Absorption-edge anisotropy in ReS2 and ReSe2 layered semiconductors

Article abstract of DOI:10.1103/PhysRevB.58.16130

Polarization-dependent absorption measurements of ReS₂ and ReSe₂ single crystals have been carried out in the temperature range between 25 and 500 K. A significant shift towards lower energies has been observed in the transmittance spectra of E∥b polarization with respect to those corresponding to E⊥ b polarization. Analysis reveals that the absorption edges of ReS₂ and ReSe₂ are indirect allowed transitions. The parameters that describe the temperature dependence of the absorption edges with different polarizations in the van der Waals plane are evaluated. The results indicate that the electron-phonon coupling constants for E∥b polarization are considerably larger than those of E⊥ b polarization.

Full text of DOI:10.1103/PhysRevB.58.16130

PHYSICAL REVIEW B
VOLUME 58, NUMBER 24
15 DECEMBER 1998-II
Absorption-edge anisotropy in ReS and ReSe layered semiconductors
2
2
C. H. Ho and Y. S. Huang*
Department of Electronic Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan, Republic of China
K. K. Tiong
Department of Electrical Engineering, National Taiwan Ocean University, Keelung 202, Taiwan, Republic of China
P. C. Liao
Department of Electronic Engineering, Kuang Wu Institute of Technology and Commerce, Peitou, Taipei 112, Taiwan, Republic of China
͑Received 11 May 1998͒
Polarization-dependent absorption measurements of ReS and ReSe single crystals have been carried out in
2
2
the temperature range between 25 and 500 K. A significant shift towards lower energies has been observed in
the transmittance spectra of E b polarization with respect to those corresponding to EЌb polarization. Analy-
sis reveals that the absorption edges of ReS and ReSe are indirect allowed transitions. The parameters that
ʈ
2
2
describe the temperature dependence of the absorption edges with different polarizations in the van der Waals
plane are evaluated. The results indicate that the electron-phonon coupling constants for E
considerably larger than those of EЌb polarization. ͓S0163-1829͑98͒02748-9͔
ʈ
b polarization are
I. INTRODUCTION
absorption curves suggests that the interband transitions in
ReS and ReSe are indirect and their band gaps are polar-
2
2
ReS and ReSe are diamagnetic semiconductors that be-
ization dependent. The polarization-dependent energy gaps
of these materials at various temperatures are determined and
their temperature dependence is analyzed by the Varshni
2
2
long to the family of the layered-type transition-metal
1
,2
dichalcogenides. The properties of the transition-meta1l
dichalcogenides have been reviewed by Wilson and Yoffe.
The layered-type transition-metal dichalcogenides crystalize
in a lattice with strong covalent bonds within a layer and
weak interactions, usually of the van der Waals type, be-
tween the individual layers. The strong anisotropy in the
chemical bonds leads to anisotropy in many material proper-
ties, such as their electrical and optical properties parallel
6
equation, an empirical expression proposed by O’Donnel
7
and Chen, and an expression containing the Bose-Einstein
occupation factor for phonons.8 The parameters that de-
scribe the temperature dependence of band gaps are evalu-
ated and discussed.
,9
II. EXPERIMENTAL DETAILS
3
and perpendicular to the layers. Unlike most of the layered
transition-metal dichalcogenides, ReS and ReSe are in a
Single crystals of ReS and ReSe were grown using the
2
2
2
2
2,4
distorted CdCl structure, leading to triclinic symmetry.
chemical-vapor transport method with Br as a transport
2
2
Crystals with triclinic symmetry are optically biaxial. There-
fore, anisotropic response of ReS and ReSe are expected
agent. Prior to the crystal growth, quartz tubes containing Br2
and the elements ͑Re, 99.95% pure; S, 99.999%; Se,
99.999%͒ were evacuated and sealed. The quartz tube was
placed in a three-zone furnace and the charge prereacted for
24 h at 800 °C with the growth zone at 1000 °C, preventing
the transport of the product. The furnace was then equili-
brated to give a constant temperature across the reaction
tube, and was programmed over 24 h to produce the tem-
perature gradient at which single-crystal growth takes place.
