Negative dielectric permittivity of nonmagnetic crystals in the terahertz waveband
Physical conditions for occurrence of the spectral bands with the negative dielectric permittivity in nonmagnetic crystalline media in the terahertz waveband have been studied in this work. It has been shown that damping the polar vibrations has a primary effect on formation of the negative diele...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2012
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| Цитувати: | Negative dielectric permittivity of nonmagnetic crystals in the terahertz waveband / S.G. Felinskyi, P.A. Korotkov, G.S. Felinskyi // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 1. — С. 83-88. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-1182822017-05-30T03:03:26Z Negative dielectric permittivity of nonmagnetic crystals in the terahertz waveband Felinskyi, S.G. Korotkov, P.A. Felinskyi, G.S. Physical conditions for occurrence of the spectral bands with the negative dielectric permittivity in nonmagnetic crystalline media in the terahertz waveband have been studied in this work. It has been shown that damping the polar vibrations has a primary effect on formation of the negative dielectric permittivity in real crystals at resonance (terahertz) frequencies, and phonon attenuation imposes significant restrictions both on the frequency range and the minimum achievable value for all dielectric tensor components. Within frameworks of the single-oscillator model, the authors have obtained: i) the criterion for the existence of the negative dielectric permittivity, which is based on physical and spectroscopic parameters of the crystal, ii) analytical expressions for calculation of the frequency band where the dielectric permittivity takes negative values. Frequency regions and the minimum value of negative dielectric permittivity are quantitatively defined in the crystal LiTaO₃. It is proved the applicability of the obtained relationships in cases of relatively complex phonon vibration spectra. 2012 Article Negative dielectric permittivity of nonmagnetic crystals in the terahertz waveband / S.G. Felinskyi, P.A. Korotkov, G.S. Felinskyi // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 1. — С. 83-88. — Бібліогр.: 9 назв. — англ. 1560-8034 PACS 77.22.Ch, 77.84.Ek http://dspace.nbuv.gov.ua/handle/123456789/118282 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
Physical conditions for occurrence of the spectral bands with the negative
dielectric permittivity in nonmagnetic crystalline media in the terahertz waveband have
been studied in this work. It has been shown that damping the polar vibrations has a
primary effect on formation of the negative dielectric permittivity in real crystals at
resonance (terahertz) frequencies, and phonon attenuation imposes significant restrictions
both on the frequency range and the minimum achievable value for all dielectric tensor
components. Within frameworks of the single-oscillator model, the authors have
obtained: i) the criterion for the existence of the negative dielectric permittivity, which is
based on physical and spectroscopic parameters of the crystal, ii) analytical expressions
for calculation of the frequency band where the dielectric permittivity takes negative
values. Frequency regions and the minimum value of negative dielectric permittivity are
quantitatively defined in the crystal LiTaO₃. It is proved the applicability of the obtained
relationships in cases of relatively complex phonon vibration spectra. |
| format |
Article |
| author |
Felinskyi, S.G. Korotkov, P.A. Felinskyi, G.S. |
| spellingShingle |
Felinskyi, S.G. Korotkov, P.A. Felinskyi, G.S. Negative dielectric permittivity of nonmagnetic crystals in the terahertz waveband Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Felinskyi, S.G. Korotkov, P.A. Felinskyi, G.S. |
| author_sort |
Felinskyi, S.G. |
| title |
Negative dielectric permittivity of nonmagnetic crystals in the terahertz waveband |
| title_short |
Negative dielectric permittivity of nonmagnetic crystals in the terahertz waveband |
| title_full |
Negative dielectric permittivity of nonmagnetic crystals in the terahertz waveband |
| title_fullStr |
Negative dielectric permittivity of nonmagnetic crystals in the terahertz waveband |
| title_full_unstemmed |
Negative dielectric permittivity of nonmagnetic crystals in the terahertz waveband |
| title_sort |
negative dielectric permittivity of nonmagnetic crystals in the terahertz waveband |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2012 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118282 |
| citation_txt |
Negative dielectric permittivity of nonmagnetic crystals in the terahertz waveband / S.G. Felinskyi, P.A. Korotkov, G.S. Felinskyi // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 1. — С. 83-88. — Бібліогр.: 9 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT felinskyisg negativedielectricpermittivityofnonmagneticcrystalsintheterahertzwaveband AT korotkovpa negativedielectricpermittivityofnonmagneticcrystalsintheterahertzwaveband AT felinskyigs negativedielectricpermittivityofnonmagneticcrystalsintheterahertzwaveband |
| first_indexed |
2025-07-08T13:40:21Z |
| last_indexed |
2025-07-08T13:40:21Z |
| _version_ |
1837086301979934720 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 1. P. 83-88.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
83
PACS 77.22.Ch, 77.84.Ek
Negative dielectric permittivity of nonmagnetic crystals
in the terahertz waveband
S.G. Felinskyi1, P.A. Korotkov2, G.S. Felinskyi3
Taras Shevchenko Kyiv National University, 4, prospect Glushkova, 03127 Kyiv, Ukraine
Phone: +380-44-526-0570; e-mail: stalisman@ukr.net1, pak@mail.univ.kiev.ua2, felinskyi@yahoo.com3
Abstract. Physical conditions for occurrence of the spectral bands with the negative
dielectric permittivity in nonmagnetic crystalline media in the terahertz waveband have
been studied in this work. It has been shown that damping the polar vibrations has a
primary effect on formation of the negative dielectric permittivity in real crystals at
resonance (terahertz) frequencies, and phonon attenuation imposes significant restrictions
both on the frequency range and the minimum achievable value for all dielectric tensor
components. Within frameworks of the single-oscillator model, the authors have
obtained: i) the criterion for the existence of the negative dielectric permittivity, which is
based on physical and spectroscopic parameters of the crystal, ii) analytical expressions
for calculation of the frequency band where the dielectric permittivity takes negative
values. Frequency regions and the minimum value of negative dielectric permittivity are
quantitatively defined in the crystal LiTaO3. It is proved the applicability of the obtained
relationships in cases of relatively complex phonon vibration spectra.
Keywords: negative dielectric permittivity, single-oscillator model, terahertz band,
LiTaO3.
Manuscript received 29.12.11; revised version received 18.01.12; accepted for
publication 26.01.12; published online 29.03.12.
