξ

Fig. 5.46 Interference of core-excited electron waves scat-

tered by surrounding atoms giving rise to EXAFS

the wavelength of the x-rays. The atomic scattering fac-

tor for neutrons is generally quite different from that

for x-rays (Sect. 5.1.1). So coupling neutron diffraction

with x-ray diffraction, we could extend the range of ele-

ments that can be studied by using diffraction methods.

Extended X-Ray Absorption Fine Structure

(EXAFS)

X-ray absorption occurs when the photon energy ex-

ceeds the threshold for excitations of core electrons

0.3

0.2

0.1

–0.1

9.6 9.8 10.0 10.2 10.4 10.6

41216

Δν(k)t

k(Å

–1

)

206

632

a-Mg

ln(I

/I)

E(keV)

r (Å)

Partial RDF (r)

Zn–Mg

Zn–Zn

|(r)|

Fig. 5.47a–d Analysis of short-range order by extended

x-ray absorption ﬁne structure. After [5.35]

Part B 5.2

Nanoscopic Architecture and Microstructure 5.3 Lattice Defects and Impurities Analysis 239

to unoccupied upper levels. The core absorption spec-

trum has a characteristic ﬁne structure extending over

hundreds of eV above the absorption edge. The undu-

lating structures within ≈ 50 eV of the edge is referred

to as x-ray absorption near-edge structures (XANES) or

near-edge x-ray absorption ﬁne structures (NEXAFS),

whereas the ﬁne structures beyond ≈ 50 eV are called

the extended x-ray absorption ﬁne structures (EXAFS).

The EXAFS is brought about by interference of the

excited electron wave primarily activated at a core

with the secondary waves scattered by the surround-

ing atoms (Fig. 5.46). Therefore, the EXAFS contains

information on the local atomic arrangement, neigh-

bor distances and the coordination numbers, around the

atoms of a speciﬁc element responsible for the core

excitation absorption. Figure 5.47a shows an experi-

mental x-ray absorption spectrum near the zinc K-edge

of an amorphous Mg

alloy, and Fig. 5.47bthe

oscillatory part of the EXAFS obtained by subtracting

the large monotonic background in (a). Assuming ap-

propriate phase shifts on electron scattering, we obtain

Fig. 5.47c, a Fourier transform of a function modiﬁed

from (b), from which we deduce Fig. 5.47d, the par-

tial distribution function around Zn atoms. The area of

deconvoluted peaks in (d) gives the coordination num-

ber for the respective atom pair. The element-selectivity

of the EXAFS method provides a unique tool that en-

ables one to investigate the local order for different

atomic species in alloys, regardless of the long-range

order or the crystallinity and amorphousness of the

sample.

X-Ray Photoemission Spectroscopy (XPS)

Though x-ray photoemission spectroscopy (XPS)is

surface sensitive, it may be used for studies of very

short-range order around atoms. The energy of the pho-

toelectrons measured relative to the Fermi level of the

sample directly reﬂects the energy of the core elec-

trons. The core level is slightly affected by how much

electronic charge is distributed on the nucleus in the

solid (chemical shift): the more negatively charged, the

higher the core level and hence the smaller the pho-

toelectron energy. The spectral peak position and its

intensity in XPS thus indicate the character of chemical

bonding or the degree of chemisorption to the selec-

tively probed atoms.

Raman Scattering

Since the principle of Raman scattering has been

described already (Sect. 5.1.2), we only mention an ap-

plication of Raman scattering to studies of structural

order in crystalline and amorphous solids. From the

energy of Raman-active optical phonons, the material

phases present in the sample can be determined. The

selection rule in infrared absorption based on the trans-

lational symmetry of the crystal allows only optical

modes near k ≈0 in the Brillouin zone. Similarly the

translational symmetry restricts the Raman scattering to

limited modes. The presence of defects or disorder in

crystals breaks the translational symmetry of the lattice,

so that translationally forbidden Raman modes become

active and detectable as an increase of the extended

lattice phonon signal. Raman scattering is one of the

standard methods for structural assessment of graphitic

or amorphous carbon [5.36]. This is owing to the fact

that the visible spectral region, the most widely used

for Raman scattering experiments, is electronically res-

onant with graphitic crystals, enhancing the scattering

signals. Two broad spectral bands, called G and D, re-

spectively corresponding to crystalline and disordered

graphitic phases are well separated for quantitative anal-

ysis of disorder.