Best results were obtained with temperature gradients of ap-
2
2
for linearly polarized light, incident normal to the basal plane
(001), kЌ(001)͔. This behavior differs from that of
transition-metal dichalcogenides with hexagonal structure
e.g., 2H-MoS , 2H-MoSe ͒, which are optically uniaxial
͓E
ʈ
͑
2
2
with the optical axis perpendicular to the van der Waals
1
plane. Although Wilson and Yoffe claimed biaxial behavior
in ReS and ReSe , to our knowledge up to now only a few
2
2
5
experimental papers have been reported. Recently, Friemelt,
Lux-Steiner, and Bucher reported anisotropy effects in the
5
proximately 1060→1010 °C for ReS and 1050→1000 °C
2
van der Waals plane of ReS single crystals and mentioned
for ReSe . Both ReS and ReSe formed thin silver-colored,
2
2
2
2
2
that ReS may be of interest for fabrication of polarization
graphitelike platelets up to 2 cm in area and 100 ␮m in
thickness. X-ray diffraction patterns confirmed the triclinic
symmetry of ReS and ReSe with all parameters consistent
2
sensitive photodetectors in the visible wavelength region.
However, to date, no detailed study on the anisotropy in the
absorption edge has been reported.
2
2
2,10
with those previously reported.
Hall effect measurements
In this paper, we report on the detailed polarization-
dependent optical-absorption study of single crystals of ReS2
reveal n-type semiconducting behavior. At room tempera-
17
ture, the carrier concentrations were 1ϫ10
and 7
ϫ1016 cm , the Hall mobilities were 20 and 10 cm /V s
Ϫ3
2
and ReSe over temperature ranging from 25 to 500 K. A
2
notable shift towards lower energies was observed in the
and the resistivities were 3 and 8 ⍀ cm for ReS and ReSe ,
respectively. The weak van der Waals bonding between the
layers of the material means that they display good cleavage
2
2
transmittance spectra of E
ʈ
b polarization with respect to
those corresponding to EЌb polarization. Analysis of the
0
163-1829/98/58͑24͒/16130͑6͒/$15.00
PRB 58
16 130
©1998 The American Physical Society

PRB 58
16 131
ABSORPTION-EDGE ANISOTROPY IN ReS AND . . .
2
FIG. 1. Top view of the van der Waals plane of ReX2 ͑XϭS,
Se͒ layered structure.
properties parallel to the layers, which can be exploited to
obtain thin-single specimens. With a razor blade, thicker
samples have been successfully thinned to a limiting thick-
ness of ϳ10 ␮m.
Measurements of the reflectance and transmittance at
near-normal incidence were made on a Bruker model IFS
Ϫ1
1
20 HR in the range of 8000–13 500 cm , with a resolution
Ϫ1
FIG. 2. Polarization-dependent absorption coefficient vs photon
of 4 cm . Transmission intensity was closely monitored to
obtain an incidence as close to 90° as possible. The reflec-
tivity studies were done on the as-grown surface ͑001͒ and
compared against an evaporated gold mirror. The Oriel near-
infrared dichroic sheet polarizers were employed for
polarization-dependent measurements. Single crystals with a
thickness of about 10 ␮m were used for transmittance mea-
surements. Plate-shaped crystals were selected and mounted
on a copper sample holder fitted into a Dewar with optical
windows. As shown in Fig. 1, the b and a axes are the short-
est and second shortest axes in the basal plane; the b axis is
parallel to the cluster chains, which corresponds to the long-
est edge of the plate.5 Measurements were made at several
temperatures between 25 and 500 K with a temperature sta-
bility of 0.5 K or better.
energy for ͑a͒ ReS and ͑b͒ ReSe at several representative tempera-
2
2
tures between 25 and 500 K. The dashed lines and solid lines are,
respectively, deduced from the transmittance spectra of E
EЌb polarizations.
ʈ
b and
of photon energy, as determined for ͑a͒ ReS and ͑b͒ ReSe2
2
at several representative temperatures between 25 and 500 K.