1. Introduction
The dispersion dependence of the dielectric permittivity
(DP) in the terahertz range in particular within the region
of negative DP values today is a subject of the increased
researchers’ interest due to prospects to create materials
with a negative refraction index (metamaterials) [1].
Negative refraction index media can be formed using the
simultaneous availability of the negative dielectric
permittivity (ε) and magnetic permeability (μ). It is able
to cause radical anomalies in propagation of
electromagnetic waves, namely: plane-parallel plate
shows focusing properties [2], light repulsion is replaced
by light attraction [3]. However, after a long history of
spectroscopic studies of the DP dispersion by using the
infrared (IR) and Raman spectroscopy methods, it has
not been given proper attention to the problem of the
quantitative determination of a real band where DP has a
negative value.
The region of negative DP values theoretically
occurs always between the transverse T and
longitudinal L phonons frequencies (T – L splitting). It
is resulted using a simplified model without damping
(Г = 0) and based on the well-known Kurosawa ratio [4].
Indeed, within this approximation () – at
T, for any frequencies pair T < < L .
However, in the case when damping is present (Г >> 0),
the real part of DP cannot reach negative values. The
effect of damping the polar vibrations on dispersive
properties of the crystalline media has been studied in
this work, and it has been shown that just the damping
action plays a crucial role in shaping of the minimum
achievable negative value of DP and width of the
frequency area of its availability.
2. Basic theory and problem statement
The light interaction with polar vibrations of the crystal
lattice is described in the semi-classical approach [5]. In
this case, the dispersion dependence () for crystals
with Nk vibration modes is determined by the general
expression:
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 1. P. 83-88.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
84
kN
i i
T
ik
T
ikik
kk
i
S
1
22
2
, (1)
where T
ik and Sik are, respectively, the transverse
phonon frequencies and oscillator strength of the thi
lattice vibrations with polarization along the thk axis;
i is the damping constant; k is the main value of
dielectric tensor at optical frequencies.
The components of the DP tensor (1) can be
expressed in general by terms of the longitudinal L
ik
and the transverse T
ik vibration frequencies as a
Kurosawa ratio [4]:
kN
i
T
ik
L
ik
kk
1
22
22
, (2)
which is actually a fair in the approximation of absence
of phonon damping i = 0. It is in this approximation:
N
j
T
i
T
j
T
i
L
j
T
i
L
i
iS
1
22
222
1 , (3)
where for simplicity it is omitted the index k for
polarization marking.
Within frameworks of the single-oscillator model,
the dispersion dependence () (1) and complex
refraction index inn~ can be written as:
i
S
n
T
T
22
2
2~ . (4)
Our analysis of the dispersion inherent to the
dielectric permittivity in the case of non-zero damping is
performed using the above Eq. (1) [6]. Exp. (4) describes
the DP dispersion within the single-oscillator model, and
it has been used as a base for our calculations. It allows
to investigate the damping action on the negative DP
region formation. The complex function ε(ω) (4) can be
taken in the following form:
i . (5)
The explicit form of and can be found from
the ratios (4) and (5):
22222
2
22222
222
T
T
T
TT
S
S
(6)
The measured reflection coefficient R at normal
incidence is associated with the refraction index n and
extinction coefficient as follows:
22
22
1
1
n
n
R . (7)
Dispersion options S, T and are measured
directly from the Raman spectra or find by the numerical
methods from the condition of the best approximation
for the reflection spectrum curve. In the approximation
when absorption is neglected, the frequency of
longitudinal phonons L can be found from the condition
of the band zero contribution into the low frequency
dielectric constant ε0.
The relationship between material and
spectroscopic parameters of the medium is described by
the following ratio:
0
T
L , (8)
which is a known Lyddane-Sachs-Teller ratio [7].
Fig. 1 shows the reflection spectra (a) and
dispersion dependence of the DP real part (b) for model
crystalline medium. The dotted line describe the function
(2) in the case when damping is absent (Г = 0). It is
damping the optical phonons that eliminates the gap in
the curves for Г > 0, as shown in Fig. 1, and, in fact, it
determines the anomalous dispersion of waves in the
crystal and appearance of the region with the negative
dielectric permittivity. The negative dielectric
permittivity region, in accordance with the Kurosawa
ratio (2), will be presented for any frequencies within the
LT splitting region, and its boundaries will be
matched to the frequencies of transverse T and
longitudinal L oscillations. However, Fig. 1b shows
that the minimum value min is increased with , and the
frequency band with negative DP is simultaneously
narrowed. The state 0min is achieved for certain
“critical” damping ( = 0.085 in Fig. 1b), and the
frequency band with negative DP completely disappears.
It should be noted that the reflection spectra (Fig. 1a)
does not contain any specific features for marking the
presence or absence of negative DP area. Therefore, the
quantitative analysis of the conditions for availability of
the negative DP and the real band of its implementation
are performed in this work using the physical and
spectroscopic parameters of the crystal medium.
3. Criterion for existence of frequency bands
with a negative DP
The dispersion dependence () can be written as (4) in
the simplest case for crystals with a single polar
vibration. The oscillator strength is defined as S = 0
- for this case.
It should satisfy the next condition at frequencies
where the real part of DP (6) reaches its extreme value:
0
. (9)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 1. P. 83-88.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
85
-20
-10
-2
0
2
4
6
8 b
0
20
40
60
80
100
R, % а
Fig. 1. Reflection spectra (a) and dispersion dependence of the
real part of DP (b) for model crystalline medium. The dotted
line illustrates an ideal case (Kurosawa ratio (2)), solid line – a
real case of non-zero damping.
According to (9), after differentiating the first
equation of system (6), and not rejecting the null
common denominator, one can obtain an intermediate
equation:
.0224
22
232242222
2322222
TTTT
TTT
SSS
SS
(10)
After simplification, the expression (10) will look as:
22222 TT . (11)
The frequency position of the () minimum is
determined by one of the equation (11) roots, namely it
is belonged to the interval of frequencies > T.
Moreover, the point will be the minimum point (min),
when it is equal:
TT
2
min . (12)
After substitution min to the first equation of the
system (6), we obtain the minimum value of the
dielectric function (). Note that by (12) the frequency
position of the minimum and the minimum value of
() (6) depend on the damping parameter (Fig. 1b).