5.3 Lattice Defects and Impurities Analysis

This section deals with methods for the study of atom-

istic structural defects and impurities in regular lattices.

Section 5.3.1 covers point defects including vacancies,

interstitial atoms, defect complexes, defect clusters, and

impurities. Section 5.3.2 deals with extended defects in-

cluding dislocations, stacking faults, grain boundaries,

and phase boundaries.

The sensitivity (or ﬁeld of view) of the methods de-

scribed here is not necessarily high (wide) enough for

detecting a low density of defects, so in many cases the

defects must be intentionally introduced into the sample

by some means. For intrinsic point defects such as va-

cancies and self-interstitials, the most common method

is to irradiate the sample with high energy particles such

as MeV electrons and fast neutrons. For dislocations, the

sample may have to be deformed plastically. However,

one should always pay careful attention to the possibil-

ity that the primary defects may react to form complexes

with themselves or impurities and the plastically de-

formed samples may contain point defects as well.

Part B 5.3

240 Part B Chemical and Microstructural Analysis

5.3.1 Point Defects and Impurities

Diffraction and Scattering Methods

X-Ray Diffuse Scattering. The lattice distortion in-

duced by the presence of point defects and impurities,

as illustrated in Fig. 5.48a, causes a diffuse scattering

in x-ray diffraction as well as a shift of the diffraction

peaks. The diffuse scattering from such imperfect crys-

tals generally has two components (Fig. 5.48b): Huang

scattering, which appears near the diffraction peaks and

represents a long-range strain ﬁeld around the defects

and impurities, and Stokes–Wilson scattering, which

extends the between diffraction peaks and represents

a short-range atomic conﬁguration and hence its dis-

tribution in the reciprocal space reﬂects the symmetry

of the strain ﬁeld associated with the defect and im-

purity. Figure 5.49 shows the Stokes–Wilson scattering

from an electron-irradiated Al sample [5.37] experi-

mentally deduced by carefully subtracting a background

of different origins one or two orders of magnitude

larger than the signal. The experimental proﬁles meas-

ured along different trajectories in the reciprocal space

agree most with the theoretical curves predicted by self-

interstitials split along the 100direction.

Rutherford Back Scattering. When light ions such as

and He

in energy of a few MeV are incident

on a solid, some of them are classically scattered by

atoms in the solid, reﬂected backward and emitted from

the sample, which is called Rutherford back scattering

(Fig. 5.50a). These backscattered ions lose an energy

Cross section (arb. units)

15 30 45 60

Cross section (arb. units)

15 30 45 60

Cross section (arb. units)

15 30 45 60

Cross section (arb. units)

15 30 45 60

Scattering angle

022 222

200

111

Experi-

mental

Scattering angle

〈100〉-split

Octahedral Tetrahedral

Fig. 5.49 Structural determination of selﬁnterstitial in Al by x-ray diffuse scattering (after [5.37, p. 87])

a) b)

I(K)

Δa

(K)

Stokes–Wilson scattering Huang scattering

Fig. 5.48 (a) A perfect lattice and (b) a lattice distortion

induced by point defects reﬂected in the x-ray diffraction

the short-range elastic ﬁelds and Huang scattering for the

long-range ﬁelds

depending on the atoms with which the ions collide

and the collision parameters: the heavier the atoms,

the larger the energy loss in an elastic collision. Fig-

ure 5.50b shows a schematic diagram of the energy

distribution of the backscattered ions. If the sample con-

Part B 5.3

Nanoscopic Architecture and Microstructure 5.3 Lattice Defects and Impurities Analysis 241

Channeling

sample

Back scattered ions

Scattered from

deep positions

Matrix

Diffusion profile

Heavy

impurity

Energy of back scattered ions

Fig. 5.50 (a) Light ions with primary energy E

are inci-

dent along a channel direction and inelastically scattered

backward (Rutherford back scattering) with a reduced en-

ergy E. (b) The energy distribution of back scattered ions.

