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Single polaron and bipolaron absorption in Ti4O7 above and below the metal-insulator transition
PhysicaC 166 (1990) North-Holland
158-166
SINGLE POLARON AND BIPOLARON ABSORPTION IN Ti407 ABOVE AND BELOW THE METAL-INSULATOR TRANSITION C. RUSCHER
and N. RUFFER
Institut ftir Mmeralogie der Universitiif Hannover, Welfengarten 1. 3000 Hannover, FRG
F.-J. SEIWERT
and R. GRUEHN
tnstitut ftir anorganische und analytische Chemie, Justus-Liebig-Umversitiit Glessen, Heinrich-Buff:Ring 58, 6300 Giessen, FRG Received 1 December 1989 Revised manuscript received
8 January
1990
The normal reflectivity of Ti,O, (T= 290 K) is measured in the range of 0.1 to 3 eV. The reflectivity shows a minimum which corresponds to the collective resonance of all Ti3+ valences. The optical absorption as calculated from the diffuse reflectivity of Ti407 is investigated in the spectral range of 0.5 to 2.5 eV and at temperatures between 10 and 300 K. A broad asymmetrical absorption band with a maximum at about 0.7 eV and an absorption sideband at about 1.5eV is observed. The overall line shape is explained by the polaronic absorption mechanism. From room temperature down to the metal-insulator transition at r, zz 150 K the total polaronic absorption cross section remains constant. It drops down in the intermediate phase (125-150 K) and then increases with decreasing temperature below the Verwey transition at T v z 125 K. The increasing absorption cross section below r, as well as the shift of the peak position to lower energy is in accordance with the gap opening at z 0.2 eV previously reported (Kaplan et al. [ 1 ] ) and is explained by the separation of single polaronic from bipolaronic energy levels. The magnetic susceptibility of Ti407 is discussed with respect to an exchange pairing mechanism above and below r,,,. It is suggested that a melting of the bipolaronic state during the metal-insulator transition results in a fluctuating valence bond system above T,.
1. Introduction The discovery of high temperature superconductivity has reinitiated the discussion of electronic ground states of crystals which cannot be understood by the common Wilson band formalism. For example, the description of pair correlation effects in the resonating valence bond (RVB) theory [ 2-41 is considered to explain the electronic properties above the superconducting transition temperature (T,) of the high-jr, systems Laz_,rSrXCu04 and YBa&u@_, [ 5,6]. However, in this theory electron-phonon coupling effects are fully ignored. It is well known that an electron can interact with phonons leading to a self trapping mechanism which is described in single polaron concepts (large polaron: [ 7-91; intermediate and small polaron: [ lo- 16 ] ) . For a random lattice, Anderson [ 171 has proposed an attractive Hubbard interaction energy of two electrons on the same lattice site. This bond was 0921-4534/90/$03.50 (North-Holland )
0 Elsevier Science Publishers
B.V.
called bipolaron. Here the repulsive Coulomb energy is overcompensated by the introduced lattice distortion due to the electron-phonon coupling. Contradictory to the “Anderson bipolaron” there are also indications for next nearest neighbour (nnn) paired electrons as described for the insulating phase of VOz [ 181. The VO,-type pair is considered as “Heitler London” bond by Mott [ 19, p. 1941. A nnn-bipolaron (n-bip) theory is based on experimental observations on Ti,O, [ 20,2 11, where the Hubbard energy is assumed to be large and repulsive. T&O, shows a semiconductor to semiconductor transition at Tyz 125 K and a semiconductor to metal transition at T,,, N 150 K [ 221. Both transitions show a steep increase in the conductivity with increasing temperature, respectively. The magnetic susceptibility of Ti407 is small and temperature independent below T,.It shows an increase at T,,, and paramagnetic behaviour towards higher temperatures [ 231. ESR investigations, electrical resistivity, specific heat,
C. Riischer et al. /Single polaron and bipolaron absorption in Ti,O,
