Materials Research Bulletin 41 (2006) 751–763
www.elsevier.com/locate/matresbu
Lattice vibrations of pure and doped GaSe
K. Allakhverdiev a,b,*, T. Baykara a, Ş. Ellialtioğlu c,
F. Hashimzade b, D. Huseinova b, K. Kawamura d, A.A. Kaya a,
A.M. Kulibekov (Gulubayov) e, S. Onari d
a
b
Materials Institute, Marmara Research Center, TÜBÌTAK, Gebze/Kocaeli 41470, Turkey
Institute of Physics, Azerbaijan National Academy of Sciences, Baku AZ1143, Azerbaijan
c
Department of Physics, Middle East Technical University, Ankara 06531, Turkey
d
Institute of Materials Science, University of Tsukuba 305-8573, Japan
e
Department of Physics, Muğla University, Muğla 48000, Turkey
Received 24 May 2005; received in revised form 27 September 2005; accepted 10 October 2005
Available online 8 November 2005
Abstract
The Bridgman method is used to grow especially undoped and doped single crystals of GaSe. Composition and impurity content
of the grown crystals were determined using X-ray fluorescence (XRF) method. X-ray diffraction, Raman scattering, photoluminescence (PL), and IR transmission measurements were performed at room temperature. The long wavelength lattice vibrations
of four modifications of GaSe were described in the framework of modified one-layer linear-chain model which also takes into
consideration the interaction of the selenium (Se) atom with the second nearest neighbor gallium (Ga) atom in the same layer. The
existence of an eight-layer modification of GaSe is suggested and the vibrational frequencies of this modification are explained in
the framework of a lattice dynamical model considered in the present work. Frequencies and the type of vibrations (gap, local, or
resonance) for the impurity atoms were calculated and compared with the experimental results.
# 2005 Elsevier Ltd. All rights reserved.
Keywords: A. Semiconductors; C. Infrared spectroscopy; C. Raman spectroscopy
1. Introduction
GaSe is a highly anisotropic material, which consists of thin layers of covalently bound gallium (Ga) and selenium
(Se) atoms with a thickness of four atoms. Between the layers, weak forces of mainly van der Waals type are present. In
the stacking direction (along the crystallographic z-axis which is in the direction of the optical c-axis), the layers can be
arranged in different ways, which leads to the existence of different polytypes. Four modifications have been described
in the literature [1]; centrosymmetric b-GaSe consists of two layers per unit cell and has the space group D46h ; noncentrosymmetric e-modification is the main component obtained from the melt, it consists of two layers, and
crystallizes with the space group D13h ; the g-type contains one layer, space group C53v often exists as stacking fault in the
melt grown e-type crystals; d-type contains four layers per unit cell, space group C46v . The existence of b-polytype was
the subject of discussion in the literature [2], although the exciton spectra of e- and g-polytypes including the b-phase
* Corresponding author. Tel.: +90 262 6412300x3496; fax: +90 262 6412309.
E-mail address: [email protected] (K. Allakhverdiev).
0025-5408/$ – see front matter # 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.materresbull.2005.10.015
752
K. Allakhverdiev et al. / Materials Research Bulletin 41 (2006) 751–763
were reported [3]. e-GaSe has a direct gap of Egd ¼ 2:020 eV (at 300 K). An indirect gap of Egi ¼ 1:995 eV is in
d
resonance with the energy position of the direct free excitons Eex
¼ 2:001 eV [1]. Optical absorption edges of e-, b-,
and d-GaSe were reported elsewhere [4].
Lattice vibrations of GaSe have been extensively studied in the past by means of the IR [5–15] and Raman
scattering [7,14,16–30] spectroscopies. The main aims of these studies were to assign the symmetries of k = 0 phonons
and to reveal the real crystal structure to understand the nature of the weak interlayer forces. Raman scattering spectra
of GaSe doped with 0.5 wt% of Mn was reported [31]. Additional bands observed at 230 cm1 and 248 cm1 and
polarized only in the plane of the layers (E-type symmetry) were assigned as due to the interlayer defects. It is
concluded that GaSe doped with iron group materials (Ni, Cr, Co, and Mn) are characterized by these two phonons.
The authors did not mention the modification of the studied crystals, neither they gave the details of the experiment,
nor did they analyze the data to consider the other possible mechanisms for these lines [31].
The 24 normal vibrations of b-GaSe decompose into irreducible representations of the D46h space group at k = 0 as
follows [7]:
G ¼ 2A2u þ 2A1g þ 2B1u þ 2B2g þ 2E1u þ 2E2g þ 2E1g þ 2E2u
where A2u and doubly degenerate E1u modes are acoustic; there are two IR modes of A2u and E1u symmetry and six
Raman active modes of A1g, E1g, and E2g symmetry. The IR and Raman active modes are mutually exclusive due to the
inversion center between the layers.
The group theoretical analysis of the lattice vibrations of the e- (b-), e- (g-), and d-polytypes of GaSe were
performed in ref. [9], ref. [9,24], and ref. [20], respectively. The symmetries of the 24 vibrational modes of the emodification (D13h space group) at k = 0 are:
G ¼ 4A01 þ 4A002 þ 4E0 þ 4E00
of which A002 þ E0 are acoustic modes; there are 11 non-degenerate Raman active 4E00 , 4A01 , 3E0 modes and 6 IR active
3A002 and 3E0 modes. The Raman and IR active modes co-exist simultaneously as the crystal does not possess an
inversion symmetry. ‘‘Rigid layer mode’’ (RLM) is expected in the low-frequency range in which the layers vibrate as
rigid units against each other and that there is no relative displacement of the Ga and Se atoms within a layer.