When heavy impurities are present in the channels, a dis-

tribution arises for smaller losses

tains impurity atoms much heavier than the atoms in the

matrix, an additional component due to these impurities

appears between the primary ion energy E

and the en-

ergy kE

(k < 1) representing the energy of an ion that

has collided only once with a matrix atom. This pro-

vides a means to measure the depth proﬁle of diffusant

atoms from the surface. Furthermore, in single crys-

tal samples, ions can penetrate, without scattered, over

an anomalously large distance along speciﬁc crystallo-

graphic directions called channels. If an interstitial atom

is present in the channel, the number of backscattered

ions incident in the channeling directions is increased.

The out-going backscattered ions also may feel the

channeling effect, which enables the method of double

channeling that allows determination of the crystallo-

graphic position of the interstitial atoms in the lattice.

Microscopic Methods

Scanning Tunneling Microscopy (STM). One of the

microscopic methods currently available for direct

observations of point defects is scanning tunneling

microscopy (STM). As explained in Sect. 5.1.2, the tun-

neling current reﬂects the local density of states (LDOS)

of electrons at the tip position above the surface. There-

fore, if the wave functions associated with defects even

–1.5 –1.0 –0.5 0.0 0.5 1.0 1.5

(dI/dV)/(I/V)

As-related defect

in LT-GaAs

Sample bias (V)

1nm

Fig. 5.51 (a) Scanning tunneling microscopic image of an

arsenic-related point defect beneath the surface and (b)

the local density of states measured at the defect posi-

tion [5.38]

beneath the surface extend out of the surface, we can im-

age the defect contrasts using STM. Figure 5.51 shows

an STM image obtained for an arsenic-related point de-

fect located below the surface [5.38]. Defect studies by

STM are in the early stages of promising applications

for various material systems.

High-Angle Annular Dark-Field STEM (HAADF-STEM).

An ambitious approach for rather indirect observa-

tions of point defects is high-angle annular dark-ﬁeld

STEM (HAADF-STEM). The progress of improve-

Part B 5.3

242 Part B Chemical and Microstructural Analysis

ment of spherical aberration has achieved successful

atomic-scale imaging of impurity atoms [5.39]andva-

cancies [5.40] in each atomic column. Nevertheless,

elaborate image simulations are necessary for decisive

conclusions to be drawn.

Spectroscopic Methods

Relaxation Spectroscopy.

Mechanical Spectroscopy. When the lattice distortion

induced by defects and impurities is anisotropic, such

as the strain ellipsoid depicted in Fig. 5.52a, the stress-

induced reorientation (ordering) of the anisotropic

distortion centers can be detected as the Snoek relax-

ation (peak), which allows one to evaluate the λ tensor

(Fig. 5.52a) and the relaxation time. The equilibrium

anelastic strain ε

(eq)

is given by

(eq)

3kT

⎡

⎣





(p)



−





(p)



⎤

⎦

(5.26)

with

(p)



(p)





(p)





(p)



, (5.27)

where C

is the concentration of the anisotropic distor-

tion centers, v

is the molar volume, and α

(p)

, α

(p)

and

(p)

are the direction cosines between the stress axis

and the three principal axes of the λ tensor. A typical

application is to carbon, nitrogen and oxygen intersti-

tial atoms in bcc metals which occupy one of three

equivalent interstitial sites (octahedral sites) inducing

tetragonal distortions in different crystallographic di-

rections. The concept of the Snoek relaxation is also

applied to the hydrogen internal friction peak in amor-

phous alloys.

A pair of differently sized solute atoms in the solid

solution in the nearest-neighbor conﬁguration can be an

a) b)

Fig. 5.52 (a) The strain ellipsoid, (b) hydrogen atoms in

a bent specimen

anisotropic distortion center. For such cases, the anelas-

tic relaxation (peak) associated with the stress-induced

reorientation (ordering) of the anisotropic distortion

centers is referred to as the Zener relaxation (peak).

The Zener relaxation provides a tool for the study

of atomic migration in substitutional alloys, without

radioisotopes, at temperatures much lower than in con-

ventional diffusion methods.

When the lattice distortion induced by an applied

stress is not homogeneous, a stress-induced redistribu-

tion of interstitial impurities takes place resulting in an

anelastic strain (Fig. 5.52b). Such relaxation is referred

to as the Gorsky relaxation and provides a tool for the

study of long-range migration of interstitial impurities.

