Fine structure in the absorption edge spectrum of NbS2Y2 (Y = Cl, Br, I)

J. Phys. Chem. Solids Vol. 41, pp. 375 384 Pergamon Press Ltd., 1980. Printed in Great Britain

FINE STRUCTURE IN THE ABSORPTION EDGE SPECTRUM OF NbS2Y 2 ( Y = C1, Br, I) J. RIJNSDORP and C. HAAS Laboratory of Inorganic Chemistry, Materials Science Centre of the University, Nijenborgh 16, 9749 AG Groningen, The Netherlands

(Received 1 March 1979; accepted 24 August 1979) Abstract--High resolution absorption spectra of single crystals of NbS2Y 2 (Y= C1, Br, I), obtained at temperatures between 4.2 and 300 K revealed extensive fine structure in the absorption edge. This structure has been analysed and interpreted in terms of allowed indirect interband transitions, involving phonons corresponding to S-S and Nb S vibrations, followed by forbidden and by allowed direct transitions. From the shape of the absorption curve associated with the phonon branches of the indirect transitions a binding energy of 23 cm - 1for indirect excitons is obtained. A binding energy of 28 cm - 1 for direct excitons is deduced from the exciton lines observed at the long-wavelength side of the direct transitions. A detailed interpretation of the optical transitions is given in terms of a molecular orbital diagram for the NbzS 4 clusters, present in these crystals. l . INTRODUCTION

A recent investigation of synthesis and crystal structures of compounds NbX2Y 2 ( Y = CI, Br, I; X = S, Se) revealed the presence of pairs of chalcogen atoms and pairs of Nb atoms in these c o m p o u n d s [ I - 3 ] . The crystal structure consists essentially ofNbzS ~ clusters, with the halide ions in the interstitial space between the clusters (Fig. 1). The stacking of the Nb2S 4 clusters and the position of the intermediate halide ions is not precisely the same in all NbS2Yz compounds. This leads to a monoclinic crystal structure for NbSzCI2 and the high-temperature form of NbS2Brz, to a triclinic structure for NbS212 and the low-temperature form of NbSEBr2.

•

Nb

©s

The compounds NbX2Y 2 are diamagnetic semiconductors with high specific resistivity [3 ]. If subjected to the chemical vapour transport method they crystallize as thin platelets of high quality, with a mosaic spread of only 0.5-0.8 ° . The high perfection makes these crystals well suited for optical investigations. However, the high quality of the strain-free platelets is lost if the crystals are cut or cleaved, as is indicated by the disappearance of the fine structure in the spectra after such treatment. In this paper we report on detailed investigations of the fine structure of the optical absorption edge of NbSEY 2 (Y = CI, Br, I) single crystals. The aim of these studies is to obtain information on the electronic energy levels and the chemical bonding in compounds with covalently bonded S-S and N b - N b pairs. In the following sections the experimental procedure, the analysis of the absorption spectra and a survey of the experimental results will be given. An interpretation of the absorption spectra, based on a molecular orbital scheme will be given, making use of recent data on the vibrational spectra [4, 5 ]. For the theoretical background of the analysis we refer to review papers, and the references mentioned there [6-9]. The results are compared with recently obtained photoelectron spectra [10].

cl 2. EXPERIMENTAL PROCEDURE

The absorption coefficient was measured with a Z high-resolution grating spectrometer (Perkin Elmer El4) at a n u m b e r of fixed temperatures between 4.2 and 300K over the range of photon energies )_ 4.000-25.000cm-~, using a tungsten strip filament lamp as a source of radiation. The detectors used were an S-10 photomultiplier (EMI 9529B) for the region X 14.500-25.000cm -~, an S-1 photomultiplier (RCA Fig. 1. Nb2S 4 cluster present in the compounds NbSzY 2 (Y 7102) operated at 210 K for the region = CI, Br, I). The cluster consists of two ($2)2- ions and a 10.000-16.500cm -a and a lead-sulfide cell for the Nb-Nb pair. The short interatomic distance in the Nb-Nb pair is due to metal-metal bonding. region 4.000-10.000cm -1. The specimen used were 375

376

J. RIJNSDORPand C. HAAS

crystals grown with the chemical transport technique[3]. All crystals used were tested with the Weissenberg technique for X-ray diffraction for singlecrystallinity. In general the selected crystals were not cleaved. The crystals were glued on a bore in the copper sample holder less than 1 mm from the second bore in which the Au (0.03~oFe) vs chromel thermocouple was placed. The glue used had the necessary property that it remained "soft" at rather low temperatures. Measurements were carried out at sufficiently small wave-number intervals in order to obtain the detailed structure of the absorption edge, the intervals being as small as 1 cm- 1 in the region of the indirect transitions and the exciton lines. During the measurements the temperature was constant within 1 K. The specimen thickness was measured with a microscope equipped with a micrometer. Of all single crystals investigated only 20~o could be used for a detailed study of the fine structure. This is presumably due to the presence of nonuniform strain in many of the crystals, which washes away the fine structure. In order to avoid the strain it is necessary to use the proper gluer. The thickness of the single crystals used for the detailed study of the fine structure is for NbSEC12 20.0/~ (42%), 19.8F, 35.9#, 280/~, for triclinic NbS2Br 2 18.61L (45%), for monoclinic NbSzBr 2 26.4~t, 54.3/~ and for NbS2I 2 12.2/~ (53~), 16.4#, 32.0kt, 56.1/z. The transmission in the transparent region (about 10.000 c m - 1) is given between brackets. 3. R E S U L T S