There exists some misinterpretation when considering the symmetry properties of g-polytype; as the elementary
unit cell is not hexagonal, sometimes a hexagonal non-elementary unit cell with three layers cell is used [20]. For gtype, the division of 12 normal modes (1 layer/cell) of vibration at the zone center are represented by the irreducible
representations of the D3h point group as [24]:
G ¼ 2A01 þ 2A002 þ 2E0 þ 2E00
of which A002 þ E0 are acoustical modes and all the remaining optical modes are both IR and Raman active. In this
polytype, no ‘‘rigid layer mode’’ is expected. For d-type, the 48 normal modes of vibration at k = 0 decompose into the
irreducible representations of the C46v space group as follows [20]:
G ¼ 8A1 þ 8B1 þ 8E1 þ 8E2
where one of A1 and E1 modes are acoustic; the remaining A1 and E1 modes as well as the E2 modes are Raman active;
E1 modes are IR allowed, whereas B1 modes are silent. In the d-polytype, a ‘‘rigid double-layer mode’’ is expected. For
this normal mode of vibration each layer moves as a rigid unit, and every pair of layers vibrates firmly together [20].
The vibrational and other physical properties of e-type are mostly investigated when comparing with the other
modifications of GaSe.
Summarizing the existing data for the Raman active phonons of e-GaSe, one can find that only 8 of the 11
theoretically predicted phonons were observed at 19 cm1 (E0 ), 59 cm1 (E00 ), 134 cm1 (A01 ), 209 cm1 (E00 ),
213 cm1 (E0 (TO)), 253 cm1 (E0 (LO)), and 308 cm1 (A01 ), where LO and TO stand for the longitudinal and
transverse optical phonons, respectively [28]. On the other hand, only four of the six predicted IR active phonons were
observed at 20 cm1 (E0 ), 37 cm1 (A1), 214 cm1 (E0 ), and 236 cm1 (A01 ) [1,5,6,9,14,15]. The major differences that
occur when comparing the Raman spectra of the e-, and the g-type are the absence of the low frequency ‘‘rigid layer
mode’’ at 20 cm1 in the g-type; appearance of a weak feature at 40 cm1 and an increase in the intensity of bands at
236 cm1 (A01 ) and 253 cm1 (E0 ) [19]. The mode at 21 cm1 shown in Table 1 for g-GaSe was very weak and
appeared due to an admixture of the e-type [19]. The major change that occurs in going from the e- to the d-polytype is
K. Allakhverdiev et al. / Materials Research Bulletin 41 (2006) 751–763
753
Table 1
Room temperature first-order Raman modes for different polytypes of e-, d-, and g-GaSe according to different authors (data of ref. [8] and ref. [9]
were measured at 77 K and 81 K, respectively)
Symmetry
Frequency (cm1)
e
[8]
[16]
[19]
[19]
[20]
[20]
[20]
[23]
[28]
[29]
[a]*
[b]*
–
–
20
–
60
134
211
–
–
–
250
309
–
–
19
–
60
135
–
213
–
–
249
308
–
–
20
–
60
134
212
214
–
–
252
308
–
–
21
–
60
134
209
214
236
247
253
307
–
–
19
–
59
–
–
–
–
–
–
–
–
–
–
–
59
–
–
–
–
–
–
–
–
13
19
37
59
–
–
–
–
–
–
–
–
–
20
–
60
135
210
214
–
–
254
310
–
–
19
–
59
134
209
213
–
–
253
308
–
–
22
–
60
135
213
217
–
–
257
311
7
–
20
–
59
134
–
213
234
–
253
307
7
13
20
37
59
134
–
213
234
–
253
307
g
E0
E0
E00
A01
E00
E0
E00
A01
E00
E0
A01
A002
E0
A01
E0
A01
In ref. [20] there was only low-frequency (<60 cm1) part of the spectra presented. All frequencies given here (in cm1) are rounded. The first and
second columns represent the symmetry of phonons for e- and g-polytypes according to ref. [9] and ref. [19], respectively. Refs. [8,16,23,28,29] are
for the crystals containing predominantly the e-type. The last two columns marked [a]* and [b]* represent our results for samples No1 and No2,
respectively.
the doubling of the masses of layers in the latter which results in the appearance of the Raman active ‘‘rigid doublelayer mode’’ at 13 cm1 (E0 ) and a weak feature at 37 cm1 (A1) [20].
To our knowledge, the only information about the Raman spectra of GaSe crystals doped with Mn (GaSehMni),
hNii, hCri, and hCoi, was presented in ref. [31]. Additional structures observed at 230 cm1 and 248 cm1 were
assigned as due to the defects in crystals. The spectra of GaSehMni were analyzed only in the range of 190–270 cm1.
On the other hand, the photoluminescence (PL) spectra and electrical properties of doped crystals have also been
reported in a number of articles [31–37]. Research on the doped GaSe crystals is of primary interest for several
reasons. Doping is necessary, e.g., for obtaining high-resistance crystals for use in non-linear optics and radiation
detectors [1,14,37–40]. It is known that the impurities can modify the vibrational spectra and in the doped crystals
local as well as resonance modes are observed [41].
The long-wavelength lattice vibrations of GaSe were interpreted by means of a linear-chain model that included the
intralayer and interlayer force constants [42]. Vibrational frequencies of the impurity atoms in GaSe were calculated in
the approximation of isolated impurity atoms [43]. An axially symmetric lattice-dynamical model including only
short-range forces was used to explain the neutron inelastic scattering measurements of the phonon dispersion relation
of GaSe [44]. It was shown that the surface vibrational properties of GaSe are very similar to those in the bulk [45]. The
surface phonons of GaSe were calculated using ab initio methods and good agreement was obtained with the
experimental surface–phonon dispersion curves investigated by high-resolution inelastic helium-atom scattering [46].