magnetic susceptibility and structural data on Ti407 [ 21,241 reveal an ordered bipolaronic state in the low temperature phase and a disordered bipolaronic state in the intermediate phase. The order-disorder transition has been discussed as a Verwey transition [ 251. It is still unclear whether there is a bipolaron liquid state in the intermediate phase of Ti407, whether the electronic system can be regarded as a “glass” state or if there occurs only a breaking of long range order into commensurate domains [ 25-27 1. Penson et al. [ 28 ] describe the stability ranges of the three Ti407 phases by order parameter theory. Using the number of pairs (n,) and the ordering of pairs (n) as corresponding order parameters the authors show that the free energy expression yields two first order phase transitions. The metallic phase corresponds in their theory to one-electron bonds. It is well known from the analysis of thermopower and magnetical susceptibility data, that there are certain deviations from the free electron approximation for the metallic phase of Ti407 [29]. Lakkis et al. [ 2 1 ] discussed the correlation enhancement of the Pauli susceptibility in terms of a Brinkman-Rice highly correlated electron gas. Penson et al. [ 28 ] showed that the concentration of bipolarons is not necessarily zero in the metallic state. In addition Riischer et al. [ 301 found polaronic like excitations by optical spectroscopy in the metallic phase of Ti407 (T=290 K) as observed in the quasi-metallic regimes of Nb02,5_-x (x>O.O2 [30]) and WO,_, (x>O.l [31]). It is the purpose of this study to investigate the metallic state of T&O,. For our analysis we have reinvestigated the temperature dependences of the electrical, magnetic and optical properties of Ti407. We present new spectroscopical results on the bipolaron state and on the metallic state of Ti407. We discuss the metallic state of Ti,O, in terms of Ti3+Ti3+ valence bond fluctuations in thermodynamical equilibrium with broken bonds instead of a free single electron approximation. This picture bases on the “van Vleck type analysis” of the magnetic susceptibility data, on our new spectroscopical results and on the knowledge of the bipolaron phases below the metal-insulator transition. The results are essential for the understanding of charge and spin fluctuations in “quasi metallic” d-metal compounds.
159
2. Experimental
Single crystals of circa 1x 1 x 1 mm3 and crystalline powders of T&O, were prepared by gaseous transport reactions with HCl in a temperature gradient. Details of the procedure are published by Seiwert and Gruehn [ 321. No impurity phases could be detected in the used samples by single crystal and powder X-ray diffraction techniques. The magnetic susceptibility and the DC conductivity were measured in the temperature range from 350 to 75 K. The experimental setups are published in detail elsewhere [ 33,341. The conductivity measurements were carried out using standard four-probe technique. The magnetic susceptibility measurements were done by the Faraday method with a field strength of Hc0.7 x lo6 A/m and calibrated by the standard method with CoHg(SCN),. Each measured point is an average of three switch on-switch off cycles of the magnetic field at a fixed temperature in the heating run. To study optical absorption we used the diffuse reflectance method [ 35 1. Spectra were taken in a single beam technique with 500 cm-’ step width. The chopped and monochromatic light beam (Zeiss M4QII) was directed onto the sample surface. The intensity of the diffuse reflected light was integrated by a white coated (MgO) shell. For temperature dependent measurements a helium closed cycle system (Leybold Heraeus) was used. The shell was placed with the sample holder on its bottom onto the top of the cold finger inside the vacuum chamber (2 X 10m6 mbar). The lOOohdiffuse standard reflection spectra was taken from MgO powder (Merck No. 5866). Each point of the reference spectra (lo) shows less than 1% deviation from the equivalent measurement as a function of temperature. Spectra in the range of 4 000 to 23 000 cm- I were measured at 100, 135,200 and 295 K. Closer temperature steps within the range of lo-300 K were taken at 6 000, 6 500, 9 000 and 11 000 cm-‘. The diffuse reflectance of Ti407 was measured from fine grained powder of the sample mixed with the white scatterer (I) relative to the pure MgO diffuse reflectivity: Rd = I/lo. The dilution was 1 mol T&O, to 300 mol MgO. The diffuse reflectivity (Rd) was recalibrated to
C. Riischer et al. /Single polaron and bipolaron absorption in Ti,O,
160
the absorption [35]:
using the Kubelka-Munk
formalism
F(&)=(l-&)2/2X&.