The dashed curves in Fig. 2 correspond to the E b polariza-
ʈ
tion, while the solid curves represent EЌb polarization. It is
noted that a significant shift towards lower energies for the
E b polarization is observed as compared to the EЌb polar-
ʈ
ization. The polarization dependence of the transmittance
curves provides conclusive evidence that both optical ab-
sorption edges are associated with the interband transitions
from different origins. Detailed theoretical study of the an-
isotropic optical properties in the van der Waals plane of
ReS and ReSe is needed and is at present beyond the scope
,11
III. RESULTS AND DISCUSSION
2
2
of this work. As is generally expected, the absorption edge
shifted toward higher energies as the temperature of the
sample is lowered. Analysis of the experimental data showed
that the absorption coefficient ␣ to be proportional to (ប␻
ϪEg)n with nϭ2Ϯ0.1. This suggests an indirect allowed
transition for these materials.
The transmittance of ReS and ReSe were measured at
near-normal incidence. The absorption coefficient ␣ was de-
2
2
termined from the transmittance T by taking into account
r
the spectral dependence of the reflectance R using the
1
2
relation
1ϪR͒2eϪ␣d
A more complete analysis, taking into account both the
absorption and emission phonons, is given as follows. For an
indirect allowed transition, the absorption coefficient ␣ for a
single-phonon process can be expressed as12
͑
Trϭ
.
͑1͒
1
ϪR2eϪ2␣d
Equation ͑1͒ assumes that there are multiple reflections
within the sample, but that they add incoherently due to
sample inhomogeneity or a sufficiently large spread of the
incident angles. Since ␣d is large for the sample crystals, the
second term in the denominator of Eq. ͑1͒ can be neglected.
Figure 2 displays the absorption coefficient as a function
2
2
A͑ប␻ϪE ϩE ͒
B͑ប␻ϪE ϪE ͒
g p
g
p
␣ប␻ϭ
ϩ
,
͑2͒
exp͑E /kT͒Ϫ1
1Ϫexp͑ϪE /kT͒
p
p
where ប␻ is the energy of the incident photon, E the band
g
gap, E the energy of the phonon assisting the transition, and
p

16 132
C. H. HO, Y. S. HUANG, K. K. TIONG, AND P. C. LIAO
PRB 58
values of E and E could be obtained by fitting a different
g
p
energy range, i.e., by rejecting some points at lower or
higher photon energies. From this selective omission of data,
an error of the order Ϯ0.02 eV can be deduced for the esti-
mation of E . Fitting data on different samples give similar
g
parameters, even though there are differences in the absorp-
tion spectra due to differences in sample thicknesses.
From our experiments, the indirect gaps at room tempera-
ture, denoted as Eg
ʈ
(EgЌ), are, respectively, determined to
be 1.35Ϯ0.02 (1.38Ϯ0.02) eV for ReS and 1.17Ϯ0.02
2
(
1.20Ϯ0.02) eV for ReSe . Here, Eg
ʈ
and EgЌ refer, respec-
b and EЌb polarizations.
It is noticed that the values for EgЌ are larger than those of
Eg , and are similar to those of our previous report of Eg ,
which determined from the absorption data of the unpolar-
2
tively, to the indirect gap of the E
ʈ
ʈ
13
ized incident light. Our values differ slightly from previous
published works.2
,14–16
The first available data, 1.33 eV for
ReS and 1.15 eV for ReSe , were obtained by Wildervanck
2
2
2
and Jellinek from transmittance measurements. Koffyberg,
Ouight, and Wold14 analyzed the photoelectrochemical
spectral-response data resulting in a lowest-energy indirect-
optical band gap of 1.4 eV for ReS . From optical-absorption
2
measurements, Marzik et al.15 determined the indirect-
optical band gaps of ReS and ReSe to be 1.32͑5͒ eV for
2
2
n-type ReS and 1.17͑5͒ eV for n-type ReSe . Recent at-
2
2
tempts to determine the energy gaps of p-type ReS and
2
p-type ReSe using the photoresponse spectra obtained at
2
semiconductor-electrolyte interfaces were carried out by
Wheeler, Leland, and Bard. The values obtained were
FIG. 3. Experimental points of (␣ប␻)1/2 vs ប␻ that are deduced
from polarization-dependent absorption measurements for ͑a͒ ReS2
16
1.31Ϯ0.01 and 1.22Ϯ0.01 eV for ReS and ReSe , respec-
2
2
and ͑b͒ ReSe at several representative temperatures between 25
2
tively.