The resemblance of the absorption curves of the monoclinic and triclinic NbSaY 2 crystals, obtained with unpolarized light, is so striking that only the experimentally determined absorption curves of NbS2C12 at a series of temperatures are shown in Fig. 2. Low absorption levels At the low absorption side of the curves several knees are visible. These knees were found in all the specimen with composition NbS2Y 2 (Y = CI, Br, I) we have studied, so we conclude that these knees are an intrinsic property of the NbS2Y 2 crystals and are not due to the presence of impurities or defects in the crystals. At low temperature up to five knees are observed for NbS2C12 and NbS2Br 2. As the temperature increases up to five other knees appear at the lowenergy side of the spectrum and the intensity of these contributions increases rapidly with increasing temperature. Whenever these new knees appear the energy differences with the knees already present at low temperature remains constant and can be related to the energy differences between the low-temperature knees. These are precisely the type of relations which one would expect if the absorption in this spectral

IVELPON registered Beverwijk, Holland.

trade-mark

of

~s00 ~c

~ooo

s00

0 15000

16000

17000

18(100

19000 20000 photon energy E (cm-~1

21000

Fig. 2. The absorption edge of NbSzC12 at various temperatures. region is caused by indirect interband transitions. The contributions (knees) present also at the lowest temperatures correspond to electronic interband transitions associated with phonon emission, the new contributions appearing only at higher temperatures are caused by transitions associated with phonon absorption [6 ]. A more detailed analysis of the curves in this region was carried out on the following basis. For energies below the beginning of the absorption in the crystals, the transmission ratio falls slowly with increasing energy and can be quite well represented by a linear law. We have assumed that this law, representing mainly the reflection losses of the specimen, but possibly also some losses due to surface imperfections, may be extrapolated into the higher energy region where absorption in the crystal occurs. The absorption coefficient K in the indirect region is made up of several contributions K,,(E), where E is the photon energy. Each of these contributions begins with a well-defined knee at photon energy E~(T) and rises at higher energies as:

E-I{E -

E ~ I T ) - A } 2.

The contribution K~,+ I(E), beginning at a threshold energy Era+ I(T), was obtained by assuming that for E > E,.+I(T), Km continued to rise as:

Ceta-Bever, E-X{E - E,~(T) - A} 2.

Fine structure in the absorption edge spectrum of NbS2Y," (Y= CI, Br, 1) Defining ~in(g, T ) = E. Ki.(E, T), the basic shape of the nth knee for n = 0, 1, 2, 3 and 5, associated with either phonon absorption (~.,,) or emission (~.,), is represented very accurately by the theoretical expression for absorption due to indirect transitions [6 ]:

no significant change in shape in the upper region of the knee c% nor in the upper region of phonon branch n in the region where the energy dependence is essentially quadratic, the threshold energy of the broadened phonon branch was calculated from:

c%(E, T) = ai,,(T). {(E - E i , ( T ) ) 1/2 + b. (E - Ei,,(T) - A + B) 1/2

E~.(T) = {X(T) - 54} cm '

+ c. (E - Ei,~(T) - A) 3/z}

(1)

where b and c are constants and a~,(T) is a function of temperature. The first and second term in this equation are due to transitions to the ground and the first excited bound exciton state with exciton binding energies A and B, respectively. The last term is due to transitions to unbound electron-hole states. These latter processes lead at higher photon energies to the relation: =~,,(E, T ) ~ {E - E I . ( T ) - A] 2.

(2)

We have not been able to determine the actual shape of phonon n = 4 contributions, because of the small energy difference I ( E i 4 ( T ) - E i a ( T ) I and the low absorption level of phonon branch n = 4 as compared with n = 3. The values of Ei. are probably not very accurate. As the temperature is raised, the knees are smoothed out due to thermal broadening (Fig. 3); a similar thermal broadening is observed in the spectra of silicon [11 ]. Assuming that the broadening produces