In the present work, the lattice vibrations of pure (especially undoped) and doped GaSe crystals have been investigated
by means of Raman scattering and IR experiments. The results were compared with the theoretical calculations of the
vibrational frequencies of the impurity atoms obtained in the frame of a modified linear-chain model.
2. Experiment
The crystals were grown by the Bridgman method. A stoichiometric mixture of Ga and Se was sealed into an
evacuated quartz ampoule (104 Torr). The grown crystals were 12–18 mm in diameter and 2–4 cm long. The crystals
were doped by adding into the ampoules some amount of metals in elemental form (from 0.1 wt% to 0.8 wt%). It was
not clear whether all weighted amount of the impurities was incorporated into the grown crystals or some of it had been
segregated during the growth process. X-ray fluorescence (XRF) measurements by Philips PW 2404 showed that the
impurity content in the grown crystals were at least two times less than the weighted.
The thermoelectric test showed that all grown crystals were p-type, with average values of the resistivity and
mobility, as measured by the conductivity and Hall effect methods, in the range of 103–107 V cm and 20–70 cm2/
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(V s), respectively. X-ray analysis showed that most of the undoped samples were e-type and very few crystals were
predominantly g-type. XRF measurements showed the next content of the initial components in pure crystals: Ga in
the range from 45.914 at% (Se 54.084 at%) to 45.742 at% (Se 54.256 at%). Some samples had the composition, Ga
47.174 at% and Se 52.826 at%. Uncontrolled impurities were detected in some of the pure crystals: Ni (0.020 at%),
Mn (0.050 at%), and O (0.020 at%). Small additions of Sn increased the resistivity by several orders of magnitude.
Heavier doping with Sn (more than 1 wt%) did not change the p-type conductivity. Heavily doped crystals were of
poor quality: they contained voids and could only be cleaved with difficulty. We therefore do not report on
measurements obtained on such crystals.
No noticeable differences were observed in the optical properties (Raman scattering and IR transmission) for the
samples with different content of chalcogenide (or metal) atoms. The Raman scattering spectra were excited by the
6328 Å line of a He–Ne laser (power 3 mW), and were recorded in backscattering geometry using a spectrometer
(JASCO TRS600) with a liquid nitrogen cooled CCD detector (PHOTOMETRICS TK512CB). The spectral resolution
was not less than 0.5 cm1. The wave number peak positions and the bandwidth (full width at half maximum, FWHM)
were determined by fitting the Lorentzian line shapes to the experimental data.
The mid-IR (4000–400 cm1) transmission spectra were measured by modified BRUKER IFS 55 Fourier transform
spectrometer. For measurements in the range of 400–15 cm1 BRUKER IFS 113v Fourier spectrometer was used. A
spectral resolution in the range of 400–15 cm1 was 1 cm1. For pure crystals, the measurements were performed
using two kinds of samples: the samples with freshly cleaved (0 0 1) surfaces (oriented in the xy-plane, which is always
perpendicular to the optical c-axis) and the slab samples prepared with the optical axis in the plane of the face [46].
These measurements allowed to separate the A- and B-type phonons. The transmission spectra at 80 K were measured
by the help of a low-temperature cryostat.
The PL spectra were excited by the E = 2.410 eV line of an Ar+ laser (power < 20 mW). The PL was dispersed
through a polychrometer and detected with a liquid nitrogen cooled CCD detector. To get the energy positions, the
FWHM, and the intensities of the PL peaks, the experimental spectra were fitted by the Gaussian function.
Two pure and ten doped GaSe and one pure GaS samples were studied using Raman scattering spectroscopy:
GaSe(S) 0.3 at%, (Cr) 0.5 at%, (Mn) 0.8 at%, (Zn) 0.8 at%—two samples, (Sn) 0.8 at%, (Sn/Nd) 0.3 at% each, (Er)
0.1 at%, (Tm) 0.1 at%, (Tl) 0.5 at%. The major difference between the two samples of GaSe (Nol and No2) is that the
sample No1 was cleaved from the part of the grown boule close to the tip of the ampoule whereas the sample No2 was
prepared from the upper part of the grown boule close to the top of the ampoule, and hence contained more
uncontrolled impurities. The Hall effect measurements showed that the free carrier concentrations for the samples No1
and No2 were p = 3.5 1015 cm3 and p = 2 1016 cm3, respectively.
All measurements, except some of the IR transmission measurements, were performed at 300 K.
3. Results and discussion
Typical first-order Raman scattering spectra for the studied crystals are shown in Fig. 1. The peaks labeled with ‘‘*’’
in the spectra are the laser plasma lines. Curves 1, 2, and 3 are for two pure GaSe samples and a pure GaS sample,
respectively. The Raman spectrum of a GaS crystal is given for visual comparison. Meaning of the notations are: 4, 5,
6, 7 = GaSe doped with hSi, hT1i, hTmi, hEri; 8, 9 = hZni; 10, 11, 12 = hSni, hMni, hCri; 13 = hSni and hNdi. The
main interest for co-doping with Sn raised in search for obtaining high-resistance crystals and possible observation of
the laser transitions between different Nd3+ levels in the GaSe matrix (the contents of the impurities in at% are given
above).