ReX ͑XϭS, Se͒ can be thought of as distorted 1T-MX
and 500 K. The hollow circles ͑solid squares͒ are data points from
2
2
the E
ʈ
b (EЌb) polarization measurements and the solid lines are
17
dichalcogenides. The 1T-MX2 phases consist of edge-
shared MX octahedra. In each MX layer, the metal-atom
the least-squares fits to Eq. ͑2͒.
6
2
sheet is sandwiched by chalcogen-atom sheets, and the metal
A and B are constants. The first term on the right-hand side
of Eq. ͑2͒ corresponds to an absorption of a photon and a
phonon, whereas the second term corresponds to an absorp-
tion of a photon and emission of a phonon and contributes
only when ប␻уE ϩE . There is a large residual absorp-
atoms of an undistorted MX layer form a hexagonal lattice.
2
3
ReS and ReSe have a d electron count, and their metal-
2
2
atom sheets exhibit a clustering pattern of ‘‘diamond
chains.’’ The atoms comprising the Re diamonds are copla-
4
nar, but each diamond unit is canted at a small angle from
the basal plane. This canting results in a small variation of
the Re atom heights perpendicular to the basal plane. The
distortion of the Re atom sheet from perfect hexagonal sym-
metry further creates a distortion of the chalcogen sheets,
both perpendicular and parallel to the basal plane. One con-
sequence of this distortion is the opening of an energy gap in
the band structure due to the mutual repulsion of the orbitals
g
p
tion at photon energies below the absorption edge. The large
values of the absorption coefficient ␣ below absorption edge
of ReS and ReSe most probably indicate the existence of
2
2
impurities or defects in the materials. At this point, we have
not considered in detail the effect of these impurity or defect
states. For simplicity, in our present study, the residual ab-
sorption is assumed to be a constant and subtracted out for
the evaluation of the band gap E and phonon energy E .
18
around the Fermi energy level. Recently, Kelty et al., and
Kertesz and Hoffmann19 employed simple tight-binding elec-
tronic band-structure calculations of the extended Huckel-
type on ReS and ReSe , respectively. The results show that
g
p
The data of ReS and ReSe at different temperatures were
2
2
then to be fitted to Eq. ͑2͒. Representative results are shown
in Fig. 3, where the hollow circles and solid squares are
representative experimental points deduced from different
2
2
the band levels around the band gap are dominated by the Re
polarization ͑E
ʈ
b and EЌb polarizations, respectively͒ trans-
metal d orbitals. The contribution of chalcogen p orbitals
z
mittance spectra and the solid lines are fitted to Eq. ͑2͒. The
are much stronger at the top portion of the valence band than
at the bottom portion of the conduction band. For the chal-
cogen p and p orbitals, the top portion of the valence band
results strongly indicate that ReS and ReSe are indirect
2
2
band-gap semiconductors, in which E
ʈ
b polarization exhibits
x
y
a smaller band gap and a single phonon makes important
contributions in assisting the indirect transitions. The non-
uniform thicknesses and unsmooth sample surface will tend
to deviate the incident angles from the normal direction, re-
sulting in some variations in the absorption spectra. Differing
has nearly equal contributions from all chalcogen atoms. The
values of the theoretical indirect gap for a single ReS2
layer,18 the three-dimensional ReS ͑Ref. 18͒ and ReSe2
2
͑Ref. 19͒ lattices are 1.27, 0.81, and 0.87 eV, respectively.
The value of 1.27 eV for a single layer of ReS ͑Ref. 18͒ is
2

PRB 58
16 133
the band gap at absolute zero; a and b are constants re-
ABSORPTION-EDGE ANISOTROPY IN ReS AND . . .