x ~2

%

,

?.e,n-a

377

where X ( T ) is the intersection of the lines calculated through the experimental points of both regions of the curve of{E. Ki.(E, T)} 112 vs E. The same values for the threshold energy, within the experimental error of about 4cm 1, are found if the unbroadened curve obtained at low temperature is scaled with the broadened curve, taking the position of the threshold energy of the broadened phonon branch equal to the threshold of the "scaled" unbroadened branch. Experimental values of the threshold energies E~n(T ) were used to calculate values ofE. = l ( E a . + Ee. ) and the phonon energies kO. = ~(E~. - Ea.). The phonon energies obtained in this way are independent of the temperature; the average values are listed in Table 1. The values of E. are almost independent of n; the average value of E. is E, which is the indirect gap reduced with the exciton binding energy A. The values of E obtained in this way were used to calculate the indirect gap E i (Table 2). The close agreement between the E. values at each temperature and the fact that kO. is independent of temperature is convincing evidence of the correctness of the analysis and its interpretation in terms of indirect interband transitions. The analysis of the low-temperature absorption curves makes it also possible to derive values of the binding energies of the exciton ground state A and the first excited exciton state B. The values obtained from the spectra are given in Table 3. It is found that the values of A and B are the same for all phonon branches. In order to be able to compare the relative contributions of the various phonons to the indirect absorption process, the ratio ~./~2 deduced from the low-temperature data, is given in Table 3. For indirect transitions one expects that the relative magnitude of the component ~a., associated with phonon absorption, and the corresponding component ~e., associated with phonon emission at the same energy above their threshold E.. and Een , respectively, is given by:

*BI ~e~(T)/~.(T)

= p . . exp {OffT}

(3)

and 20

20

40

60 80 100 • (E~Ein(r)) [crn4]

120

Fig. 3. The decomposition of one of the phonon branches of NbS2C12 at 6.2K and the variation of the basis shape with temperature due to thermal broadening. K and E are givenin cm - L The data at the various temperatures have been aligned in energy using the value of E~n(Table i ) associated with each curve and each set has been given an arbitrary vertical displacement.

(4)

Ee, , - Ea, , = 2kO.

k being Boltzmann's constant, 0. the phonon energy in degrees Kelvin and

P" = [AE,.

-- kO.3

"

(5)

378

J. RUNSDORPand C. HAAS Table 1. Phonon energies kO. (in c m - 1) obtained from the threshold energies for indirect transitions. The uncertainty of these phonon energies is about 1 ~o T

kO0

kOI

k@2

k8 3

k@4

k@5

NbS2Cl2

6-300 K

76

NbS2Br2

4-300 K

75

154

239

308

337

585

153

227

298

329

NbS212

4-300 K

75

575

146

216

291

322

567

Table 2. Values of the indirect gap E~, obtained from the relation E~ =/~ + A, the forbidden direct gap Ey and the position of the exciton lines E1 r and Ea (in cm-1) at various temperatures T(degrees Kelvin) T NbS2CI 2

Ei

Elf

Ef

Ea

6.2

16630

17235

17242

19960

31.4

16598

17180

19927

61.5

16541

17080

19854

97.4 156.8

16427 16284

16950 16810

19760 19540

196.8 241.9 288.1 302.3

16187 15892 15686 15537

16690 16380 16160 16000

19380 19020 (18800)

17398

19695

17230 17070 16740 16020

19556 19420 19060 (18210)

15798 15670

17582 17468

4.8

16733

71.5 121.3 204.8 295.6

16610 16453 16153 15435

4.6 68.9

15380 15275

126.8 188.3

15099 14861

(15430) 15180

17320 17030

295.2

14404

14660

16440

NbS2Br2

NbS212

17391

15791

Table 3. Values of the exciton binding energies in the ground state (A) and the first excited state (B) (in cm - 1) The relative intensity of the phonon contributions is given by the ratio %/ct2 T

A

B

%/m2

ml/m2

NbS2CI2

6.2

23

9

3.6

7.0

NbS2Br2

4.8

23

9

3.8

NbS212

4.6

23

9

3.9

m2/m2

%/m2

%/a2

1.0

1,5

1.7

6.9

1.0

1.7

1.7

7.2

1.0

1.6

Table 4. Values of p. and AE. (in cm-1) obtained from the absorption data. The values are practically independent of the temperature

NbS2CI2 NbS2Br2 NbS212

T = 6 - 300 K T = 4 - 300 K T = 4 - 300 K

PO

Pl

P2

P3

P5

1.07 1.08 1.10

1.17 1.18 1.22

1,11 1,11 1,14

1.33 1.41 1.46

2.31

Values of p, and AE, obtained from the experimental data, show that these quantities are practically temperature independent (Table 4). It is found that AE, has approximately the same value for n = 0, 1, 3, 5, but a much larger value for n = 2. The decomposition of the absorption curve ofNbSzC12 at 6.2 K is given in

&Eo

AE1

&E2

AE3

AE5

4500 3900 3200

3900 9200 4300 3700 8700 3500 2800 2900 6600 3100

Fig. 4. We remark that although some of the changes of slope in the curve are quite small, they are very reproducible and have been observed at all temperatures and in several crystals. The difference between the values of the threshold energies found for different crystals of monoclinic (triclinic) NbS2Y 2 was

Fine structure in the absorption edge spectrum of NbSzY2 (Y= CI, Br, I)

/

379

i

~ E

J-

~7220

1260f

16680

16750

16840

16920

17000 170~3 17160 17240 • photonenergyE[cmJ]

Fig. 4, The decomposition of the absorption curve of NbS2C12 at 6.2 K in the indirect transition region. K and E are given in cm- t. The insets show an enlarged part of the curve on a linear scale in K.