3.1. Pure crystals
Measured frequencies are listed along with the frequencies published by different authors in Table 1. Eight and ten
Raman bands were clearly recorded for crystal Nol (curve 1) at 7.0 cm1, 19.9 cm1, 59.3 cm1, 134.1 cm1,
212.6 cm1, 233.5 cm1, 252.9 cm1, and 307.3 cm1 and for the crystal No2 (curve 2) at 7.3 cm1, 13.2 cm1,
19.9 cm1, 37.1 cm1, 59.3 cm1, 134.1 cm1, 212.6 cm1, 233.5 cm1, 252.9 cm1, and 307.3 cm1, respectively.
The band at 7 cm1 was very weak in the spectrum 1. Hereafter, rounded values of frequencies will be given.
A careful inspection of Fig. 1 revealed the major differences between the spectra of samples No1 and No2 and a
similarity between the spectra of GaSe sample No2 and GaS (always crystallizes in the b-type), if features at the
K. Allakhverdiev et al. / Materials Research Bulletin 41 (2006) 751–763
755
Fig. 1. Room temperature Raman spectra of undoped (pure) and doped GaSe crystals. The spectra were excited with the 633 nm line of a He–Ne
laser with power 3 mW. The lines denoted by ‘‘*’’ are the laser plasma lines. Meaning of the notations: 1, 2 – two different pure GaSe crystals; 3 –
GaS; 4, 5, 6, 7 – hSi 0.3 at%, hTli 0.5 at%, hTmi 0.1 at%, hEri 0.1 at%; 8, 9 – hZni 0.8 at%; 10, 11, 12 – hSni 0.8 at%, hMni 0.8 at%, hCri 0.5 at%; 13
– hSni 0.3 at%, hNdi 0.3 at%.
frequencies higher that 50 cm1 are ignored. This similarity should be expected because of the very similar crystal
structure and the almost equal inter-atomic distances of both the crystals. Four low-frequency bands were observed in
the spectrum of sample No2 at 7 cm1, 13 cm1, 20 cm1, and 37 cm1. For sample No1, two low-frequency bands
were recorded at 20 cm1 (RLM, typical for e-type) and 7 cm1 (very weak; origin will be discussed later). Relative
intensity of the ‘‘rigid layer mode’’ at 20 cm1 is higher for sample Nol, whereas the intensity of band at 234 cm1 is
higher for sample No2. The band recorded at 37 cm1 (polar A-type) together with an increased intensity of the band
at 234 cm1 evidenced that the relative content of the g-type is higher in crystal No2. These results are consistent with
other Raman data [16,18,19].
Diffraction diagrams of samples Nol and No2 showed three additional peaks at 2u = 31.28, 32.38, and 35.88 for the
latter. Besides, it showed that the diffraction line at 2u = 21.58, (0 0 4) reflection for sample No2 were broadened by
stacking faults. The (0 0 2) and (0 0 6) reflections, however, were not broadened. Broadening of bands was also
observed in the Raman spectra of sample No2. FWHM of the Raman bands at 134 cm1 (A-type), 59 cm1 (E-type),
and 20 cm1 (E-type, RLM) were 2.85 cm1, 1.3 cm1, and 1.75 cm1 and 3.5 cm1, 1.34 cm1, and 1.86 cm1 for
the samples Nol and No2, respectively. We believe that these broadenings give the characteristic differences between
the ordered and disordered structures of GaSe. These results show that sample No2 has a high number of stacking
faults (including the dislocations) [47,48]. Broadening is the result of faulty stacking and was also observed in the
X-ray diffractograms of GaSe [49].
3.2. Doped crystals
The band at 244 cm1 was recorded in the Raman spectra of GaSehCri, GaSehZni, and GaSehSi samples (Fig. 1).
This band has not been recorded in the spectra of pure and other doped crystals. Most probably this band corresponds
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to the A002 ðLOÞ phonon at 247 cm1 reported in ref. [19]. According to the symmetry selection rules, this mode is
Raman inactive. We do believe that this mode appears as a result of breaking the symmetry selection rules due to
disorder introduced by the impurity. This result is in accordance with that published in ref. [19] where the resonance
behavior of this mode was reported.
The low-frequency bands at 7 cm1, 13 cm1, 20 cm1, and 30 cm1 were recorded in the spectra of samples
doped with hZni, hSi, and hTmi. These bands have been recorded also in the Raman spectrum of sample No2. As it was
mentioned above (Section 3.1), sample No2 may be considered as containing a high density of defect states. Metal
atoms can replace the Ga atoms in the lattice of GaSe and they may occupy the voids between the layers. For example,
it was supposed that the Tm3+ ions might occupy the voids between the layers (the number of voids are quite high)
[50]. This supposition was supported by the fact that the cleavage of single crystals with increasing concentration of
Tm3+ became more difficult [50]. Inclusion of metal atoms between the layers will result in a high number of stacking
faults in the crystal lattice. We suppose that the samples doped with hZni, hSi, and hTmi contain a high density of
defect states, which may result in the formation of a mixture of e- and other possible modifications. The spectrum 6
(Fig. 1) shows an evidence for this: in addition to four low-frequency bands at 7 cm1, 13 cm1, 20 cm1, and
30 cm1, two weak bands at about 50 cm1 were recorded.
Besides the bands at 7 cm1, 13 cm1, and 20 cm1, a weak band at 92 cm1 was recorded in the spectra of the
samples doped with hCri and hSi (Fig. 1).
Two weak bands at about 80 cm1 and 195 cm1 in the E ? c geometry were recorded in the spectra of GaSe
sample doped with Tl. These lines were absent in E jj c geometry.
No new bands were recorded in the spectra of GaSe hSni, hMni, and hSn/Ndi, when comparing with the pure
samples.