(1)
This method was used before in optical absorption investigation on systems like MO’ _,WXOJ, W03_X, Nb02.s_, and Nb-W-oxides and was shown to give excellent results on localised electron effects [36,31,30,37]. The normal (regular) reflectivity was measured from polished crystal surfaces. For calibration an Almirror was used. The measurements were performed using a standard microscope technique (Bruker IFS 25 and IR-microscope A590: 600-5 000 cm-‘; Zeiss spectrometer, microscope: 5 000-22 000 cm-’ ).
3. Results 3.1. Conductivity and magnetic susceptibility For comparison with results previously reported by other authors [ 2 l-24,38,39] we have depicted our results of the conductivity and magnetic susceptibility in figs. 1 and 2, respectively. For the bipolaron order-disorder transition we find a hysteresis lag over a temperature range of 105-135 K, the main jump in the conductivity for this transition can be observed at 117- 120 K. We attribute deviations of the bipolaron order/disorder transition regime with other authors (Schlenker et al. [ 391, Schlenker and Marezio [ 241, Lakkis et al. [ 2 1 ] : 133- 142 K, Inglis et al. [38]: 125-139 K, Bartholomew et al. [22]: 126-140 K) to kinetical effects. We found different temperatures in TV measuring on the same crystal with different cooling/heating times. The onset of metallic behaviour (fig. 1) occurs at about 153 K in agreement with data reported previously (149-154 K: [21,22,24,38,39] ). Inglis et al. [ 381 observed anisotropic behaviour above T, with metallic or semiconducting behaviour for different crystallographic directions. We shall not go into further details about anisotropy effects, because they are not significant for our results - although they are of some importance for optical properties as well as for localisation effects of charge carriers [ 40,41 1. Systematical deviations of the specific conductivity with
results presented by others could be a result of uncertain geometrical form factors. Our magnetic susceptibility data of Ti407 are in good agreement with results previously reported [ 2 1,23,39]. The magnetic susceptibility of the semiconducting phases are related to van Vleck orbital paramagnetism [ 2 11, indicating the bipolaronic state below T,. At the metal-insulator transition temperature T,,, (see fig. 1) there is a jump in the susceptibility (see fig. 2 ) which indicates the increase of the number of Bohr’s magnetons due to the destabilization of singlet states of bipolarons [ 2 1,25 1. The high temperature phase susceptibility is attributed to a Pauli paramagnetism with a correlation enhancement of moments [ 2 11. We shall discuss below an order-disorder model for the magnetic susceptibility. 3.2. Rejlection measurements (T= 290 K) The specular or regular reflectance (fig. 3 ) reveals a broad minimum in the range 12 000- 16 000 cm- ‘. There is some soft structure near 10 000 cm- ’ and below 5 000 cm- ‘. In the inset of fig. 3 we show the theoretical dependence of the reflectivity for normal incidence as given by the Drude model: e’=e,(1-‘4$/(y2+w2)), t”=E,(yw;/(y*w+W3)), (2) R(w) 1+
(E’*+f”*)
= 1-t (t’2+e”*)
‘I*_
[2(E’*+en2)“2+2e’]“2 l/Z+ [2(e’*+e”*) ‘/2+2e’]‘/2
( wp = plasma frequency, y= damping constant, eoa= high frequency dielectric constant). Assuming a Drude free carrier model for the description of the reflectivity of Ti407 the observed minimum in the reflectivity corresponds to the plasma frequency of approximately all Ti3+ valences. On the other hand a single Drude model cannot explain the observed line profile. The low frequency side of the minimum resembles a highly damped Drude case (see fig. 3 ), whereas for the high frequency reflectivity a weakly damped Drude case has to be assumed. Furthermore taking into account the specific conductivity (a) of the high temperature phase of Ti,O, together with a charge carrier con-
C. R&her et al. /Single polaronand bipolaronabsorptionin T&O,
-5
I
I
3
I
I
5
I
1
7
I
I
9
l/T
,
11
161
I
1
13
(1O-3 K-')
Fig. I. Arrhenius plot of the conductivity of Ti,O, in the temperature range 290-75 K. Open circles represent the heating up measurement within the hysteresis. All data below the metal-insulator transition were taken within 1 h.