800cm-1 or less. Apart from a constant shift of less than 800 cm-1, there was no difference between the threshold energies of the different phonon branches found for the triclinic and/or monoclinic form of NbS2Y 2. The largest differences were observed between monoclinic crystals and for NbS2I 2 indicating that these differences might be attributed to the observed variation of the unit-cell parameters [3 ] as well as to strain in the crystals.

Intermediate absorption levels As the energy increases, a point will be reached at which the photon energy is sufficient to produce direct transitions Ak = kc -k,, = 0 across the energy gap between valence and conduction band states; kc and kv are the wave vectors of the electronic states of the conduction and valence band, respectively. The absorption due to these direct transitions has been studied in detail as a function of the temperature. In order to obtain the contribution of the direct transition, it is assumed that the contribution of the indirect transitions continues to rise as E - I { E -- Ei,, -- A } 2 in the region of interest. At low temperatures the absorption first rises to a peak at El: and then continues to rise more slowly (Fig. 5). At temperatures above 30 K this exciton peak of the direct transitions virtually disappears as a result of thermalbfoadening. The experimental values of El: and the threshold energy of the direct transitions E: are listed in Table 2. The absorption level of the direct transitions increases with temperature and has a (AE) 3/2 energy dependence for energies larger than about E: + 80 cm- 1, indicating that forbidden direct interband transitions are governing the absorption in this region [6 ].

20

0

20

z,O 60 80 100 120 140 150 photon energy (cm-1)

Fig. 5. The contribution of forbidden direct transitions to the absorption coefficientK of NbS2C12.The data at the various temperatures have been aligned in energy using the shape of the curve at higher energy. Each curve has been given an arbitrary vertical displacement.

High absorption levels As the photon energy increases the absorption begins to increase very rapidly due to the onset of transitions across a second direct gap. The absorption due to this second set of direct transitions is obtained, assuming that the absorption due to the forbidden direct transitions continues to rise as (AE) 3/2 which describes the observed absorption coefficient for energies just below the second direct transition region (Fig. 6). At low temperatures the absorption rises to a peak at E.. At temperatures above 100 K the peak E, disappears as a result of thermal broadening. Extrapolated values of E, at higher temperatures (Table 2) were obtained by shifting absorption curves so that they approximately coincide in the highabsorption region where thermal broadening presumably has only a small effect. The absorption starts to rise very rapidly for photon energies larger than about E~ + 200 cm- 1 and the crystals become opaque. The absorption level in the region of the peak has a value of about 170 cm- 1. The low absorption level and the value of the photon energy at E. suggests that we have to attribute the line at E~ to ($2) 2- exciton formation[12,13]. The very rapid rise of the absorption above E~ + 200cm -1 indicates that allowed direct interband transitions are superimposed on the forbidden direct interband transitions. Figure 7 shows the variation of the indirect gap E~, the forbidden direct gap E: and the ( $ 2 ) 2 - exciton peak E. as function of the temperature. The temperature dependence of E~, E: and E, is similar to the temperature dependence of energy gaps of other semiconductors [8 ].

380

J. RUNSDORPand C. HAAS transitions are (almost) forbidden for this polarization direction. A detailed analysis shows that the p h o n o n branches n = 0, 1, 2, 3 and 5 of the indirect transition in NbS2Br 2 are forbidden for light polarized parallel to the N b - N b direction. The p h o n o n branch n = 4 is too weak to allow an unambiguous determination of the polarization behaviour, but it is possible that this branch is allowed also for light polarized parallel to the N b - N b direction.

I

I 6, SK

u ~UgK

T

I

2aa,~

T

/ -100

-50

50 phot o1~ energy

cml00-1) (

Fig. 6. The contribution of allowed direct transitions to the absorption constant K of NbS2C12. The data at the various temperatures have been aligned in photon energy using the shape of the curve at higher energies. Each curve has been given an arbitrary vertical displacement.

NbS21~

i o ; ....

ioI r0sm

r~co

4. DISCUSSION The similarity of the shape of the absorption curves, the relative intensities and the p h o n o n energies of NbSEC12, NbS2Br 2 and NbSEI 2 indicate that the wave functions of the top of the valence band and the m i n i m u m of the conduction band are mainly determined by the NbzS 4 clusters, which are essentially the same in all three compounds. We shall try to explain the absorption spectrum in terms of a molecular orbital scheme of this NbES 4 cluster. The approximate symmetry of the NbES 4 cluster is D2h,the actual symmetry is C2h in the monoclinic crystals and C 1 in the triclinic crystals. The influence of this lowering of the symmetry is small as indicated by the similarity of the spectra of monoclinic and triclinic NbS2Br 2. Therefore, we will use D2h symmetry for the analysis. The NbzS 4 cluster of DEh symmetry is sketched in Fig. 1. A coordinate system X, y Z is defined such that Z is parallel to the N b - N b line and X is parallel to the S-S bonds. The representations of the symmetry group Dzh are defined in such a way, that X, Y, Z transform a s B3u, B2u and BI., respectively. The molecular orbitals for the Nb2S 4 cluster are given in Fig. 8. The sulfur 3px orbitals of an ($2) z- ion combine to give an occupied bonding orbital ag(3px ) and an unoccupied antibonding orbital (r* (3px), both directed

Ag{Ldx2 y2 ) O" (3p) u x

Fig. 7. The variation of the indirect gap E,, the forbidden direct gap E I and the spin-forbidden exciton peak Eo of NbS2C12, NbS2Br 2 and NbS2I 2 as a function of temperature.