The relaxation time τ and the relaxation strength Δ

for

a specimen of rectangular section are given by

τ =d

/π

D (5.28)

and

=(C

E/9kTβ)(tr λ)

, (5.29)

where D is the diffusion coefﬁcient of the impurity, β

is the number of interstitial sites per host atom, and d

and E are the thickness and the Young’s modulus of

the specimen, respectively. This method is applied to

hydrogen diffusion because of the rapid diffusion rate.

Dielectric or Magnetic Relaxation. Quite similar re-

laxation processes occur in the dielectric or magnetic

response of defects and impurities. Tightly bound pairs

in ionic crystals have in some cases, such as in F

centers (pairs of anion vacancy and isovalent cation im-

purity atom) in alkali halides, an electric dipole moment

along the bond orientation [5.41]. Therefore, respond-

ing to the application and the removal of an electric

ﬁeld, the dipole moment reorients by a movement of

constituent atoms, which is detected as a dielectric re-

laxation. Magnetic aftereffects are also brought about

by movements of point defects and impurities. Since

interstitial atoms in bcc metals diffuse in response to

an external magnetic ﬁeld through magnetostriction, es-

sentially by the same mechanism as the Snoek effect,

the Snoek relaxation can be studied by measurements

of the magnetic relaxation.

Noise Spectroscopy. Relaxation and ﬂuctuation repre-

sent two aspects of statistical behaviors of a physical

quantity q that is under stochastic random forces [5.42,

43]. Debye-type relaxation behavior, which is charac-

terized by a time constant τ

with which the quantity

Part B 5.3

Nanoscopic Architecture and Microstructure 5.3 Lattice Defects and Impurities Analysis 243

responds to a stepwise stimulation, is described by an

autocorrelation function

(τ) =



q(t)q(t +τ)



∝exp

(

−τ/τ

)

. (5.30)

Generally, the autocorrelation function in the time

domain is related to the ﬂuctuation power spectrum

(ω) in the frequency domain through the Wiener–

Khintchine theorem

(ω) =4

∞



(τ)cosωτ dτ. (5.31)

Therefore, from measurements of the ﬂuctuation

power spectrum without intentional stimulation, we can

deduce the relaxation time. The ﬂuctuations may be de-

tected in the form of electrical noise. The source of the

noise varies from case to case. In semiconductors con-

taining impurities or defects forming deep levels in the

band gap, we may observe a generation–recombination

(g–r) noise arising from the ﬂuctuations of carrier den-

sity on random exchange of carriers between the deep

levels and the band states. Since the ﬂuctuation or re-

laxation time constant is determined, in many cases, by

the rate of thermal activation of carriers from the deep

level to the band, measurements of the noise intensity as

a function of temperature give spectroscopic informa-

tion about the depth of the deep level from the relevant

band edge. Since such noise becomes more signiﬁcant

as the number of carriers becomes smaller, noise spec-

troscopy is useful for studies of small systems such as

nanostructures.

Infrared Absorption Spectroscopy (IRAS). The optical

photon energy that IRAS detects is determined by the

atomic mass and the strength of the bonds with the

surrounding atoms. Therefore, the local vibrational fre-

quency of impurity atoms of mass similar to the matrix

easily overlap with the lattice phonon band and form

a resonance mode that is hard to detect in IRAS due

to the strong background. Since the local vibrational

modes (LVM) associated with intrinsic point defects be-

have similarly, few studies have been reported for IRAS

experiments of LVM of intrinsic point defects. (This is

also due to the fact that simple intrinsic point defects

are usually easily reconstructed to secondary defects

such as complexes with impurities.) Instead, IRAS ex-

periments on light elements such as oxygen in Si have

been intensively carried out and the experimental data

documented. Hydrogen is an exceptionally light ele-

ment which is used indirectly to detect point defects,

such as self-interstitials and vacancies in Si for exam-

ple [5.44], through the LVM associated with hydrogen

defect complexes. The detection limit of impurities by

IRAS is ≈0.01%.