So, for availability of the negative permittivity region,
where () < 0, it is necessary that:
(min) < 0, where
2
2
minmin
2
T
TS
, (13)
whence the inequality directly follows:
02 22
TT
S
. (14)
Changes of the minimum value of DP ( min ) as a
function of the damping constant Г observed in Fig. 1
are fully described by the analytical dependence (13). In
particular, if Г 0 in Eq. (13), then min like to
that in the Kurosawa ratio (2).
The inequality (14) for values of Г > 0 leads to the
following conditions:
10
T , (15)
and it gives a simple quantitative criterion for existence
of frequency bands with negative DP within the single-
oscillator crystal model. It is based on the ratio of
damping constant phonon with the frequency T to such
material parameters of crystal as 0 and .
The criterion (15) can be represented through
spectroscopic parameters of the crystal medium, if we
apply to it the ratio of Lyddane-Sachs-Teller (8). Then,
the condition of availability of the negative DP region
takes the following form:
TL or 1
TL
. (16)
In fact, existence of the negative dielectric
permittivity region until the phonon damping constant
does not exceed LT splitting for this phonon is the
immediate consequence of (16).
Thus, the critical damping cr should be determined
using (15) and (16). It corresponds to the upper
boundary phonon damping and restricts the existence of
negative DP in the crystal:
110
T
L
T
cr or TLcr . (17)
The dispersion dependence () for critical
damping at /T = 0.085 is illustrated by the curve in
Fig. 1b, if the minimum of real part for DP is equal to
zero.
If < cr, then the frequency band with negative
DP values is formed around the minimum of DP within
LT splitting frequencies of these oscillations. And
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 1. P. 83-88.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
86
vice versa, when > cr (/T = 0.1 in Fig. 1b) the
function () is positive, including the band of residual
rays ( LT ).
4. Real frequency band with negative DP
The frequency range of negative DP area really does not
match to the LT splitting, unlike the idealized case of
no damping, which is described by the Kurosawa
ratio (2). The negative DP area is shrunk relative to
LT splitting up to its complete disappearance due to
increase of the damping constant as shown in Fig. 1b.
Therefore, we have studied behavior of the frequency
band with negative DP as a function of the damping
constant.
Let us introduce the notation of the frequencies:
is the frequency when DP begins to take negative
values, and is the frequency when DP goes out from
negative region. If for expression (3) consider the case of
single-oscillator model (N = 1) and substitute the
resulting value for the oscillator strength in the first
equation of the system (6), then:
22222
2222
1
T
LTT
. (18)
The frequencies and are found from terms
of equality of the real part of DP, described with the
expression (18), to zero 0 . After simple
mathematical transformations, rejecting no physical
variations, we obtain:
21
222222222 4
2
1
TLTLTL
. (19)
Let is the real frequency band of
negative DP. An explicit expression for Δω may be
easily found using the relations (19), so we have:
2
1
TL
TL . (20)
The expression (20) is the generalization of the no
damping idealized case as if Г = 0, then TL .
Similarly, if in expressions (19) to direct the damping
constant to zero, we get: T , and L .
Our analysis shows that a large number of crystals
with one pronounced vibration in the infrared region,
such as classic items of alkali-halide crystals (NaCl,
KBr, NaF, LiF, and many others) in the anomalous
dispersion region reaches sufficiently high in modulus
negative values of DP. For these crystals, the difference
TL reached tens of THz, whereas the damping
constant was typically less then one THz at room and
low temperatures. Therefore, the ratio 1.0 TL
and according to (16) ensures the existence of high in
modulus negative values of DP observed in practice,
moreover the frequency range with () < 0 almost
coincides with the interval of LT splitting, according
to (20). However, the absence of negative DP with
relative damping 1 TL has been recently
found by us in some crystalline modifications of boron
nitride [8, 9].
In addition, there are a class of non-magnetic
media, in particular nonlinear crystals, where the
parameters , T, L may be directly measured from the
Raman spectra. The existence and width of range of
negative DP in LiTaO3 were quantitatively investigated
in this work.
5. Negative dielectric permittivity in LiTaO3 crystal
LiTaO3 has 30 vibrational degrees of freedom in the
ferroelectric phase, which are distributed by the types of
symmetry: 5A1+5A2+10E, moreover E variations are
double degenerate. Acoustic phonons belonging to the
representations A1- and E type, and vibrations from
2A type do not appear both in IR and Raman spectra.
The feature of the 1A type and E type
vibrations are as follows: they are active both in Raman
spectra and IR reflection spectra, and this crystal was
chosen for our research.
The polar vibration parameters of crystal lithium
tantalate for parallel ( 1A type) and perpendicular
( E type) polarizations are shown in Table, and they are
measured from the Raman spectra. Our calculated
parameters are presented in columns 95 for each
oscillation, namely: LT splitting, normalized to
LT splitting damping (16), frequencies and
(19), the frequency width of the negative DP area (20).
The minimum negative DP value ( min , column 10) is
calculated directly from (6), as in (13) parameter ε is
not monosemantic in the case of complex phonon
spectrum.
The frequency dispersion of DP for LiTaO3 crystal
is shown in Fig. 2 for both polarizations. The oscillation
number in Fig. 2 corresponds to the number in Table. All
vibrations of the parallel polarization show negative DP
areas, in accordance to quantitative data calculated using
our criteria (16), and they can be seen in Fig. 2. The
vibrations A1 and A2 have very “deep” negative DP
areas: 119min and 40min , respectively, and in
their absolute values are higher than the DP values at
low (ε0 = 30.24) and high (ε = 4.15) frequencies. The
vibrations A1 and A2 create the common region of
negative DP from 6.06 to 10.39 THz as it is also shown
in Fig. 2. The real frequency bands of negative DP
values Δ (see Table) are calculated using Eq. (20) for
all the vibrations in LiTaO3 crystal ( 1A type), and they
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 1. P. 83-88.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
87
Table. Polar vibration parameters for crystal LiTaO3.