~

- • TLg(3py; 3Pz)

.....

'

Polarisation effects The transmission spectra of monoclinic and triclinic NbS2Br 2 single crystals were also studied with light polarized parallel and perpendicular to the N b - N b direction (monoclinic axis or perpendicular to the aaxis for the triclinic unit cell given in [3]). The perpendicular direction makes an angle of about 34 ° with the X-axis (the (S2)-direction) of the Nb2S 4 cluster (Fig. 1) in monoclinic (triclinic) NbS2Br 2. The observed strong reduction of the absorption in the indirect and second direct transition region for light polarized parallel to the N b - N b direction for monoclinic and triclinic NbS2Br z indicates that these

-

B3u(3Px]

-

-

B1Q 13pyl A 13pi) u z B2gl3pz )

~

B3u(3py)

~ 1 1

~ 3 e V , ,

(52)2"

22($2)- (S 2)

~leV--

Ag{4d 2) z

i t

Bgu(3pyl

B2 u{3Px*J Ag{3p) -

--~

B19 g'dx/} B 2 u (4dy)z B3u (Ldxz} • •

B3g{3p~l -B1u(3Pz} Agl]py} -

Xu[3Py; 3Pz)

Og(3Px)

BIg {3Px) -

, • ~ • I ~-

-r~6eV

• • :,~ 7 eV • , - _1 ~* (Nb2Sz)

N[. ~*_ Nb z~

Fig. 8. Molecular orbital diagram for the (Nb2S4) 4+ cluster with D2h symmetry. The line connecting the two niobium atoms is taken as the Z-direction, the line connecting the two sulfur atoms of a ($2)2- pair is taken as the X-direction. Only the bonding molecular orbital combinations of the Nb(4d) orbitals are represented. The approximate binding energies (in eV) of groups of molecular orbitals as obtained from photoelectron spectra (maxima of peaks) are indicated.

Fine structure in the absorption edge spectrum of NbS2Y_, (Y = C1, Br, II along the S-S line. The sulfur 3pr and 3p~ orbitals produce bonding and anti-bonding orbitals g.(3pr; 3p~) and n*(3py: 3p~), which are all occupied. The interaction of the two (S2)2- ions leads to a further splitting, as indicated in Fig. 8. This further splitting is easily obtained by constructing bonding and antibonding combinations of the molecular orbitals of the two interacting ( $ 2 ) 2 - ions in one (NbzSJ '*+ cluster. For example, the combination of the two 7ru(3py)(S2) 2- orbitals leads to a molecular orbital B2,(3p*) which is anti-bondingbetween the two (S 2)2 ions and a molecular orbital A~(3pr) which is bonding between the two ($2) 2- ions. The splitting of Pr P* orbitals is due to a-type bonding between the ($2) zions and is expected to be larger than the ~z-type bonding which leads to the p~-p* and the p~ p* splittings. The niobium 4d orbitals split under the influence of the low-symmetry crystal field and covalent bonding between the Nb orbitals. The interaction between the Nb(4d~2) orbitals directed along the N b - N b line is responsible for the metal-metal bonding in the Nb, pair. This interaction leads to a bonding level A~(4d~a) occupied by two electrons, the other d-orbitals remain unoccupied. This molecular orbital scheme explains the diamagnetic character of the NbSzY 2 compounds. The molecular orbital scheme (Fig. 8) gives only a qualitative description of the nature of the wave functions. In the solid the actual wave functions will be combinations of Nb, S and Y orbitals, forming energy bands and the symmetry is lowered to C2h (monoclinic crystals) or C a (triclinic crystals). The energy of the empty Nb 4d orbitals relative to the empty sulfur 3p orbitals is unknown. Photoelectron spectra of the valence band show that the occupied bonding Nb(4d~ 2), with Ag symmetry, lies just above the highest occupied S orbital, with Bag symmetry.