Raman Spectroscopy. Since Raman scattering is a two-

photon process, the scattering signal is generally very

weak, as mentioned in Sect. 5.1.2. Therefore, Raman

scattering has so far rarely been used for studies of lo-

cal vibrations associated with defects at low density.

However, when resonant Raman scattering is used, the

detection limit could be reduced down to several ppb

under good conditions if the ﬂuorescence or lumines-

cence background is low. Figure 5.53 demonstrates the

power of resonant Raman scattering, which enables

detection of the LVM of As-related point defects in

GaAs [5.45]. Generally, for LVM to be detected clearly

by Raman scattering, the vibrational energy of the LVM

must be well separated from those of the resonance

modes (the local mode in resonance with the lattice

phonon bands) which have relatively large intensities.

The F-center (a neutral anion vacancy trapping an elec-

tron) in KI [5.46] is such an exceptional case in which

the masses of potassium and iodine are so different that

the gap between the acoustic and the optic bands is

wide enough to accommodate the gap mode induced by

defects.

4000

3000

2000

1000

Raman intensity (units)

150 200 250 300 350

Raman shift (cm

–1

)

LT-GaAs at 200K

Local

vibrational

modes

(TO)(LO)

Nd-YAG

(1064 nm)

He–Ne

(514.5 nm)

Fig. 5.53 Defect-selective resonant Raman spectra of

a low-temperature grown GaAs crystal measured at 200 K

by using two different excitation lasers. The wavelength

of the Nd:YAG laser (1064 nm) is resonant with the intra-

center electronic excitation of the As-related EL2 centers.

(Courtesy of Dr. A. Hida)

Part B 5.3

244 Part B Chemical and Microstructural Analysis

Visible Photoabsorption and Photoluminescence (PL).

In contrast to IRAS and Raman scattering, photoab-

sorption and photoluminescence (PL) in the visible to

near-infrared region are widely used for studies of de-

fects in nonmetallic solids. If the sample yields strong

PL, the sensitivity can be as high as ≈ 0.1 ppm. The

spectrum of PL is complementary to the photoabsorp-

tion spectrum, with both representing electronic dipolar

transitions. Defects in semiconductors and insulators

often introduce electronic states in the band gap. The

classic examples are color centers in alkali halides.

When the electronic state is deep in the gap, the elec-

tronic wave function tends to be localized in space. In

such cases, the electronic energy is sensitive to the local

arrangement of atoms at the defect. This effect is re-

ferred to as strong electron lattice coupling, the degree

of which is reﬂected in the optical transition spectra.

Figure 5.54a,b illustrates the situations by using conﬁg-

uration coordinate diagrams: each curve indicates the

(adiabatic) potential energy of the system, consisting of

the electronic energy and the nuclear potential energy,

drawn as a function of the nuclei positions in the sys-

tem that are expressed by one-dimensional coordinate

Adiabatic potential

Excited state

Ground state

Configuration coordinate

Adiabatic potential

Excited state

Ground state

Configuration coordinate

Intensity

Photon energy

Intensity

Photon energy

Photo-

luminescence

Photo-

absorption

Photo-

luminescence

Photo-

absorption

Stokes shift

Fig. 5.54a–d Conﬁguration coordinate diagrams and corresponding spectra of photoabsorption and photoluminescence

(PL) for two cases in which electron–lattice coupling is weak (a),(c) and strong (b),(d). Note that the absorption and PL

spectra are almost mirror symmetric, regardless of the electron–lattice coupling

(conﬁguration coordinate). The lower curves represent

the adiabatic potentials for the electronically ground

state while the upper curves the adiabatic potentials for

an electronically excited state. The ladder bars indi-

cate the vibrational states in each electronic state. If the

electron–lattice coupling is weak (Fig. 5.54a), the stable

conﬁgurations do not differ much between the ground

state and the excited state. In this case, the optical tran-

sitions take place with spectra characterized by a sharp

zero-phonon line reﬂecting transitions with no phonon

involved, as shown in Fig. 5.54c. If the electron–lattice

coupling is strong (Fig. 5.54b), the stable conﬁgura-

tions differ considerably between the ground state and

the excited state. The consequences are a broad band-

width and a large Stokes shift (red shift) as shown in

Fig. 5.54d.