№
ωL
(THz)
ωT
(THz)
Г
(THz)
TL
(THz) TL
(THz)
(THz)
Δ
(THz) min
1 2 3 4 5 6 7 8 9 10
A1-type ε = 4.15, ε0 = 30.24
A1 7.35 6.03 0.45 1.32 0.34 7.31 6.06 1.25 –119
A2 10.41 7.59 0.45 2.82 0.16 10.39 7.61 2.78 –40
A3 11.97 10.68 0.24 1.29 0.19 11.96 10.69 1.27 –7.6
A4 25.92 18 0.24 7.92 0.03 25.92 18.00 7.92 –107
E-type ε = 4.13, ε0 = 35.78
E1 2.4 2.22 0.57 0.18 3.17 No negative DP area
E2 4.89 4.2 0.18 0.69 0.26 4.87 4.21 0.66 –101
E3 7.44 6.18 0.18 1.26 0.14 7.43 6.19 1.24 –130
E4 8.34 7.53 0.3 0.81 0.37 8.31 7.56 0.75 –9.5
E5 9.54 9.48 0.18 0.06 3.00 No negative DP area
E6 13.56 11.49 0.36 2.07 0.17 13.54 11.51 2.03 –47
E7 14.22 13.86 0.18 0.36 0.50 14.20 13.88 0.32 –2.7
E8 19.44 17.88 0.33 1.56 0.21 19.42 17.90 1.52 –53
E9 26.1 19.86 0.45 6.24 0.07 26.09 19.87 6.22 –17
Note. Column #1 is the vibration number; #2 and #3 are frequencies of longitudinal and transverse vibrations, respectively; #4
and #5 are the damping constant (Г) and T – L splitting; #6 is damping normalized to T – L splitting; #7 and #8 are the
frequencies at which the DP comes from the negative region ( ) and begins to take negative values ( ); #9 is the range
width for negative DP; #10 is the minimum negative DP value.
>0
LiTaO3 A - type
LiTaO3 E - type
THz
cm-1A1
A2
A3
A4
E1
E2
E3
E4
E5
E6
E7
E8
E9
-100
-50
0
50
100
3 6 9 12 15 18 21 24
100 200 300 400 500 600 700 800
Fig. 2. Frequency dispersion of the real part of DP for LiTaO3 crystal in the cases of parallel (A1 – type, dotted line) and
perpendicular (E – type, solid line) polarizations.
are very close to the LT splitting due to the fact that
relative damping is much smaller than unity. So, as
Eq. (20) was derived for the single-oscillator crystal
model, the vibrations with common negative DP areas
can cause significant uncertainties. Let us consider, for
example, vibrations E3 and E4 in LiTaO3 crystal for
perpendicular polarization. These vibrations have a
common negative DP area in accord with Fig. 2,
however, quantitative data (Table, columns 97 ) are
indicative of the splitting of negative DP for E3 and E4
vibrations by the positive DP band with the 0.12-THz
width. Exact determination of the frequency band where
DP gets negative values for the merge areas case can be
calculated using Eq. (19) considering for the former
oscillation and for the latter oscillation, respectively.
Absence of the negative DP regions for
ωT = 2.22 THz and ωT = 9.48 THz of perpendicular
polarization (see Fig. 2, vibrations E1 and E5) also
corresponds to the table data and criterion (16), because
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 1. P. 83-88.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
88
the relative damping value in the order of 3. In this case,
the ratio (20) returns the complex roots, which is logical
because the frequency dispersion of the real part of DP
does not cross zero. According to (16) and quantitative
data in Table, all other phonon oscillations have negative
DP area, as we can see in Fig. 2.
The A1-type has no phonon vibrations in the low-
frequency region 2.1-5.1 THz, unlike E type that has
two vibrations, one of them (E2) reaches large negative
values ( 101min ). It should be noted that both
polarizations have two common frequency regions with
negative DP. There are two characteristic vibrations for
1A type (A1, A2) and two vibrations for E type
(E3, E4) in the range from 6.06 to 8.28 THz. One
oscillation of 1A type has a pronounced negative DP
area ( 107min ), E type is separated by two close
vibrations with two negative bands with minimum
values 35 and 17 in the high-frequency region 17.88
up to 26.07 THz, respectively.
6. Conclusions
The research results of physical conditions providing
existence of the region with negative DP values in real
crystalline medium have been presented in this work. It
has been shown that the existence of negative DP in
crystals essentially depends on damping the polar
phonons. The quantitative criterion (16) for the existence
of frequency bands with negative DP has been obtained
on the basis of the single-oscillator crystal model. The
analytical expressions (19), (20) for calculation of the
frequency bands with negative DP have been obtained.
The estimation of narrowing the negative DP area
relatively to the LT frequency splitting as a function
of damping has been presented.
There are two areas (from 6.06 to 8.28 THz and
from 17.88 to 26.07 THz) in LiTaO3 crystal where both
components of DP tensor acquire negative values. The
analysis confirms the applicability of the criterion (16)
for crystals with a relatively complex spectrum of polar
phonons. It has been shown that the negative DP band
can be calculated using (20) with a high accuracy in the
case of separate areas and allows in the first
approximation to estimate the frequency band for the
common areas of negative DP.
The analytical dependences between the basic
parameters of crystals expressed by Eqs (16), (19), and
(20) may be useful when analyzing the negative regions
of DP in crystalline media with rather complicated
phonon spectrum.
References
1. V.G. Veselago, Electrodynamics of materials with
negative refractive index // Physics – Uspekhi, 173
(7), p. 790-794 (2003).
2. V.G. Veselago, Formulating Fermat’s principle for
light traveling in materials with negative refraction
// Physics – Uspekhi, 172 (10), p. 1215-1218
(2002).
3. V.G. Veselago, Energy, momentum and mass
transfer by an electromagnetic wave in a negative
refraction medium // Physics – Uspekhi, 179 (6),
p. 689-694 (2009).
4. T. Kurosawa, Polarization waves in solids // J.
Phys. Soc. Jap. 16 (9), p. 1288-1308 (1961).
5. P.A. Korotkov, and G.S. Felinskyi, Research of
negative dielectric permeability area in the media
without inversion center // Bull. Kyiv Univ.: ser.
Phys. & Math. No.2, p. 162-171 (2008), in
Ukrainian.
6. S.G. Felinskyi, P.A. Korotkov, and G.S. Felinskyi,
Criterion of the existence region of negative
dielectric permittivity at the polar oscillations
frequencies in crystals // Bull. Kyiv Univ.: ser.
Phys. & Math. No.1, p. 191-196 (2010), in
Ukrainian.