All metal atoms can, in principle, substitute for either Ga or Se atoms, and if their masses are heavier than the
masses of Ga and Se, this should result in a decrease in frequency of the RLM. Indeed, careful examination of the peak
positions of the RLM for pure and doped GaSehTli, hTmi, and hSni samples showed a slight decrease (about 1–
1.5 cm1) in the frequency in doped crystals.
3.3. Photoluminescence
The PL spectra of pure and doped samples, together with the PL peak positions (bottom), are shown in Fig. 2. The peak
at 623.2 nm (1.988 eV) for pure GaSe has FWHM 0.02482 eVand is due to the emission of the direct free excitons [51].
The peak position and the FWHM for pure GaSe and GaSe doped with all other impurities, except Tm, was nearly the
same such that both values for all crystals, except GaSehTmi, start to differ in the 4th digit after the decimal point. For
instance, the peak positions (and FWHM) for GaSehSi and GaSehSn/Ndi were 1.993 eV (and 0.02484 eV) and 1.988 eV
(and 0.0234 eV), respectively. The PL peak for GaSe hTmi was substantially shifted to a longer wavelength (633.6 nm,
1.957 eV) and was much broader (FWHM 0.0707 eV). The PL peak broadening of GaSe crystals at 300 K was typical for
highly disordered and mechanically strained samples [51]. This is in accordance with the above statement that the
GaSehTmi sample has a high number of stacking faults. We believe that the PL data may be the subject of a separate
investigation. For this reason we will not discuss these results further in the present work.
3.4. IR absorption
Two pure samples with thicknesses t = 8.22 mm (freshly cleaved surface) and t = 3.12 mm (surface containing the
optical c-axis) and several samples containing the impurities (T1, Mn, Zn, S, and only with the freshly cleaved
surfaces) were measured. The absorption bands at 20 cm1, 37 cm1, 83 cm1, 362 cm1, 420 cm1, 450 cm1,
512 cm1, and 620 cm1 were clearly seen in the transmission spectra of pure samples. No new bands were observed
for doped crystals. Measurements for pure crystals in non-polarized light using the samples with freshly cleaved
surfaces and comparison with the spectra of the slab samples with the optical axis in the plane of the face allows one to
distinguish between the A- and E-type phonons. The absorption lines at 37 cm1 and 20 cm1 were assigned as onephonon bands with the A- and E-type symmetries. By decreasing the temperature to 80 K, the intensity of these bands
decreased slightly and the peak positions shifted to higher wave numbers to about 2–3 cm1. The far IR absorption
band at 83 cm1 was observed in both polarizations of the incident light and was attributed to the multi-phonon
processes.
K. Allakhverdiev et al. / Materials Research Bulletin 41 (2006) 751–763
757
Fig. 2. Room temperature PL spectra of the direct free excitons for pure (No1) and doped samples (top) and the PL peak positions (bottom). The PL
spectra were excited by the E = 2.410 eV line of an Ar+ laser (power 20 mW). To get the PL peak energy positions, the bandwidth (full width at half
maximum) of the experimental spectra were fitted by the Gaussian function.
The absorption bands at 450 cm1 and 512 cm1 were attributed to the localized mode due to the presence of
uncontrolled impurities [52]. The band at 420 cm1 is nearly twice as large in energy as the IR active mode at
211 cm1 (E-type) [6]. For this reason, this band is assigned to a two-phonon process. By decreasing the temperature
from 290 K to 80 K, the intensity of this band and the band at 620 cm1 decreased slightly and the absorption peak
positions shifted by 3 cm1 to higher wave numbers. Although the band at 620 cm1 is nearly twice as large in
energy as the Raman active band at 307 cm1, we do not assign this band to a two-phonon process. According to the
Raman data [7,14,16–30], the line at 307 cm1 has A-type symmetry and should be observed in the IR absorption of
the samples with the surfaces containing the optical c-axis, whereas in our case this line was recorded in the ~
e?x
geometry and corresponds to E-type vibration. For this reason, we attribute this line to the multi-phonon process. The
intensity of the absorption band at 362 cm1 evidently decreased with decreasing temperature and for this reason we
assigned this band to multi-phonon processes.
Simultaneous activity of the bands at 20 cm1 and 37 cm1 in the Raman and IR spectra implies that the studied
GaSe samples do not predominantly contain the b-type, although they may include the three different polytypes (b, d,
and e) [53].
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K. Allakhverdiev et al. / Materials Research Bulletin 41 (2006) 751–763
Fig. 3. Two-layer unit cell of e-GaSe (dashed box) and the non-zero force constants of the model.
3.5. Model
GaSe crystals occur in different polytypes. All modifications have a structurally identical layer, which consists of
four sublayer atoms. The differences among the four studied modifications arise from the number of layers in the unit
cell (one for g- and four for d-type) and from the stacking of layers in the z-direction (b- and e-types, each containing
two layers, but different stacking leads to a different space group). There are reasons to think of the existence of eightand even more-layer polytypes of GaSe.
The low-frequency (v < 60 cm1) Raman spectra of d-type were explained in terms of a rigid layer model by
doubling the unit cell along the c-axis when comparing with the e-type [20]. Using this model, the authors [20]
explained the origin of the lowest frequency RLM at 13 cm1 for the d-type and described also the phonon modes in
the e-, g-, and d-modifications.