Generally, the dipolar optical transition is not as ef-

ﬁcient as competitive nonradiative processes, including

Auger processes, multiphonon emission processes, and

thermally activated opening of different recombination

channels. The last process is common in many systems,

so samples for PL measurements must usually be cooled

to a low temperature (≤20 K) to obtain a detectable

Part B 5.3

Nanoscopic Architecture and Microstructure 5.3 Lattice Defects and Impurities Analysis 245

PL intensity. The sample cooling is also beneﬁcial for

both photoabsorption and PL measurements to resolve

the spectral ﬁne structures.

Scanning electron microscopy using cathodolumi-

nescence as the signal (SEM-CL) allows one to observe

the distribution of radiative centers in semiconducting

crystals. Figure 5.55 shows monochromatic SEM-CL

images of semiconductor quantum dots [5.47] from

which we can determine from where in the dots each

luminescence arises. Similar images can be obtained

by STEM equipped with a light collecting system. An

advantage of STEM-CL is that one can also obtain crys-

tallographic information about the defect.

Electron Paramagnetic Resonance (EPR). The second

term −2SJ



S in (5.19) represents the spin–spin ex-

change interaction that is of fundamental importance

in magnetic (ferromagnetic and antiferromagnetic) ma-

terials. In paramagnetic substances where the exchange

interaction is absent, the effective spin Hamiltonian rel-

evant to electron paramagnetic resonance (EPR)is

H = B



S−λ

SS+SAI . (5.32)

Here the term −λ

SS, though formally similar to

−2SJ



S, now represents the magnetic dipole interac-

tion between electronic spin S and orbital momentum L

(spin–orbit coupling) of magnitude λ(L ·S). For the sys-

tems with S =1/2 encountered in most cases, this term

is absent. Still, however, the effective magnetogyric ra-

tio γ



≡γ

(1 −λ) is modiﬁed from that of an isolated

electronic spin γ

=μ

(μ

≡e/2m

c: Bohr magne-

ton) to



≡γ

(1 −λ) =μ

(1 −λ) ≡μ

g . (5.33)

Since a static magnetic ﬁeld B

induces a Zee-

man splitting of unpaired electrons much larger than

that of nuclear spins, a measurable microwave ab-

sorption is observed when the microwave frequency

becomes resonant with the Larmor frequency. Unlike

Fourier-transform NMR (Sect. 5.4.2), an EPR spectrum

is measured by sweeping the external magnetic ﬁeld

applied to the sample placed in a microwave cavity

resonant at a ﬁxed frequency of ≈ 10 GHz. Usually

the signal is detected by a magnetic ﬁeld modulation

technique so that the resonance ﬁeld is accurately deter-

mined as the ﬁelds at which the signal crosses the zero

base line (Fig. 5.56a).

The symmetry of the defect can be determined

by the dependence of the electronic g-tensor or the

corresponding resonance ﬁeld on the direction of the

1.30 1.35 1.40 1.45 1.50

Photon energy (eV)

1.37 eV 1.43 eV

5 μm

CL intensity (arb. units)

Fig. 5.55 Monochromatic SEM-CL images of semiconductor

quantum dots and the Cl spectrum (after [5.47])

a) b)

–

hω

–

+ 5/2

+ 1/2

– 3/2

+ 3/2

– 1/2

– 5/2

– 3/2

+ 1/2

+ 5/2

– 5/2

– 1/2

– 3/2

Fig. 5.56a,b Spin resonance of isolated electrons (a) and

electrons interacting at the hyperﬁne level with a nucleus

with I =5/2 (b)

crystallographic axis with respect to the direction of the

external magnetic ﬁeld. Some point defects in semicon-

ductors form electronic levels in the band gap and could

change their charge states according to the Fermi level.

If the levels are degenerate with respect to the orbitals

due to the point symmetry of the center, a symmetry-

breaking lattice distortion (Jahn–Teller distortion) may

occur depending on the charge state so as to lift the

degeneracy and consequently lower the electronic en-

Part B 5.3

246 Part B Chemical and Microstructural Analysis

Fig. 5.57

Oxygen vacancy

in MgO

ergy. An historical example in which the EPR method

shows its power is seen in structural determination of

vacancies in Si in different charge states [5.48].