Indirect transitions In the crystal the top of the valence band and the minimum of the conduction band lie at different k values in the Brillouin zone. From indirect transition theory [6] it is clear, that exciton formation in the absorption process produces well-defined knees at the low-energy side of phonon branches. The initial rise of cq,,(E, T) as the square root of the energy is due to allowed transitions to the ground state of the exciton. The difference between the threshold energies of first and second term in cq,(E, T) (eqn 1) is independent of n, therefore the second term is ascribed to transitions to the first excited state of the exciton lying 14 _+ 4cm a above the ground state. The third term of eqn (1) arises from allowed transitions producing unbound electron-hole pairs in the crystal. The threshold for these transitions is 23 _+ 2 cm-1 above the threshold E.(T) for transitions to the ground state, so the binding energy of an indirect exciton in all three NbS/Y 2 ( Y = C I , Br, I) compounds is E~ = 2 3 _+ 2 cm- a. By increasing the values of E with the value E~ we obtain the values for the indirect gas E~ as a Pcs 41/4-

381

function of temperature (Table 2). The uncertainty in the values of the indirect gap E i is determined by the error of Eex and the error of/~ (about 4 cm- 1) and is therefore estimated to be about 6 cm- 1. The indirect transitions n = 0, 1, 2, 3 and 5 are allowed so the matrix element describing the indirect transition of an electron from the initial state Ii, k~, > with wave vector k~.to the final state If kc > with wave vector k~.contains non-zero elements of the form:

~,i, k~lH~,dlj, k~,) ~,j, k~]Hh~[f k~) ~j - sf - ( - 1 )tkO,

(6)

and/or

( f k~lH ~a[j, k~~ Cj, k ~lHh, Ii, k~,~ ei - e,j - ( - 1)tk0,

(7)

where lJ) is some intermediate state, the Hamiltonian Hr~a describes the interaction between the radiation and the electrons and H~,t the interaction between electrons and the lattice vibrations; l = 0 or 1 corresponds to phonon emission or phonon absorption, respectively. Thus, the contribution of each possible intermediate state to the transition matrix element is proportional to {AEj - ( - 1)tkO,}- 1, where AEj = sj - ~f or ~i - - ~ j ' The contribution of the state IJl) with the lowest AE value AEj~ will dominate if AEja << AE~, and this results in eqn (5) for p, if we take AE, in eqn (5) equal to the energy of the intermediate state ]J~) relative to the bottom of the conduction band or the top of the valence band. lfwe assume that AE = E, + 200 - E, we calculate values for p. which are in good agreement with the experimental values for n = 0, 1, 3 and 5. The conclusion is, that the allowed direct transitions, beginning at about E. + 200cm 1, serve as an intermediate state for the indirect transitions for the phonons n = 0, 1, 3 and 5. The contribution of phonon n = 2 has a different intermediate state or it is the sum of at least two contributions with different AE. The next step is that we try to correlate the phonon frequencies deduced from the indirect absorption with phonon frequencies determined directly by IR and Raman spectra. A study of the vibration spectra of the compounds NbSzY 2 (Y = C1, Br, I) showed that the frequencies of the Nb S and S-S vibrations are practically independent of the halide ion Y. Apparently, there is little interaction of the Nb2S 4 cluster vibrations with the halide ions and the Nb2S 4 cluster can be treated as an independent entity as far as vibrations are concerned. This justifies an analysis in terms of vibrations of Nb2S 4 clusters with (approximate) D2h symmetry. The normal modes of this cluster have been analysed by Perrin et al. [4, 5 ] and consist of two S-S vibrations (Ag + B2u) with frequencies about 600cm 1 and a number o f N b - S ; S-S vibrations with frequencies ranging from 140-380cm a. All vibrations have been observed in IR or Raman spectra. A survey of the observed vibrational frequencies is given in Table 5. The modes are sketched in Fig. 9.

382

J. RIJNSDORPand C. HAAS Table 5. Comparison of the phonon frequencies (cm-1) derived from indirect transitions (T) with vibrational frequenciesobserved in infrared (IR) and Raman (R) spectra. The symmetry refers to the normal modes of a Nb2S4 cluster with D2h symmetry NbS2CI 2 n

IR

i

B3u

2 3

R

NbS212

T

IR

R

T

IR

R

T

IR

Ag

154

166

188

153

174

183

146

169

Au

BIg

239

243

216

B3u

Ag

308

320

310

291

315

328

(322)

340 348 358

585

567

570

Blu 4 5

NbS2Br2

323

252

227

317

298

362 (337)

B2u

B2g B39

B2u

Ag

5a5

358 336

(329)

592

575

377 582

370

588

The vibrational modes observed in IR and Raman spectra are essentially the lattice modes with wave vector q = 0. The indirect transitions involve lattice modes with q = kc - k,. If the interaction between the clusters is small, then the dispersion of the cluster vibrations will be small and the phonon frequencies at q :~ 0 will not differ much from the corresponding q = 0 frequencies. Indeed the phonon energies derived from the analysis of the indirect transitions closely correspond with the fundamental vibrations of the Nb:S 4 cluster (Table 5). The phonons n = 1-5 correspond to optical modes transforming as indicated in Table 5. Phonon n = 0 cannot be ascribed to one of the normal modes of the cluster, it probably corresponds to a low-frequency lattice mode of the crystal. An attempt will be made to interpret the observed optical transitions in terms of the molecular orbitals of the Nb2S 4 cluster. In the crystal the interaction between the clusters will lead to energy bands, mixing of wave functions and lowering of the symmetry. Especially at general points in the Brillouin zone, this will lead to a relaxing of selection rules. If the interactions between the clusters are weak, these effects will also be relatively weak and transitions which are forbidden in the isolated cluster are expected to be weakly allowed in the crystal.