To explain the experimental results obtained in the present study, we used the idea described in ref. [20] and
combined it with the Wieting’s [42] model. Besides, we have calculated the vibration frequencies of the impurity
atoms in the framework of our one-layer approximation model. Such an approximation seems to be correct because the
interaction between the layers are of weak van der Waals type and even inclusion of the interlayer interaction will not
have considerable influence on the results of calculations. In our model (Fig. 3), GaSe was represented as a linearchain of atoms Se(l)-Ga(2)-Ga(3)-Se(4)–Se-Ga-Ga-Se (the atom numbers are given in parentheses). To construct the
dynamical matrix, we introduced the force constants which considered the interactions between the nearest neighbor
atoms: Cw —between Se and Ga, Cg—between Ga and Ga atoms, and Cb—between Se and Se atoms. There are
Table 2
Characteristics of the bonds and force constants included in our modified linear-chain model, as depicted in Fig. 3. Cw , Cg, Cb, and Ct are the
calculated values for the compression (superscript ‘‘c’’) and shear (superscript ‘‘s’’) force constants between Se–Ga, Ga–Ga, Se–Se, and Se–Ga pairs
Bond
Se–Ga
Ga–Ga
Se–Se
Se–Ga
Location
Same layer
Same layer
Adjacent layers
Same layer, 2nd nn
Bond-length (Å)
2.473
2.520
3.850
4.186
Force constant (104 dyn/cm)
x
x=c
Cwx
Cgx
Cbx
Ctx
10.64
12.19
0.605
0.975
x=s
(12.3)
(10.8)
(0.924)
(–)
9.26 (9.98)
1.02 (1.53)
0.175 (0.161)
0.994 ()
For comparison, the corresponding values of Wieting’s model [42], which does not include the force constant Ct, are given in parenthesis.
K. Allakhverdiev et al. / Materials Research Bulletin 41 (2006) 751–763
759
compression force constants (hereafter marked with the superscript ‘‘c’’), which describe the atomic vibrations along
the chain (perpendicular to the layers) and shear force constants (hereafter marked with the superscript ‘‘s’’), which
characterize the vibrations in the direction perpendicular to the chains (in the layer plane). The force constants
obtained in the Wieting model [42] do not describe the highest shear Raman active mode well enough, and conversely
the fine adjustment of the highest compression mode resulted in no good description of the low-frequency compression
mode. These circumstances decrease the accuracy of the results obtained in the Wieting model [42]. For a better
description of the vibration frequencies of GaSe, we introduced one more force constant Ct that considered the
interaction of the Se atom with the second nearest Ga atom in the same layer. The force constants obtained by fitting to
the experimental frequencies are shown in Table 2, where for comparison the results of [42] are also presented. The
calculated frequencies for different modifications obtained by using the force constants given in Table 2 are
summarized in Table 3 where the experimental frequencies used for fitting the force constants are shown in
parentheses. The frequencies for the d-GaSe and the possible eight-layer polytype of GaSe are also shown in Table 3.
The lowest Raman active frequency mode at 7 cm1 is assigned as a rigid quadruple-layer mode (RQLM) of the eightlayer modification of the a-polytype.
The dynamical matrix considered in our calculations for one layer is as follows:
0
Cw þ Cb þ Ct
B
MA
B
B Cw
B pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
B MA MC
B
B Ct
B pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
B MA MC
B
@ Cb iqc
e
MA
Cw
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
MA MC
Cw þ Cg þ Ct
MC
Cg
MC
Ct
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
MA MC
Ct
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
MA MC
Cg
MC
Cw þ Cg þ Ct
MC
Cw
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
MA MC
1
Cb iqc
e
C
MA
C
C
Ct
C
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C
MA MC
C
C
Cw
C
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C
MA MC
C
Cw þ Cb þ Ct A
MA
(1)
Solution of this matrix assuming q = 0 and p/c, gives 8 frequencies for each shear and compression modes, which
describe the two-layer modification of GaSe. If we add the solution for q = p/2c and 3p/2c, then we obtain 16
frequencies for d-GaSe and finally, by adding the solutions for q = p/4c, 3p/4c, p/4c, and 3p/4c, we obtain 32
frequencies for the eight-layer modification of GaSe, which are shown in Table 3. Values marked by ‘‘*’’ should be
accounted twice. The reason for this accidental coincidence for some frequencies is because we supposed the same
values of the force constants for different polytypes.
Fig. 4 shows the solution of the secular equation of the above matrix along the GA-direction in the Brillouin zone.
The results of our model (dotted lines) are compared with those of Wieting’s model (solid lines). For both models, the
bold lines are for the compressional modes and thin lines are for the shear modes. The open circles at q = 2p/c depict
the experimental values given in Table 3. The right panel in Fig. 4 shows the density of modes calculated by Jandl et al.
[44] using an axially symmetric, three-dimensional Born–von Karman force constant model.
Table 3
The results of calculations using the modified linear-chain model for different modifications of GaSe
Type
Shear frequencies (cm1)
Compression frequencies (cm1)
e, 2 layers
213.4 (213), 212.6, 210.0 (210),
209.2, 62.9, 59.3 (59), 19.9 (20), 0
213.4, 209.2, 62.9, 0
213.4, 213.0*, 212.6, 210.0, 209.6*,
209.2, 62.9, 61.2*, 59.3, 19.9, 13.6*, 0
213.4, 213.3*, 213.0*, 212.8*, 212.6,
210.0, 209.9*, 209.6*, 209.3*, 209.2,
62.9, 62.4*, 61.2*, 59.9*, 59.3, 19.9,
18.2*, 13.6*, 7.2* (7), 0
307.9, 307.5 (308), 233.5 (234), 230.8,
142.6, 134.2 (134), 36.7 (37), 0
307.9, 230.8, 142.6, 0
307.9, 307.7*, 307.5, 233.5, 232.2*,
230.8, 142.6, 138.5*, 134.2, 36.7, 25.2*, 0
307.9, 307.8*, 307.7*, 307.6*, 307.5, 233.5,
233.1*, 232.2*, 231.2*, 230.8, 142.6, 141.5*,
138.5*, 135.5*, 134.2, 36.7, 33.6*, 25.2*, 13.4*, 0
g, 1 layer
d, 4 layers
a, 8 layers
The experimental frequencies used for fitting the force constants are shown in parenthesis. Values marked by ‘‘*’’ should be accounted twice. The
reason for accidental coincidence of some of the frequencies is because we assumed same values of the force constants for different polytypes.