The chemical environment of the center or the lat-

tice position of the defect can be identiﬁed from the

knowledge of hyperﬁne structures arising from the last

term SAI in (5.32) that originates in the magnetic

dipole–dipole interaction between the electronic spin

and the nuclear spins over which the electron cloud is

mostly distributed. Figure 5.56b illustrates how the hy-

perﬁne structure is brought about by the interaction of

the electron with a nucleus of I = 5/2. As a concrete

example [5.49], we consider oxygen vacancies in ionic

MgO crystals as shown in Fig. 5.57. The vacancy of

the oxygen ion O

2−

is double-positively charged and

EPR insensitive because all the electrons are paired.

But if it traps an electron and becomes single-positively

charged, EPR signals arise from the unpaired S =1/2

spin. Since no nucleus is present at the center of the

vacancy, the hyperﬁne interaction is due only to the sur-

rounding six equidistant Mg

ions. Among abundant

isotopes, only the

Mg nucleus (10.11% abundance)

has nonzero nuclear spin of I = 5/2. Therefore, the rel-

ative intensity of hyperﬁne signals is determined by

the occupation probability of

Mg ions among the

six Mg ion sites. For example, the probability that

all the Mg sites are not occupied by

Mg ions is

(0.8989)

=52%, so 52% of the vacancies should show

no hyperﬁne splitting, as shown in Fig. 5.56a. Similarly

the probability of ﬁnding one

Mg ion (I =5/2) in

the neighborhood is 6×(0.8989)

×(0.1011) =35.6%,

so that 35.6% of vacancies should exhibit (2I +1) =6

hyperﬁne splitting signals, as illustrated in Fig. 5.56b

following the selection rule in EPR Δs

=±1and

ΔI

= 0. The probability of ﬁnding two

Mg ions

(I = 5) is (6!/4!2!)×(0.8989)

×(0.1011)

= 10%, so

= –

= +

Δm

= ± 1

= –

Micro-

wave

Fig. 5.58 Electron nuclear double resonance (ENDOR)in

I = 1/2 nuclei

that 10% of vacancies should exhibit (2I +1) =11 hy-

perﬁne splitting signals. The intensity of the hyperﬁne

signals in this case differs depending on the degener-

acy of the spin conﬁgurations: only one conﬁguration

(5/2, 5/2) contributes to the I

= 5 signal, two con-

ﬁgurations, (5/2, 3/2) and (3/2, 5/2), to the I

= 4

signal, three conﬁgurations, (5/2, 1/2), (3/2, 3/2) and

(1/2, 5/2), to the I

= 3 signal, and so on, resulting in

the relative intensity ratio 1 :2 :3 :4 :5 : 6 : 5 : 4 :3 :

2 :1, which is veriﬁed experimentally.

The electron–nuclear interaction represented by the

term SAI in (5.32) can be extended to include indi-

rect interactions



with nuclei k beyond the

nearest neighbors, giving rise to the super-hyperﬁne

structures around each hyperﬁne signals. One major

drawback of EPR spectroscopy is the low resolution

(Δν ≈ 10 MHz) compared to NMR (Δν ≈ 50 kHz),

which is a disadvantage in the analysis of crowded

or overlapped signals. Due to the electron–nuclear in-

teraction, the EPR intensity of each super-hyperﬁne

structure changes when nuclear magnetic resonance oc-

curs in the nucleus responsible for the super-hyperﬁne

EPR signal (Fig. 5.58). This EPR-detected NMR spec-

troscopy is called electron nuclear double resonance

(ENDOR)[5.50] which also improves the low sensi-

tivity of conventional NMR [5.51]. ENDOR is used

to investigate the spatial extent of the electron cloud

probed by EPR.

In semiconductors, unpaired spins as sparse as 10

to 10

−3

can be detected by EPR provided that the

density of free carriers is below 10

−3

so that the

microwave can penetrate the sample well. In metals, the

presence of free electrons limits the penetration depth

below the sample surface due to the skin effect. It should

be stressed that EPR signals can be detected only when

unpaired electrons are present.

Part B 5.3

Nanoscopic Architecture and Microstructure 5.3 Lattice Defects and Impurities Analysis 247

Optically Detected Magnetic Resonance (ODMR).