l/&JO4
(emu/mol) 4
-1 -
50
100
150
200
250
300
350
Temperature (K)
Fig. 2. Inverse magnetical susceptibility of Ti,,O, as a function of temperature.
centration n x 10 x 102’ cmm3) which corresponds to the observed plasma frequency, we deduce mobilities (Jo) p=a/ne
(3)
of the order l-4 cm2/Vs (150-300 K). The same order of magnitude was found by Bartholomew and Frankl [ 22 J taking into account Hall effect measurements. These mobilities are intermediate be-
162
C. Riischer et al. /Single polaron and bipolar-on absorption in TilO,
1
F(Rd)
1.5 eV +
++
0.09
0
I
1
100
K
0 135 K , 200 K 0.08
A 290 K
+ 0.07
4+ ‘b+ lt
U.06
l
0.05
0.04
+
0’
0.03
Fig. 3. Reflectivity of Ti407 as a function of wavenumber. Two theoretical curves of a Drude free carrier model (see formula 2) for parameters t-=5.8, w,,= 12 760 cm-‘, y= 10 000 cm-’ (-) and e,=5.8. wP= 12 760 cm-r, y=3 000 cm-’ (---) are plotted in the inset.
tween those acceptable for a free electron model and small polaron picture [ 12,16 1. The free electron picture is, therefore, at the borderline of being applicable. 3.3. Optical absorption The Kubelka-Munk function of Ti407 is plotted in the spectral range of 4 000-16 000 cm-’ in fig. 4. The spectra of four different temperatures are shown. The overall feature is the appearance of a broad and asymmetric absorption band with a maximum in the range of 6 000-8 000 cm-’ and a second absorption structure at 12 000- 14 000 cm- ’ . The spectra measured at 135 K and 100 K indicate a small shift of the peak maximum towards lower energies as compared to the band maximum of the spectra taken above 150K. The temperature dependences are given in detail for 6000, 6 500, 9000 and 11000 cm-’ in fig. 5(a-d). Using the data shown in fig. 5 an increasing absorption in the low temperature phase of Ti407 is evident. In the intermediate phase between T, and T,the intensity drops whereas above T, the whole spectra remain constant with a scatter of the values on experimental reasons. Nearly no correlation to the phase transitions is observed in the high
0.02
L
I
t
I
I
1
t
4
6
8
10
12
14
Wavenumber
I lfi
(lo3 cm -')
Fig. 4. The Kubelka-Munk function of Ti407 at four different temperatures. Bars indicate position of peak maxima at 100 K ( 1) and 290 K (2) and sideband maxima (3). The arrow indicates the shift of peak maxima for decreasing temperature.
frequency tail of the peak, although there is a small increase below TV.The highest amount of increase in intensity is observed for the measurement at 6 000 cm-‘. Kaplan et al. [ 1 ] observed an optical gap of about 0.2 eV ( 1 613 cm- ’ ) below 140 K by measuring the transmission in the range of about 1 4002 500 cm-‘. Therefore, the absorption intensity somewhere below 5 000 cm-’ clearly decreases to lower intensity values crossing the high temperature curves.
4. Discussion The idea of the “next nearest neighbour” bipolaron state is well accommodated by structural details of T&O, [24,27,42,43]. The structure is built up of (Ti06)8octahedra in a rutile-type manner (fig. 6 (a) ). The oxygen array is approximately hexagonal close-packed and the Ti-ions occupy sites such that the octahedra share corners, edges and faces. The corner- and edge-shared octahedra form sheets or blocks which are truncated by the face shearing oc-
. . . . I
C. Riischeret al. /Single polaron and bipolaronabsorptionin Ti,O,
r(Rd) . l
a.
-I GO00 cm
l
. . .
“,*
.
0
. .
.
. .
0.07 L
JI_1
?O
(K)
260
60
-1
I,. 6500
Cl11
FW,,
I
0.00
F(R<,: d.
‘0 0.06
. .
.
.
.
. 0.05 J-l----_-L
20
60
r*‘, l*
-4
100
11000 cm-1
. a
.