2
i
i
ferred to as Varshni coefficients. The constant a is related to
i
the electron-phonon interaction and b is closely related to
i
the Debye temperature. For comparison purposes we have
also listed numbers for the polarization-dependent direct
2
0
3
band edge excitonic-transition energies of ReS and ReSe ,
2
2
1
and the unpolarized results of absorption measurements.
The Debye temperature was estimated to be approximately
21
1
96 K for ReS and 181 K for ReSe . Our values of b
ʈ
2
2
equals 133Ϯ50 (150Ϯ50) K and bЌ equals 128Ϯ50 (140
Ϯ50) K for ReS (ReSe ) are in reasonable agreement with
2
2
the theoretical estimation. It is noted that the values of a are
i
much larger than those obtained from the direct band-edge
excitonic transitions.21 We will discuss the differences in a
later section.
The temperature dependence of the indirect band gap in
two different polarizations have also been fitted ͑dotted
curves in Fig. 3͒ by an empirical expression proposed by
O’Donnel and Chen,7
FIG. 4. The temperature dependence of polarization-dependent
indirect band gaps of ReS and ReSe . Representative error bars are
2
2
shown. The solid lines are least-squares fit to the Varshni equation
see Ref. 6͒. The dotted curves are least-squares fit to an empirical
͑
expression proposed by O’Donnel and Chen ͑see Ref. 7͒. The
dashed curves are fitted by an expression containing the Bose-
Einstein occupation factor of phonon ͑see Ref. 8͒.
E ͑T͒ϭE ͑0͒ϪSi͗ប⍀
͘
͓coth͑
͗
ប⍀
͘
/2kT͒Ϫ1͔, ͑4͒
gi
gi
in approximate agreement with the absorption band gap of
1
.35Ϯ0.02 (1.38Ϯ0.02) eV. However, the calculated value
18,19
where iϭʈ
, Ќ, E (0) is the indirect band gap at 0 K, S is a
gi i
for the three-dimensional lattice
that actually obtained in our optical-absorption measure-
ments. The calculated value of 0.81 eV for ReS is smaller
than that of ReSe , which is exactly in the opposite trend of
the experimental values. These discrepancies indicate the in-
adequacy of the tight-binding method for studying semicon-
ductors with a layer structure and a more effective method is
needed in this area.
Plotted in Fig. 4 are the temperature variations of the
polarization-dependent indirect energy gaps of ReS2 and
are much smaller than
dimensionless coupling constant related to the strength of
electron-phonon interaction, and is an average phonon
͗
ប⍀i
͘
2
energy. The obtained values of the various parameters are
given in Table II. For comparison purposes the numbers
from previous reports on direct band-edge excitonic transi-
2
2
0
tion energies of ReS and ReSe , indirect band gaps of
2
2
22
Si, RuS and RuSe ͑Ref. 23͒ are also included in Table II.
2
2
As shown in Table II, our values of the electron-phonon
coupling parameter S are much larger than the excitonic
i
transition energies of ReS and ReSe as reported in Ref. 20.
ReSe with representative error bars. A least-squares fit ͑the
solid line͒ to the Varshni semiempirical relationship
2
2
2
6
The values of S are also much larger than the indirect gaps
i
of Si,22 RuS , and RuSe . From Eq. ͑4͒, with kT much
23
2
2
2
larger than
ues for dEg
Ϫ0.68 ͑Ϫ0.66͒ meV/K are in good agreement with the val-
͗
ʈ
ប⍀i
͘
gi i
, dE (T)/dTϭ2kS . The calculated val-
/dT and dEgЌ /dT equal Ϫ0.66 ͑Ϫ0.62͒ and
E ͑T͒ϭE ͑0͒Ϫa T /͑b ϩT͒
͑3͒
gi
gi
i
i
yields parameters as given in Table I. Here iϭ
ʈ
, Ќ, E (0) is
gi
TABLE I. Fitting parameters that describe the temperature dependence of polarization-dependent indirect
band gaps and direct band-edge excitonic-transition energies of ReS and ReSe using the Varshni equation.
2
2
Also listed are the numbers determined with absorption of the unpolarized incident light.