7. R. Lyddane, R.G. Sachs, E. Teller, On the polar
vibrations of alkali halides // Phys. Rev. 59, p. 673-
676 (1941).
8. S.G. Felinskyi, P.A. Korotkov, and G.S. Felinskyi,
Negative dielectric function in anisotropic
modifications of boron nitride // New technologies,
No.2, p. 51-57 (2010), in Ukrainian.
9. S.G. Felinskyi, P.A. Korotkov, G.S. Felinskyi,
Terahertz properties and the negative dielectric
regions in boron nitride // IEEE Intern. Workshop
on THz Radiation: Basic Research & Applications
(TERA 2010), Sevastopol, 2010, September 12-14,
p. 265-266.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 1. P. 83-88.
PACS 77.22.Ch, 77.84.Ek
Negative dielectric permittivity of nonmagnetic crystals
in the terahertz waveband
S.G. Felinskyi1, P.A. Korotkov2, G.S. Felinskyi3
Taras Shevchenko Kyiv National University, 4, prospect Glushkova, 03127 Kyiv, Ukraine
Phone: +380-44-526-0570; e-mail: stalisman@ukr.net1, pak@mail.univ.kiev.ua2, felinskyi@yahoo.com3
Abstract. Physical conditions for occurrence of the spectral bands with the negative dielectric permittivity in nonmagnetic crystalline media in the terahertz waveband have been studied in this work. It has been shown that damping the polar vibrations has a primary effect on formation of the negative dielectric permittivity in real crystals at resonance (terahertz) frequencies, and phonon attenuation imposes significant restrictions both on the frequency range and the minimum achievable value for all dielectric tensor components. Within frameworks of the single-oscillator model, the authors have obtained: i) the criterion for the existence of the negative dielectric permittivity, which is based on physical and spectroscopic parameters of the crystal, ii) analytical expressions for calculation of the frequency band where the dielectric permittivity takes negative values. Frequency regions and the minimum value of negative dielectric permittivity are quantitatively defined in the crystal LiTaO3. It is proved the applicability of the obtained relationships in cases of relatively complex phonon vibration spectra.
Keywords: negative dielectric permittivity, single-oscillator model, terahertz band, LiTaO3.
Manuscript received 29.12.11; revised version received 18.01.12; accepted for publication 26.01.12; published online 29.03.12.
1. Introduction
The dispersion dependence of the dielectric permittivity (DP) in the terahertz range in particular within the region of negative DP values today is a subject of the increased researchers’ interest due to prospects to create materials with a negative refraction index (metamaterials) [1]. Negative refraction index media can be formed using the simultaneous availability of the negative dielectric permittivity (ε) and magnetic permeability (μ). It is able to cause radical anomalies in propagation of electromagnetic waves, namely: plane-parallel plate shows focusing properties [2], light repulsion is replaced by light attraction [3]. However, after a long history of spectroscopic studies of the DP dispersion by using the infrared (IR) and Raman spectroscopy methods, it has not been given proper attention to the problem of the quantitative determination of a real band where DP has a negative value.
The region of negative DP values theoretically occurs always between the transverse (T and longitudinal (L phonons frequencies (T – L splitting). It is resulted using a simplified model without damping (Г = 0) and based on the well-known Kurosawa ratio [4]. Indeed, within this approximation ((() ( –( at ( ( (T, for any frequencies pair (T < ( < (L . However, in the case when damping is present (Г >> 0), the real part of DP cannot reach negative values. The effect of damping the polar vibrations on dispersive properties of the crystalline media has been studied in this work, and it has been shown that just the damping action plays a crucial role in shaping of the minimum achievable negative value of DP and width of the frequency area of its availability.
2. Basic theory and problem statement
The light interaction with polar vibrations of the crystal lattice is described in the semi-classical approach [5]. In this case, the dispersion dependence ((() for crystals with Nk vibration modes is determined by the general expression:
(
)
(
)
(
)
å
=
¥
G
w
-
w
-
w
w
+
e
=
w
e
k
N
i
i
T
ik
T
ik
ik
k
k
i
S
1
2
2
2
,
(1)
where
T
ik
w
and Sik are, respectively, the transverse phonon frequencies and oscillator strength of the
th
-
i
lattice vibrations with polarization along the
th
-
k
axis; (i is the damping constant;
¥
e
k
is the main value of dielectric tensor at optical frequencies.
The components of the DP tensor (1) can be expressed in general by terms of the longitudinal
L
ik
w
and the transverse
T
ik
w
vibration frequencies as a Kurosawa ratio [4]:
(
)
(
)
(
)
Õ
=
¥
w
-
w
w
-
w
e
=
w
e
k
N
i
T
ik
L
ik
k
k
1
2
2
2
2
,
(2)
which is actually a fair in the approximation of absence of phonon damping (i = 0. It is in this approximation:
(
)
(
)
(
)
(
)
Õ
¹
¥
w
-
w
w
-
w
ú
ú
ú
û
ù
ê
ê
ê
ë
é
-
÷
÷
ø
ö
ç
ç
è
æ
w
w
e
=
N
j
T
i
T
j
T
i
L
j
T
i
L
i
i
S
1
2
2
2
2
2
1
,
(3)
where for simplicity it is omitted the index k for polarization marking.
Within frameworks of the single-oscillator model, the dispersion dependence ((() (1) and complex refraction index
k
-
=
i
n
n
~
can be written as:
(
)
G
w
-
w
-
w
w
+
e
=
w
e
=
¥
i
S
n
T
T
2
2
2
2
~
.
(4)
Our analysis of the dispersion inherent to the dielectric permittivity in the case of non-zero damping is performed using the above Eq. (1) [6]. Exp. (4) describes the DP dispersion within the single-oscillator model, and it has been used as a base for our calculations. It allows to investigate the damping action on the negative DP region formation. The complex function ε(ω) (4) can be taken in the following form:
(
)
(
)
(
)
w
e
¢
¢
+
w
e
¢
=
w
e
i
.