Possible existence of an eight-layer a-polytype was suggested in the present work.
760
K. Allakhverdiev et al. / Materials Research Bulletin 41 (2006) 751–763
Fig. 4. Left panel: Compressional (bold) and shear (thin) phonon dispersions for GaSe in our chain model (dotted lines), for which Ct 6¼ 0, fitted to
the experimental measurements (open large circles at the zone edge) which are displayed in parentheses in Table 3, compared with the Wieting’s
model [42] (solid lines), for which Ct = 0. Right panel: Phonon density of states [44] with the three gaps. Our solutions do not fall into the gaps,
however, since the linear-chain models do not correctly describe the E-type vibrations and their dispersion, extra structures displayed by the phonon
density of states are not expected to be reproduced by the linear-chain models.
We also used the above-described model to calculate the vibration frequencies of the point defects like a vacancy or
substitutional impurity. For example, let us consider a case when one of the Se(l) atoms, e.g., the one in the elementary
unit cell marked as ‘‘0’’ is replaced by a foreign atom:
½1
Seð1Þ-Gað2Þ-Gað3Þ-Seð4Þ
½0
½1
Ãð1Þ-Gað2Þ-Gað3Þ-Seð4Þ Seð1Þ-Gað2Þ-Gað3Þ-Seð4Þ
where the number in the square brackets denote the index of the elementary unit cell.
Let us consider the 12 equations of motion:
MA ü1;1 Cw ðu1;2 u1;1 Þ Ct ðu1;3 u1;1 Þ Cb ðu2;4 u1;1 Þ ¼ 0
MC ü1;2 Cw ðu1;1 u1;2 Þ Ct ðu1;4 u1;2 Þ Cg ðu1;3 u1;2 Þ ¼ 0
MC ü1;3 Cw ðu1;4 u1;3 Þ Ct ðu1;1 u1;3 Þ Cg ðu1;2 u1;3 Þ ¼ 0
MA ü1;4 Cw ðu1;3 u1;4 Þ Ct ðu1;2 u1;4 Þ C̃b ðu0;1 u1;4 Þ ¼ 0
M̃A ü0;1 C̃w ðu0;2 u0;1 Þ C̃t ðu0;3 u0;1 Þ C̃b ðu1;4 u0;1 Þ ¼ 0
MC ü0;2 C̃w ðu0;1 u0;2 Þ Ct ðu0;4 u0;2 Þ Cg ðu0;3 u0;2 Þ ¼ 0
MC ü0;3 Cw ðu0;4 u0;3 Þ C̃t ðu0;1 u0;3 Þ Cg ðu0;2 u0;3 Þ ¼ 0
MA ü0;4 Cw ðu0;3 u0;4 Þ Ct ðu0;2 u0;4 Þ Cb ðu1;1 u0;4 Þ ¼ 0
MA ü1;1 Cw ðu1;2 u1;1 Þ Ct ðu1;3 u1;1 Þ Cg ðu0;4 u1;1 Þ ¼ 0
MC ü1;2 Cw ðu1;1 u1;2 Þ Ct ðu1;4 u1;2 Þ Cg ðu1;3 u1;2 Þ ¼ 0
MC ü1;3 Cw ðu1;4 u1;3 Þ Ct ðu1;1 u1;3 Þ Cb ðu1;2 u1;3 Þ ¼ 0
MA ü1;4 Cw ðu1;3 u1;4 Þ Ct ðu1;2 u1;4 Þ Cb ðu2;1 u1;4 Þ ¼ 0
(2)
K. Allakhverdiev et al. / Materials Research Bulletin 41 (2006) 751–763
761
where M is the mass of the atom lying in the nth plane of the unit cell; subscripts A and C are for the anion and the
cation, respectively; u is the normal displacement of the atoms and numbered subscript shows the number of the
elementary unit cell and the mass of the impurity atom; changes of the force constants due to the presence of the
defects is indicated by ‘‘’’.
When writing down these equations, we have assumed through out our discussion that a vacancy is determined as an
atom with zero mass and that the periodicity of the crystal lattice is preserved both towards left and towards the righthand sides from the unit cell which contains this defect [54]. Solution of Eq. (2) may be calculated in the following
form:
un;l ¼ Al eivt kjnj ;
(3)
where n is the index for the elementary unit cell, l the index for the basis atom, v the frequency of vibration, and Al is
the amplitude of vibration of the lth atom.
Expression K was found from the solution of the equations of motion for an ideal crystal. In this case, the expression
eiqc was substituted for K in the dynamical matrix (1). Then, by substituting the values obtained for K into the
determinant of the above given 12 equations (Eq. (2)), we have found the sought solution for the vibration spectra. For
this case, an indispensable condition should be implemented as jKj < 1. For convenience, we assumed that the force
constants do not change when the impurity atom is introduced. In principle, it is reasonable to consider these changes
in the framework of our model.
Similarly, we have also considered cases of impurity atoms substituted for Ga atoms. Calculated frequencies are
shown in Table 4.