E (0)
g
a
b
Materials
Polarization
͑eV͒
͑meV/K͒
͑K͒
ReS a
E
b
1.51Ϯ0.02
1.52Ϯ0.02
1.35Ϯ0.02
0.73Ϯ0.05
0.68Ϯ0.05
0.81Ϯ0.05
0.76Ϯ0.05
0.37Ϯ0.05
0.39Ϯ0.05
0.45Ϯ0.05
0.51Ϯ0.05
0.62Ϯ0.05
0.75Ϯ0.05
133Ϯ50
128Ϯ50
150Ϯ50
140Ϯ50
175Ϯ75
180Ϯ75
175Ϯ75
170Ϯ75
115Ϯ50
135Ϯ50
ʈ
2
EЌb
ʈ
ReSe2a
E
b
EЌb
ʈ
1.36Ϯ0.02
ex b
ReS (E )
E
b
1.554Ϯ0.005
1.588Ϯ0.005
1.387Ϯ0.005
1.415Ϯ0.005
1.52Ϯ0.02
2
1
ex b
ReS (E )
EЌb
ʈ
E b
EЌb
2
2
ex b
ReSe (E )
2
1
ex b
2
ReSe (E )
2
ReS2c
c
ReSe2
1.36Ϯ0.02
a
This work.
Reference 20.
Reference 13.
b
c

16 134
C. H. HO, Y. S. HUANG, K. K. TIONG, AND P. C. LIAO
PRB 58
TABLE II. Values of the fitting parameters of the expression
non temperature. The fitted values for EiB , aiB , and ⌰iB are
proposed by O’Donnel and Chen that describe the temperature de-
pendence of polarization-dependent indirect band gaps and direct
band-edge excitonic transition energies of ReS and ReSe , and
given in Table III, and the corresponding values for the di-
rect band-edge excitonic transition energies of ReS and
2
21
2
2
ReSe2, GaAs ͑Ref. 8͒ and ZnSe ͑Ref. 24͒ are also listed for
indirect band gaps of Si, RuS , and RuSe .
2
2
comparison. The parameter a of Eq. ͑3͒ is related to a and
i
iB
⌰iB of Eq. ͑5͒ by taking the high-temperature limit of both
expressions. This yields a Ϸ2a /⌰iB . Comparison of
Tables I and III shows within error bars this relation is sat-
isfied.
Eg0
͑eV͒
͗ប⍀͘
͑meV͒
i
iB
Materials
S
ReS (Eg
ʈ
)a
1.50Ϯ0.02
1.51Ϯ0.02
1.34Ϯ0.02
1.35Ϯ0.02
1.554Ϯ0.005
1.588Ϯ0.005
1.387Ϯ0.005
1.415Ϯ0.005
1.17
4.3Ϯ0.5
4.1Ϯ0.5
4.5Ϯ0.5
4.34Ϯ0.5
2.0Ϯ0.1
2.0Ϯ0.1
2.8Ϯ0.1
2.8Ϯ0.1
1.49
23Ϯ3
22Ϯ2
18Ϯ4
16Ϯ3
19Ϯ3
20Ϯ3
25Ϯ3
21Ϯ4
25.5
2
a
As shown in Tables I, II, and III, the results of the
ReS (EgЌ)
2
ReSe (Eg
ʈ
)a
ReSe (E )a
electron-phonon coupling constants a in Eq. ͑3͒, S in Eq.
i
i
2
͑4͒, and aiB
in Eq. ͑5͒, which describe the temperature de-
2
gЌ
ex b
pendence of the indirect band gaps for ReS and ReSe are
ReS (E )
ReS (E )
2
2
2
1
ex b
2
larger than those for the direct band-edge excitonic-transition
2
energies.2
0,21
These results are similar to that of the previous
ex b
ReSe (E )
2
1
25
ex b
report on 2H-MoS and MoSe layered crystals. Goldberg
et al. reported a detailed study of the low-energy absorp-
2
2
ReSe (E )
2
2
25
c
Si (Eind
)
tion in layered crystals of MoS and MoSe . The experimen-
RuS (E )d
2
2
1.43Ϯ0.01
0.89Ϯ0.01
2.41Ϯ0.2
1.94Ϯ0.2
19.6Ϯ2.0
12.5Ϯ2.0
2
ind
tal results showed that the temperature shift of the absorption
tails is larger than those of the excitons. The phenomena can
be interpreted as follows: as the temperature is lowered, the
reduction in the thermal broadening of the exciton causes a
faster shift of the absorption tail. In addition, it is also no-
RuSe (E )d
2
ind
a
This work.