(5)
The explicit form of
e
¢
and
e
¢
¢
can be found from the ratios (4) and (5):
(
)
(
)
(
)
(
)
(
)
ï
ï
ï
î
ï
ï
ï
í
ì
G
w
+
w
-
w
G
w
w
=
w
e
¢
¢
G
w
+
w
-
w
w
-
w
w
+
e
=
w
e
¢
¥
2
2
2
2
2
2
2
2
2
2
2
2
2
2
T
T
T
T
T
S
S
(6)
The measured reflection coefficient R at normal incidence is associated with the refraction index n and extinction coefficient ( as follows:
(
)
(
)
2
2
2
2
1
1
k
+
+
k
+
-
=
n
n
R
.
(7)
Dispersion options S, (T and ( are measured directly from the Raman spectra or find by the numerical methods from the condition of the best approximation for the reflection spectrum curve. In the approximation when absorption is neglected, the frequency of longitudinal phonons (L can be found from the condition of the band zero contribution into the low frequency dielectric constant ε0.
The relationship between material and spectroscopic parameters of the medium is described by the following ratio:
¥
e
e
=
w
w
0
T
L
,
(8)
which is a known Lyddane-Sachs-Teller ratio [7].
Fig. 1 shows the reflection spectra (a) and dispersion dependence of the DP real part (b) for model crystalline medium. The dotted line describe the function (2) in the case when damping is absent (Г = 0). It is damping the optical phonons that eliminates the gap in the curves for Г > 0, as shown in Fig. 1, and, in fact, it determines the anomalous dispersion of waves in the crystal and appearance of the region with the negative dielectric permittivity. The negative dielectric permittivity region, in accordance with the Kurosawa ratio (2), will be presented for any frequencies within the
L
T
-
splitting region, and its boundaries will be matched to the frequencies of transverse (T and longitudinal (L oscillations. However, Fig. 1b shows that the minimum value
min
e
¢
is increased with (, and the frequency band with negative DP is simultaneously narrowed. The state
0
min
=
e
¢
is achieved for certain “critical” damping (( = 0.085 in Fig. 1b), and the frequency band with negative DP completely disappears. It should be noted that the reflection spectra (Fig. 1a) does not contain any specific features for marking the presence or absence of negative DP area. Therefore, the quantitative analysis of the conditions for availability of the negative DP and the real band of its implementation are performed in this work using the physical and spectroscopic parameters of the crystal medium.
3. Criterion for existence of frequency bands
with a negative DP
The dispersion dependence ((() can be written as (4) in the simplest case for crystals with a single polar vibration. The oscillator strength is defined as S = (0 ‑ (( for this case.
It should satisfy the next condition at frequencies where the real part of DP (6) reaches its extreme value:
0
=
w
¶
e
¢
¶
.
(9)
- 20
- 10
- 2
0
2
4
6
8
b
0
20
40
60
80
100
R , %
а
Fig. 1. Reflection spectra (a) and dispersion dependence of the real part of DP (b) for model crystalline medium. The dotted line illustrates an ideal case (Kurosawa ratio (2)), solid line – a real case of non-zero damping.
According to (9), after differentiating the first equation of system (6), and not rejecting the null common denominator, one can obtain an intermediate equation:
(
)
(
)
.
0
2
2
4
2
2
2
3
2
2
4
2
2
2
2
2
3
2
2
2
2
2
=
G
w
w
+
G
w
w
-
w
-
w
w
w
+
+
G
w
w
-
w
-
w
w
w
-
T
T
T
T
T
T
T
S
S
S
S
S
(10)
After simplification, the expression (10) will look as:
(
)
2
2
2
2
2
G
w
=
w
-
w
T
T
.
(11)
The frequency position of the (((() minimum is determined by one of the equation (11) roots, namely it is belonged to the interval of frequencies ( > (T. Moreover, the point ( will be the minimum point ((min), when it is equal:
G
w
+
w
=
w
T
T
2
min
.
(12)
After substitution (min to the first equation of the system (6), we obtain the minimum value of the dielectric function ((((). Note that by (12) the frequency position of the minimum and the minimum value of (((() (6) depend on the damping parameter ( (Fig. 1b). So, for availability of the negative permittivity region, where (((() < 0, it is necessary that:
((((min) < 0, where
(
)
2
2
min
min
2
G
+
G
w
w
-
e
=
e
¢
=
w
e
¢
¥
T
T
S
,
(13)
whence the inequality directly follows:
0
2
2
2
<
w
e
-
G
w
+
G
¥
T
T
S
.
(14)
Changes of the minimum value of DP (
min
e
¢
) as a function of the damping constant Г observed in Fig. 1 are fully described by the analytical dependence (13). In particular, if Г ( 0 in Eq. (13), then
-¥
®
e
¢
min
like to that in the Kurosawa ratio (2).
The inequality (14) for values of Г > 0 leads to the following conditions:
÷
÷
ø
ö
ç
ç
è
æ
-
e
e
w
<
G
¥
1
0
T
,
(15)
and it gives a simple quantitative criterion for existence of frequency bands with negative DP within the single-oscillator crystal model. It is based on the ratio of damping constant phonon with the frequency (T to such material parameters of crystal as (0 and ((.
The criterion (15) can be represented through spectroscopic parameters of the crystal medium, if we apply to it the ratio of Lyddane-Sachs-Teller (8). Then, the condition of availability of the negative DP region takes the following form:
(
)
T
L
w
-
w
<
G
or
1
<
w
-
w
G
T
L
.
(16)
In fact, existence of the negative dielectric permittivity region until the phonon damping constant does not exceed
L
T
-
splitting for this phonon is the immediate consequence of (16).
Thus, the critical damping (cr should be determined using (15) and (16). It corresponds to the upper boundary phonon damping and restricts the existence of negative DP in the crystal:
1
1
0
-
w
w
=
-
e
e
=
w
G
¥
T
L
T
cr
or
T
L
cr
w
-
w
=
G
.
(17)
The dispersion dependence (((() for critical damping at (/(T = 0.085 is illustrated by the curve in Fig. 1b, if the minimum of real part for DP is equal to zero.
If ( < (cr, then the frequency band with negative DP values is formed around the minimum of DP within
L
T
-
splitting frequencies of these oscillations. And vice versa, when ( > (cr ((/(T = 0.1 in Fig. 1b) the function (((() is positive, including the band of residual rays (
L
T
-
).
4. Real frequency band with negative DP
The frequency range of negative DP area really does not match to the
L
T
-
splitting, unlike the idealized case of no damping, which is described by the Kurosawa ratio (2). The negative DP area is shrunk relative to
L
T
-
splitting up to its complete disappearance due to increase of the damping constant as shown in Fig. 1b. Therefore, we have studied behavior of the frequency band with negative DP as a function of the damping constant.