580 585

575

This is in accordance with the expectation of small interactions between Nb2S 4 clusters, and thus provides indirect evidence for the small phonon dispersion in these compounds. A tentative assignment of the observed transitions is given in Fig. 10. The initial states of the indirect transitions are states of the highest occupied band, i.e. the band composed mainly of Nb(4d=2) orbitals with Ag symmetry. This is a rather narrow band, photoelectron spectra indicate a band width of about 1 eV [10]. Photoelectron spectra also suggest that°the top of the Ag band lies about 0.8-1.00 eV above the top of lower lying bands. Because the differences E,, - E i and E s - E i are considerably less than 1 eV, the initial state of the direct transitions with threshold energies Ey and E, + 200 cm- 1 must also be the Nb(4dz2 ) band with Ag symmetry. The lowest unoccupied states of both the sulfur and the Nb orbitals lead to molecular orbitals of B3, symmetry, i.e. B3,(3px ) and B3u(4dx=) (Fig. 8). These states are expected to hybridize strongly and therefore the lowest unoccupied band of B3, symmetry will have

,

Big

-- (sp=nforbidden ~rans~ion Eo ) ~ 4 d z 2 }

EO

-v

-c wove vector

e3g

Fig. 9. The optical modes of the Nb2S 4 cluster (D2h) corresponding with the phonons of the indirect transition.

_k

Fig. 10. Tentative assignment of observed optical transitions in a band structure diagram (energyE vs wavevector k). The symbols A~, Bls and Ba. indicate the symmetry of the dominant molecular orbital of the Nb2S4 cluster in the wave function. The energy of the intermediate state of the phonons n = 0, 1, 3 and 5 is denoted by ej, of phonon n = 2 by gj,,.

Fine structure in the absorption edge spectrum of NbS2Y 2 (Y = C1,Br, I) a mixed Nb/S character. The final state of the indirect transitions will consist of states of this lowest B3u band. It follows from the observation of indirect transitions that the maximum of the Ag band at k~ and the minimum of the B3. band at k~ are at different points in the Brillouin zone, kc ~ kv. If was found that the allowed direct transitions at about E. serve as an intermediate state for the indirect transitions. This suggests that these transitions correspond to direct transitions from the Nb(4d~z)-Ag band to the lowest B3. band at k~. In the cluster B3, LAg transitions are allowed only for light polarized with the electric vector parallel to the Xdirection. Therefore, in first approximation the direct transitions B3, ~- Ag are forbidden for light polarized with the electric vector parallel to the Z-direction, in agreement with experimental observations. The absorption due to indirect transitions is related to the strength of the electron-phonon coupling. Phonons n = 1 (Ag) correspond to vibrational modes which produce a strong modulation of the N b - N b distance in the cluster (Fig. 9). It is expected that this vibrational mode will couple strongly with electronic transitions involving N b - N b bonding and antibonding molecular orbitals. Indeed the largest contribution to the indirect transitions comes from phonon branch n = 1 (Table 3). In contrast with the phonons n = 2 and n = 4 , the n = l , 3 and 5 phonons have a component of Ag symmetry. Therefore, the indirect transitions of phonon branches n = 1, 3 and 5 are presumably due to the emission or absorption of phonons of A~ symmetry, with the Nb(4d~)-A~ band state at k~ (ej in Fig. 10) as an intermediate state. The energy difference between the top of the Ag band at k~ and the intermediate state ej is AE. For NbS2C12, NbS2Br 2 and NbS2I 2 AE = 4200, 3500 and 3100 c m - 1, respectively. These values are reasonable if compared with the width (about 1 eV) of the Nb(4d~)Ag band, as obtained from photoelectron spectra. Indirect transitions between Ag and B3~ bands involving phonons n = 1, 3 and 5 with Ag symmetry are expected to be allowed only for light polarized with the electric vector parallel to X. Therefore, these indirect transitions will be forbidden for light polarized with the electric vector parallel to Z, as is observed. Phonons n = 2 do not have a component of A~ symmetry, therefore the indirect transitions with n = 2 type phonons involve a different intermediate state. Phonon n = 2 has a Bag component, therefore the Nb(4dy~)-B2~ band states at k~, and the S(3p*)-Bxg band states at k~ may serve as intermediate states for indirect transitions involving phonon n = 2. If the contribution through S(3p*)-Bag band states dominates, then the energy difference between the top of the Nb(4d~:)-Ag band at k~ and the top of the S(3p*)Bag band at k¢ (e~,, in Fig. 10) is 9200, 8700 and 6600cm -a for NbSECI2, NbS2Br 2 and NbSEI2, respectively. Both the trend and the values are reasonable if compared with values of the difference between the top of the Nb (4d~)-A~ band and the top of

383

the S(3p*)-Bag band estimated from photoelectron spectra; these values are - 1 eV, ~ 1 eV and ~0.8eV for NbS2CI2, NbSzBr 2 and NbSzI2, respectively. Indirect transitions between Ag and B3u bands involving phonons n = 2 with Bag symmetry are expected to be allowed only for light polarized with the electric vector parallel to Y Therefore, these indirect transitions will be forbidden for light polarized with the electric vector parallel to Z, as is observed.