Reference 20.
Reference 22.
Reference 23.
b
c
d
ticed that the electron-phonon coupling constants a , S , and
i i
aiB for Eg
suspected that the electron-phonon coupling constants for
b polarization are larger than those of EЌb polarization,
ʈ
are larger than those of EgЌ . At this point we
ues of Ϫ0.74 ͑Ϫ0.7͒ and Ϫ0.77 ͑Ϫ0.74͒ meV/K for ReS2
ReSe ) as obtained from the linear extrapolation of the ex-
E
ʈ
(
2
and may be general characteristics of the crystals with layer
structure of triclinic symmetry. However, a more systematic
experimentation should be carried out to verify this property.
perimental data.
The temperature dependence of the polarization-
dependent indirect band gaps can also be fitted ͑dashed
curve͒ by an expression containing the Bose-Einstein occu-
pation factor of phonon:8,9
IV. SUMMARY
2
In summary, the polarization-dependent optical-
absorption study of ReS and ReSe single crystals was car-
E ͑T͒ϭE Ϫa 1ϩ
,
͑5͒
gi
iB
iB
ͭ
ͮ
2
2
exp͑⌰iB /T͒Ϫ1
ried out over temperatures ranging from 25 to 500 K. The
transmittance spectra of E b polarization shift towards lower
ʈ
where iϭ
ʈ
or Ќ, aiB represents the strength of the electron-
energies with respect to those corresponding to EЌb polar-
phonon interaction, and ⌰iB corresponds to the average pho-
ization. Analysis of the absorption curves suggests that the
TABLE III. Values of the Bose-Einstein-type fitting parameters that describe the temperature dependence
of band gaps and excitonic transition energies of ReS , ReSe , GaAs, and ZnSe.
2
2
EB
aB
⌰B
Feature
Eg
Materials
ReS2a
͑eV͒
͑meV͒
͑K͒
ʈ
1.57Ϯ0.02
1.41Ϯ0.02
1.58Ϯ0.02
1.42Ϯ0.02
1.583Ϯ0.008
1.428Ϯ0.01
1.619Ϯ0.008
1.462Ϯ0.010
1.512Ϯ0.005
2.800Ϯ0.005
66Ϯ21
73Ϯ22
62Ϯ18
68Ϯ21
32Ϯ10
45Ϯ15
34Ϯ10
53Ϯ20
57Ϯ29
73Ϯ4
194Ϯ50
196Ϯ50
193Ϯ50
193Ϯ50
200Ϯ50
224Ϯ75
200Ϯ50
225Ϯ75
240Ϯ102
260Ϯ10
a
a
b
b
ReSe2
ReS2
a
EgЌ
ReSe2
ReS2b
ReSe2
ReS2b
ReSe2
ex
E1
ex
E2
d
c
E
E
GaAs
ZnSed
g
d
g
a
This work.
Reference 21.
Reference 8.
Reference 24.
b
c
d

PRB 58
interband transitions in ReS and ReSe are indirect and their
16 135
ABSORPTION-EDGE ANISOTROPY IN ReS AND . . .
2
the electron-phonon coupling constants for E
ʈ
b polarization
2
2
band gaps are polarization dependent. The polarization-
dependent temperature variation of the energy gaps are ana-
lyzed by the Varshni equation, an empirical expression pro-
posed by O’Donnel and Chen, and a Bose-Einstein-type
expression. The parameters that describe the temperature de-
pendence of band gaps are evaluated. The results show that
are considerably larger than those of EЌb polarization.
ACKNOWLEDGMENT
The authors acknowledge the support of the National Sci-
ence Council of the Republic of China under Projects Nos.
NSC87-2112-M-011-001 and NSC88-2112-M-011-001.
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