The linear-chain model has one essential shortcoming—it does not take into consideration the phonon dispersions
in the direction perpendicular to the chain axis. As a result, many of the calculated local vibrations may get into the
allowed range of the phonon distribution function g(v) calculated for an ideal crystal [43,44]. In this case, the local
Table 4
Calculated shear and compression frequencies of the impurity modes along with the experimental measurements (all in cm1)
Impurity
Shear frequencies
Compression frequencies
For Se
N
O
P
S
373.9(L), 211.3, 72.9, 20.1
353.8(L), 211.3, 72.3, 20.0
275.5(G), 211.2, 68.8
272.3(G), 211.2, 68.6
420.5(L),
399.5(L),
330.1(L),
328.1(L),
263.3,
214.5,
211.4,
211.3,
211.2, 67.9
210.5
192.5(G), 56.5
77.08, 20.3
323.2(L), 264.0(G), 160.7
308.1(L), 234.1, 142.8
307.7, 221.9, 123.2
293.7(G), 179.7(G), 37.4
303.2, 211.07, 66.7
286.6(G), 211.1, 66.2
230.7, 211.0, 63.6
440.9(L), 248.2(G), 143.1
415.4(L), 247.3(G), 143.0
332.6(L), 239.5
Mn
Zn
226.8, 210.9, 63.4
215.8, 210.5
327.2(L), 238.4
312.3(L), 234.4
Sn
Tl
Sn/Nd
Er
Tm
211.2,
211.2,
211.2,
211.2,
211.2,
283.2(G),
274.0(G),
278.8(G),
276.4(G),
276.3(G),
Vacancy
211.1, 70.8, 20.2
Cl
As
Te
Vacancy
For Ga
Mg
Si
Cr
185.6(G), 57.2
169.0, 53.8
178.7(G), 56.0
174.2(G), 55.0
174.0(G), 55.0
287.6(G), 172.8(G), 37.0
286.2(G), 171.7(G), 36.9
269.8(G), 163.2
268.4, 162.6
212.4, 133.8
192.4(G), 131.4
204.7(G), 133.1
199.1(G), 132.5
198.8(G), 132.2
Experimental Raman lines
305, 253, 244, 233, 213,
135,92, 60, 30, 19, 13, 7
305, 253, 244, 213,
60, 19, 13, 7
305, 253, 213, 134,
305, 253, 244, 233,
60, 30, 19, 13, 7
305, 253, 233, 213,
305, 253, 213, 134,
305, 253, 213, 134,
305, 253, 213, 134,
305, 253, 233, 213,
30, 19, 13, 7
134, 92,
60, 19
213, 134,
134, 60, 19
60, 19
60, 19
60, 19
134, 50,
253.0, 143.8, 37.4
The atoms which are to be replaced by the impurities are shown underlined. The capital bold letters (L) and (G) denote the local modes and gap
frequencies, respectively. All other frequencies are of resonance type.
762
K. Allakhverdiev et al. / Materials Research Bulletin 41 (2006) 751–763
vibrations become the so-called resonance ones. This circumstance decreases the quality of the results obtained within
the framework of a linear-chain model. At the same time, this model satisfactorily predicts possible impurity
vibrations. The type of these vibrations (local, gap, or resonance) will depend on the real spectral density of the phonon
states which, in principle, may be taken from the experiment [44] or from the results of the three-dimensional
calculations [43].
There are three gaps (forbidden frequency ranges) in the phonon density distribution function of an ideal two-layers
modification of GaSe and they are shown in Fig. 1 of ref. [43]. The first gap is in the range of 5.1–6.3 THz which
corresponds to 170–210 cm1. It means that the frequencies of the impurity atoms, which will find themselves within
this range, will be considered as local vibrations (in the literature they are also called ‘‘gap vibrations’’). The second
gap in the phonon density of states is in the very narrow range of 7.35–7.5 THz which corresponds to 245–250 cm1.
The third gap is in the range of 7.9–9.05 THz and corresponds to 263–302 cm1. All frequencies of the impurity
atoms, which will coincide with these three ranges, will be considered as local (gap) modes. All frequencies above
9.2 THz (308 cm1) are called as ‘‘real’’ local modes. The rest of the impurity frequencies, which will coincide with
the allowed frequency range of the real crystal, will be considered as resonance frequencies. The types of the impurity
vibrations classified in the present work are shown in Table 4 (G: gap, L: local, and all the rest are resonance).
As one can see there are reasonable agreements between the calculated and experimental values of the frequencies
for some of the impurity atoms. A weak band at 195 cm1 observed in the Raman spectra of GaSehTli is in accordance
with that calculated at 192.4 cm1 and identified in the present work as a gap mode, whereas a band at 80 cm1 is
classified as a resonance (Table 4). The weak bands at 92 cm1 and 30 cm1 recorded in the spectra of GaSehCri,
GaSehSi, and GaSehTmi are attributed to resonance modes of vibration.
4. Conclusions
The Raman scattering, PL, mid-IR, and far-IR absorption spectra of pure and doped p-GaSe were measured. The
stoichiometry of GaSe and the impurity content were determined by X-ray fluorescence method. It was shown that a
modified linear-chain model satisfactorily describes the lattice dynamics of the existing four modifications as well as
the vibration frequencies and the type of the impurity atoms. To explain the results of the optical measurements, the
existence of a new eight-layer polytype was suggested and the lattice dynamics was calculated in the framework of a
modified one-layer linear-chain model. The absorption band at 83 cm1 was assigned to the multi-phonon process.
Acknowledgments
This material is based upon work supported by the Marmara Research Center of the Turkish Scientific and
Technical Research Council (TÜBITAK) in a frame of a joint project no. 5035601 with the Institute of Materials
Science, University of Tsukuba. Partial supports through TBAG – U/28 (101T168) and TBAG – 2220 (102T113) are
acknowledged.
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