0
II-1 180
l
.
220
’
.
260
(K)
Temperature
Fig. 5. The Kubelka-Munk function of Ti407 as a function of temperature at 6 000 (a), 6 500 (b), 9 000 (c) and I 1 000 cm-’ (d). Tv=125KandT,,,=150Kareindicatedbyarrows.
tahedra to give a fourfold modulation of the pseudorutile c-axis. Four crystallographical independent Ti sites, Ti ( 1) and Ti ( 2 ) inside the blocks and Ti ( 3 ) and Ti (4) at the end of the blocks can be outlined. The room temperature structure is assumed to consist of an average of Ti3.*+ valences (see fig. 6(b) )
163
distributed over the four Ti positions. In the low temperature phase (see fig. 6 (c) ) a separation into Ti3+ and Ti4+ can be attached to the Ti( 1 ), Ti( 3) and Ti (2), Ti (4) sites, respectively, as was concluded from changes in the Ti-0 and Ti-Ti distances. The peak profile of the room temperature absorption spectra was analysed earlier [ 301 in terms of Bryksins analytical expression for polaronic FranckCondon transitions. The superposed absorption structure in the high frequency wing ( 1.4- 1.7 eV, see fig. 4) was discussed [ 301 as a possible indication for the bipolaron state even in the metallic phase of Ti,O,. As is observed from the temperature dependence of the optical absorption the increase in the absorption intensity and the shift of the main peak maximum towards the region 0.3-0.6 eV correlates with the ordering of the Ti3+Ti3+ array. As there are no “drastical” changes in the line profile of the absorption spectra, we conclude that the main absorption mechanism remains unchanged in the metallic state as well as in the insulating phases. It is primarily the absorption cross section which changes, i.e. the probability of finding an empty nnn site. Kaplan et al. [ 1] termed transitions within the Ti( l)Ti( 3) slabs as intrinsic transitions, whereas transitions between the bipolaronic states and empty Ti( 2) and Ti (4) sites were denoted as extrinsic. They have measured only a small spectral range (0.18-0.32 eV ) and observed a small section of the low frequency part of the polaronic absorption band which they identified as extrinsic transitions. As the spectra are now known within the range of 0.18-2.8 eV as a function of temperature we explain the bandgap together with the increasing absorption intensity and the lowering of the position of the peak maximum by the separation of single polaronic levels from bipolaronic ground states. The 0.5 eV peak then has to be attributed to extrinsic transitions. Conclusions about the 1.5 eV sideband of the optical absorption of Ti,O, can be drawn from spectroscopical investigations on IWJO~.~_~[ 301 and NbOz [ 45 1. The line profiles of the optical absorption bands of Nb02.5_-x and Ti407 were found to follow the same nnn transition mechanism [ 301. For NbOz,s_, a crossover from mixed Nb5+Nb4+ to the “pure” Nb4+ valence state for NbOz can be followed by optical absorption spectroscopy. The mixed valence state shows both the 0.6 eV and the 1.5 eV ab-
164
C. R&her
et al. /Single palaron and bipolaron absorption in Ti407
Fig. 6. Structure principle of Ti407 replotted from Schlenker and Marezio [24] (a). The “valence disordered” state and the “valence ordered” insulating low temperature phase are outlined in (b) and (c), respectively (after Marezio et al. [43] ).