Optically detected magnetic resonance (ODMR)is

applicable to studies of electronic states in which a mag-

netic ﬁeld lifts the degeneracy of energy levels differing

only in magnetic spin. ODMR is a double resonance

technique that allows highly sensitive EPR measure-

ments and facilitates the assignment of the spin state

responsible for the magnetic resonance. The most com-

mon experimental schemes are either detecting circular

polarized photoluminescence (PL) or magnetic circular

dichroic absorption (MCDA) in response to magnetic

resonance. Figure 5.59 illustrates a simple case in which

the ground state and an excited state are spin degener-

ate in the absence of a magnetic ﬁeld. When a static

magnetic ﬁeld is applied, these states are split to Zee-

man levels. Due to the selection rule, optical transitions

(PL and photoabsorption) occur either between the ex-

cited down-spin state and the ground up-spin state with

a right-hand circular polarization (σ

), or between the

excited up-spin state and the ground down-spin state

with a left-hand circular polarization (σ

−

). In the re-

laxed excited state in which the occupancy of the

up-spin level is higher than that of the down-spin level,

the σ

−

PL is stronger in intensity than σ

PL.How-

ever, if we apply an electromagnetic ﬁeld in resonance

with a spin ﬂip transition between the excited Zee-

man levels, the occupancy of the down-spin excited

state becomes somewhat increased, which is detected

by an increase of the σ

emission at the expense of

the σ

−

emission. Thus, the PL-detected magnetic reso-

nance or magnetic circular polarized emission (MCPE)

measurements (Fig. 5.59a) allow the detection of EPR

in the electronically relaxed excited state. MCPE is also

possible when the nonradiative recombination is spin-

dependent. An example is found in studies of dangling

bonds in amorphous Si :H solids [5.52] and of relaxed

excited state of F-centers in alkali halides [5.50]. MCPE

can be more sensitive than ordinary EPR as long as

the PL intensity is high. However, the concentration of

the centers studied successfully by MCPE is limited by

the concentration quenching effect, a decrease of PL

intensity due to an energy transfer mechanism (cross re-

laxation) starting to operate between nearby centers in

close proximity (Sect. 5.4.3).

Similarly, ODMR based on magnetic circular

dichroic absorption (MCDA) shown in Fig. 5.59b facil-

itates the assignment of the ground state spin. MCDA

has advantages over MCPE in its applicability to non-

luminescent samples. Other signals used in ODMR

include such spin-dependent quantities as spin-ﬂip

Raman scattering light, resonant changes of electric

Excited state

EPR Microwave

EPR

–

Microwave

Ground state

Excited state

Ground state

–

Fig. 5.59 (a) Magnetic circular polarized emission (MCPE)

and (b) Magnetic circular dichroic absorption (MCDA)

current affected by the spin state of the defect cen-

ters that determines the carrier lifetime. Especially the

last electrically detected magnetic resonance (EDMR)

technique allows one to detect selectively a very few

paramagnetic centers that are located along the narrow

current path [5.7].

Muon Spin Rotation (μ

SR). A muon is a mesonic par-

ticle which is created during the decay of a π-meson

generated by a high energy particle accelerator. The

muon is a Fermion having spin 1/2, whose mass is 207

times the electron mass. There are two types of muon,

and μ

−

, differing in the electric charge +e and −e,

respectively. The muon disintegrates to an electron (or

positron) and two neutrinos in a mean lifetime of ≈2 μs

emitting a γ -ray with a characteristic angular distribu-

tion relative to the spin direction, as shown in Fig. 5.60a.

Therefore, the rotation of spins under a static magnetic

ﬁeld during the lifetime can be observed as a signal os-

cillating at the Larmor frequency detected by a γ-ray

counter placed in an appropriate direction (Fig. 5.60b).

Since the negatively charged μ

−

are strongly attracted

to nuclear ions, they behave as if forming a nulcleus

of atomic number Z −1. In contrast, since the posi-

tively charged μ

are repelled by nuclear ions in solids,

they migrate along interstitial sites or become trapped

at vacancies, behaving like a light isotope of protons.

In ferromagnetic solids, μ

feel an internal magnetic

ﬁeld speciﬁc to the site on which the μ

spend their

Part B 5.3