Let us introduce the notation of the frequencies:
-
w
is the frequency when DP begins to take negative values, and
+
w
is the frequency when DP goes out from negative region. If for expression (3) consider the case of single-oscillator model (N = 1) and substitute the resulting value for the oscillator strength in the first equation of the system (6), then:
(
)
(
)
(
)
(
)
ú
ú
ú
û
ù
ê
ê
ê
ë
é
w
-
w
+
w
G
w
-
w
w
-
w
+
e
=
w
e
¢
¥
2
2
2
2
2
2
2
2
2
1
T
L
T
T
.
(18)
The frequencies
-
w
and
+
w
are found from terms of equality of the real part of DP, described with the expression (18), to zero
(
)
0
=
w
e
¢
. After simple mathematical transformations, rejecting no physical variations, we obtain:
(
)
2
1
2
2
2
2
2
2
2
2
2
4
2
1
÷
÷
ø
ö
ç
ç
è
æ
w
w
-
G
-
w
+
w
±
G
-
w
+
w
=
w
±
T
L
T
L
T
L
.
(19)
Let
-
+
w
-
w
=
w
D
is the real frequency band of negative DP. An explicit expression for Δω may be easily found using the relations (19), so we have:
(
)
2
1
÷
÷
ø
ö
ç
ç
è
æ
w
-
w
G
-
w
-
w
=
w
D
T
L
T
L
.
(20)
The expression (20) is the generalization of the no damping idealized case as if Г = 0, then
T
L
w
-
w
=
w
D
. Similarly, if in expressions (19) to direct the damping constant to zero, we get:
T
w
=
w
-
, and
L
w
=
w
+
.
Our analysis shows that a large number of crystals with one pronounced vibration in the infrared region, such as classic items of alkali-halide crystals (NaCl, KBr, NaF, LiF, and many others) in the anomalous dispersion region reaches sufficiently high in modulus negative values of DP. For these crystals, the difference
T
L
w
-
w
reached tens of THz, whereas the damping constant ( was typically less then one THz at room and low temperatures. Therefore, the ratio
(
)
1
.
0
<
w
-
w
G
T
L
and according to (16) ensures the existence of high in modulus negative values of DP observed in practice, moreover the frequency range with (((() < 0 almost coincides with the interval of
L
T
-
splitting, according to (20). However, the absence of negative DP with relative damping
(
)
1
>
w
-
w
G
T
L
has been recently found by us in some crystalline modifications of boron nitride [8, 9].
In addition, there are a class of non-magnetic media, in particular nonlinear crystals, where the parameters (, (T, (L may be directly measured from the Raman spectra. The existence and width of range of negative DP in LiTaO3 were quantitatively investigated in this work.
5. Negative dielectric permittivity in LiTaO3 crystal
LiTaO3 has 30 vibrational degrees of freedom in the ferroelectric phase, which are distributed by the types of symmetry: 5A1+5A2+10E, moreover E variations are double degenerate. Acoustic phonons belonging to the representations A1- and
-
E
type, and vibrations from
-
2
A
type do not appear both in IR and Raman spectra.
The feature of the
-
1
A
type and
-
E
type vibrations are as follows: they are active both in Raman spectra and IR reflection spectra, and this crystal was chosen for our research.
The polar vibration parameters of crystal lithium tantalate for parallel (
-
1
A
type) and perpendicular (
-
E
type) polarizations are shown in Table, and they are measured from the Raman spectra. Our calculated parameters are presented in columns
9
5
-
for each oscillation, namely:
L
T
-
splitting, normalized to
L
T
-
splitting damping (16), frequencies
+
w
and
-
w
(19), the frequency width of the negative DP area (20). The minimum negative DP value (
min
e
¢
, column 10) is calculated directly from (6), as in (13) parameter ε( is not monosemantic in the case of complex phonon spectrum.
The frequency dispersion of DP for LiTaO3 crystal is shown in Fig. 2 for both polarizations. The oscillation number in Fig. 2 corresponds to the number in Table. All vibrations of the parallel polarization show negative DP areas, in accordance to quantitative data calculated using our criteria (16), and they can be seen in Fig. 2. The vibrations A1 and A2 have very “deep” negative DP areas:
119
min
-
=
e
¢
and
40
min
-
=
e
¢
, respectively, and in their absolute values are higher than the DP values at low (ε0 = 30.24) and high (ε( = 4.15) frequencies. The vibrations A1 and A2 create the common region of negative DP from 6.06 to 10.39 THz as it is also shown in Fig. 2. The real frequency bands of negative DP values Δ( (see Table) are calculated using Eq. (20) for all the vibrations in LiTaO3 crystal (
-
1
A
type), and they T
L
w
-
w
are very close to the
L
T
-
splitting due to the fact that relative damping is much smaller than unity. So, as Eq. (20) was derived for the single-oscillator crystal model, the vibrations with common negative DP areas can cause significant uncertainties. Let us consider, for example, vibrations E3 and E4 in LiTaO3 crystal for perpendicular polarization. These vibrations have a common negative DP area in accord with Fig. 2, however, quantitative data (Table, columns
9
7
-
) are indicative of the splitting of negative DP for E3 and E4 vibrations by the positive DP band with the 0.12-THz width. Exact determination of the frequency band where DP gets negative values for the merge areas case can be calculated using Eq. (19) considering
-
w
for the former oscillation and
+
w
for the latter oscillation, respectively.
Absence of the negative DP regions for ωT = 2.22 THz and ωT = 9.48 THz of perpendicular polarization (see Fig. 2, vibrations E1 and E5) also corresponds to the table data and criterion (16), because the relative damping value in the order of 3. In this case, the ratio (20) returns the complex roots, which is logical because the frequency dispersion of the real part of DP does not cross zero. According to (16) and quantitative data in Table, all other phonon oscillations have negative DP area, as we can see in Fig. 2.