Direct transitions The forbidden direct transitions are assigned to transitions from the Nb(4d~:)-Ag band to the S (3p*)Big band. These transitions are parity forbidden. Moreover, the splitting between B3,(3px ) and Bag(3p*) levels is due to the direct interaction between orbitals of the t w o ( 5 2 ) 2 - ions in one cluster and to interaction between these ions due to covalent interaction with Nb orbitals. These interactions are small, as is indicated by the small frequency difference (about 4 cm- 1) between the Ag and B2u components of the S-S vibration. This explains that the difference between E s and E i is small. The peak at the low-energy side of the forbidden direct transitions, observed at low temperature, is due to the formation of direct excitons (Ak = 0), the absorption at higher energies (E > EI) arises from transitions leading to unbound direct excitons. Using the theory of direct transitions [6] and assuming a hydrogenic model, it is possible to estimate the binding energy of the direct exciton from the positions of the exciton peak relative to the threshold energy. The value of the exciton binding energy for transitions to the ground state at low temperature for the forbidden direct transitions is 28 4-_ 8 cm-a and is in reasonable agreement with the value of about 4 0 c m - i , found from the shape of the continuum close to the threshold energy of the forbidden direct transitions. The hydrogenic model allows to estimate the radius a of the excitons from the equation a 2 = hE(8~zEpEex)-a, /1 being the effective mass of the exciton E6,9 ]. Taking the electron rest mass for p, a radius of 40-50A is calculated, indicating that the electron-hole orbit of the exciton extends over several times the Nb2S 4 cluster radius. The absorption coefficient of the line at E a is (probably) too low for an exciton of allowed direct transitions. We think that the line at E a, in analogy to the absorption spectrum of ZRS3[12,13 ], may be attributed to spin-forbidden transitions 3B3,[S(3p~,)/ Nb(4dx~)] ~ IAag[Nb(4d~2)]. 5. CONCLUSIONS In this paper experimental data on the optical absorption of single crystals ofNbS2Y 2 (Y = CI, Br, I) are reported. A detailed interpretation of the data was possible in terms of indirect transitions and allowed and forbidden direct transitions between electronic energy bands. The transitions show features due to phonon fine structure and direct and indirect excitons. An interpretation of this type is characteristic for solids

384

J. RIJNSDORPand C. HAAS

with electrons in delocalized wave functions (Bloch states) forming energy bands. For the assignment of the electronic transitions and the phonons we have made use of a cluster model. It was assumed that the electronic states and phonons can be considered in first approximation as being derived from NbzS 4 clusters El4]. The interaction between the clusters, which leads to the formation of energy bands, is assumed to be a small perturbation on the cluster states. It was found that the polarization behaviour of the transitions can be interpreted quite satisfactorily with such a cluster model. This intermediate character of the interpretation with elements from band theory, with extended electronic states and localized molecular orbitals of the clusters indicates that the compounds NbS2Y 2 (Y = C1, Br, I) can be considered as salt-like ionic compounds consisting of complex (Nb2S4) 4+ ions and halide ions. The atoms in an (Nb2S4) 4+ ion are tightly bound by covalent bonds and the clusters interact weakly with one another. REFERENCES

1. Sch~iferH. and Beckmann W., Z. Anorg. Allg. Chem. 347, 225 (1966).

2. yon Schnering H. G. and Beckmann W., Z. Anorg. Allg. Chem. 347, 231 (1966). 3. Rijnsdorp J., de Lange G. J. and Wiegers G. A., to be published. 4. Perrin C., Perrin A. and Prigent J., Bull. Soc. Chim. France 8, 3086 (1972). 5. Perrin C., PerrinA. and Caillet P., J. Chim. Phys. 70, 105 (1973). 6. McLean T. P., Progress in Semiconductors 5, 55 (1960). 7. Optical Properties of Solids (Edited by S. Nudelman and S. S. Mitra), Plenum Press, New York (1969). 8. Optical Properties of Solids (Edited by F. Abel6s). North Holland, Amsterdam (19721. 9. Knox R., Theory of Excitons, Solid State Physics, Supplement 5, Academic Press, New York (1963). 10. Rijnsdorp J., to be published. 11. Macfarlane G. G., McLean T. P., Quarrington J. E. and Roberts V., Phys. Rev. I l l , 1245 (1958). 12. Schairer W. and Shafer M. W., Phys. Status Solidi At7, 181 (1973). 13. Jellinek F., Pollak R.A. and Sharer M. W., Mat. Res. Bull. 9, 845 (1974). 14. King R. B., Transition metal cluster compounds, Progr, Inorg. Chem. 15, 287 (1973).