The A1-type has no phonon vibrations in the low-frequency region 2.1-5.1 THz, unlike
-
E
type that has two vibrations, one of them (E2) reaches large negative values (
101
min
-
=
e
¢
). It should be noted that both polarizations have two common frequency regions with negative DP. There are two characteristic vibrations for
-
1
A
type (A1, A2) and two vibrations for
-
E
type (E3, E4) in the range from 6.06 to 8.28 THz. One oscillation of
-
1
A
type has a pronounced negative DP area (
107
min
-
=
e
¢
),
-
E
type is separated by two close vibrations with two negative bands with minimum values
3
5
-
and
17
-
in the high-frequency region 17.88 up to 26.07 THz, respectively.
6. Conclusions
The research results of physical conditions providing existence of the region with negative DP values in real crystalline medium have been presented in this work. It has been shown that the existence of negative DP in crystals essentially depends on damping the polar phonons. The quantitative criterion (16) for the existence of frequency bands with negative DP has been obtained on the basis of the single-oscillator crystal model. The analytical expressions (19), (20) for calculation of the frequency bands with negative DP have been obtained. The estimation of narrowing the negative DP area relatively to the
L
T
-
frequency splitting as a function of damping has been presented.
There are two areas (from 6.06 to 8.28 THz and from 17.88 to 26.07 THz) in LiTaO3 crystal where both components of DP tensor acquire negative values. The analysis confirms the applicability of the criterion (16) for crystals with a relatively complex spectrum of polar phonons. It has been shown that the negative DP band can be calculated using (20) with a high accuracy in the case of separate areas and allows in the first approximation to estimate the frequency band for the common areas of negative DP.
The analytical dependences between the basic parameters of crystals expressed by Eqs (16), (19), and (20) may be useful when analyzing the negative regions of DP in crystalline media with rather complicated phonon spectrum.
References
1. V.G. Veselago, Electrodynamics of materials with negative refractive index // Physics – Uspekhi, 173 (7), p. 790-794 (2003).
2. V.G. Veselago, Formulating Fermat’s principle for light traveling in materials with negative refraction // Physics – Uspekhi, 172 (10), p. 1215-1218 (2002).
3. V.G. Veselago, Energy, momentum and mass transfer by an electromagnetic wave in a negative refraction medium // Physics – Uspekhi, 179 (6), p. 689-694 (2009).
4. T. Kurosawa, Polarization waves in solids // J. Phys. Soc. Jap. 16 (9), p. 1288-1308 (1961).
5. P.A. Korotkov, and G.S. Felinskyi, Research of negative dielectric permeability area in the media without inversion center // Bull. Kyiv Univ.: ser. Phys. & Math. No.2, p. 162-171 (2008), in Ukrainian.
6. S.G. Felinskyi, P.A. Korotkov, and G.S. Felinskyi, Criterion of the existence region of negative dielectric permittivity at the polar oscillations frequencies in crystals // Bull. Kyiv Univ.: ser. Phys. & Math. No.1, p. 191-196 (2010), in Ukrainian.
7. R. Lyddane, R.G. Sachs, E. Teller, On the polar vibrations of alkali halides // Phys. Rev. 59, p. 673-676 (1941).
8. S.G. Felinskyi, P.A. Korotkov, and G.S. Felinskyi, Negative dielectric function in anisotropic modifications of boron nitride // New technologies, No.2, p. 51-57 (2010), in Ukrainian.
9. S.G. Felinskyi, P.A. Korotkov, G.S. Felinskyi, Terahertz properties and the negative dielectric regions in boron nitride // IEEE Intern. Workshop on THz Radiation: Basic Research & Applications (TERA 2010), Sevastopol, 2010, September 12-14, p. 265-266.
Table. Polar vibration parameters for crystal LiTaO3.
�
�
№�
ωL
(THz)�
ωT
(THz)�
Г
(THz)�
� EMBED Equation.3 ���
(THz)�
� EMBED Equation.3 ����
� EMBED Equation.3 ���
(THz)�
� EMBED Equation.3 ���
(THz)�
Δ(
(THz)�
� EMBED Equation.3 ����
�
1�
2�
3�
4�
5�
6�
7�
8�
9�
10�
�
A1-type ε( = 4.15, ε0 = 30.24�
�
A1�
7.35�
6.03�
0.45�
1.32�
0.34�
7.31�
6.06�
1.25�
–119�
�
A2�
10.41�
7.59�
0.45�
2.82�
0.16�
10.39�
7.61�
2.78�
–40�
�
A3�
11.97�
10.68�
0.24�
1.29�
0.19�
11.96�
10.69�
1.27�
–7.6�
�
A4�
25.92�
18�
0.24�
7.92�
0.03�
25.92�
18.00�
7.92�
–107�
�
E-type ε( = 4.13, ε0 = 35.78�
�
E1�
2.4�
2.22�
0.57�
0.18�
3.17�
No negative DP area�
�
E2�
4.89�
4.2�
0.18�
0.69�
0.26�
4.87�
4.21�
0.66�
–101�
�
E3�
7.44�
6.18�
0.18�
1.26�
0.14�
7.43�
6.19�
1.24�
–130�
�
E4�
8.34�
7.53�
0.3�
0.81�
0.37�
8.31�
7.56�
0.75�
–9.5�
�
E5�
9.54�
9.48�
0.18�
0.06�
3.00�
No negative DP area�
�
E6�
13.56�
11.49�
0.36�
2.07�
0.17�
13.54�
11.51�
2.03�
–47�
�
E7�
14.22�
13.86�
0.18�
0.36�
0.50�
14.20�
13.88�
0.32�
–2.7�
�
E8�
19.44�
17.88�
0.33�
1.56�
0.21�
19.42�
17.90�
1.52�
–53�
�
E9�
26.1�
19.86�
0.45�
6.24�
0.07�
26.09�
19.87�
6.22�
–17�
�
Note. Column #1 is the vibration number; #2 and #3 are frequencies of longitudinal and transverse vibrations, respectively; #4 and #5 are the damping constant (Г) and T – L splitting; #6 is damping normalized to T – L splitting; #7 and #8 are the frequencies at which the DP comes from the negative region (� EMBED Equation.3 ���) and begins to take negative values (� EMBED Equation.3 ���); #9 is the range width for negative DP; #10 is the minimum negative DP value.
� EMBED Word.Picture.8 ���
Fig. 2. Frequency dispersion of the real part of DP for LiTaO3 crystal in the cases of parallel (A1 – type, dotted line) and perpendicular (E – type, solid line) polarizations.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
83
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