sorption bands. For NbOz only the 1.5 eV band becomes visible [ 45 1. NbOz has to be considered as a half tilled band case, where all sites are occupied by the “Nb4+ valence”. Therefore nnn Franck-Condon transitions are only possible into final states which are already filled. The 1.5 eV peak might be an indication for this situation in NbO*. We argue that the 1.5 eV absorption band observed in the Ti407 spectra (see fig. 4) indicates a nnn transition onto occupied sites, i.e. intrinsic transition. We shall now consider the magnetic behaviour of Ti407. From our experimental data we have concluded that there are no drastical changes in the electronic excitation mechanism above and below the metal-insulator transition. Therefore the electronic ground state configuration “as seen by light quanta” remains nearly unchanged and mainly the absorption intensity indicates the differences of the three Ti407 phases (see figs. 1, 5 (a-c) ). We now assume that the spins of two valence electrons of adjacent Ti-ions are coupled antiferromagnetically by the Heisenberg exchange effect. Within this picture only the exchange breaking mechanism can contribute to
the paramagnetic susceptibility. To analyse this effect as a possible explanation for the paramagnetic behaviour of Ti407 we have recalculated the measured magnetic susceptibility of fig. 2 in a “van Vleck” type manner. Thereby the square of the effective number of Bohr’s magnetons ( (Pan)*) is calculated from the measured molar susceptibility (xm) by:
<~r)~=xm3kTINa~~
(4)
(N, = Avogadro number, p = Bohr’s magneton ). The results shows a linear increase of ( ,u~~)* with increasing temperature together with the jump at T,. In order to show this effect compared to the contribution of all possible spins ( ( p,,,_ ) * ) as is given by the “spin only value” of Ti407, ( P”,X > =sJxs
(5)
(gz 2, gyromagnetical number; S= l/2; x=0.5, number of spins per Ti01.75) we have plotted in fig. 7 ((~Umax>2-(lleff)2)/(~,,,)2
(6)
C. R&her et al. /Single polaron and bipolaronabsorptionin T&O,
200
100
300
TEMPERATURE
165
400 (Kl
Fig. 7. Normalized square of the effective number of Bohrs magnetons ( + ) (see formulae (4-6) ) as a function of temperature. The explanation of T,, T2 and the solid and dashed lines are given in the text.
as a function of temperature. The observed temperature dependence of the normalized number of Bohr’s magnetons as calculated by eq. (6) obviously fits into a Landau order parameter behaviour (7) ( Q= order parameter) with two different transition temperatures T, and T2 above and below T,,,, respectively. From this observation we are motivated to denote formula (6) by Q2. From the linear extrapolation of Q2 down to Q2 = 0 (see arrows in fig. 7) the “virtual” transition temperatures T, and T2 are found to be 3 636 K and 1 350 K, respectively. The meaning of Q above T,,, could be the symmetry breaking of the local s= l/2 state, which is a constant local symmetry and orders into valence bonds already in the metallic state. Below T,,Q describes the single polaron/bipolaron ordering. It should be noted that a linear extrapolation of Q2 down to T= 0 K.points to 100% of bipolarons, whereas the “metallic pairs” would saturate below 100%. Lakkis et al. [ 2 1] have shown that the Pauli susceptibility data of Ti407 corresponds to a density of states of about 12 eV-’ per electron. Schlenker and Buder [29] calculated from the thermopower data
of the metallic state of Ti407 a density of states of about 20 eV-’ per electron within the free electron picture. In an ideal case of a free electron model for the metallic state of Ti407 both quantities for the density of states should be identical. Whether or not our bond model, where the nnn bonds are in thermodynamical equilibrium with broken bonds, can explain the above considered deviations from the free electron model or even reveals a description of the transport properties need further investigations.
5. Conclusion The average concentration of singlet paired states as a function of temperature can be read out of fig. 7. There are about 85O/oof pairs at 290 K and there is a jump from ca. 92% of “metallic pairs” in the metallic state to circa 98% bipolarons in the semiconducting state at 150 K. The picture that there is no drastic change in the electronic configuration has been demonstrated by optical absorption spectroscopy and is supported by the magnetic susceptibility data. As a result electron-phonon coupling is expected to be present above T,,although a complete
166
C. Riischer et al.
/Single polaron and bipolaron absorptron rn T&O,
decoupling cannot be ruled out by our experimental work. If there would be no electron-phonon coupling above T, then the valence bond state of Ti407 could resemble the “RVB”-liquid with the appearance of spinons and holons for charge and spin transport, respectively. If bipolarons are assumed to be stable above T,,, a “bipolaron liquid” or a “bipolaron gas” has to be considered for the metallic properties of T&O,. The jump of Q at 150 K is correlated with the lattice instability and the effective exchange interaction energy is increased by the separation of single polaronic levels from the bipolaronic one.
Acknowledgements The magnetic and conductivity measurements were done using the facilities of the Arrhenius Laboratory of Stockholm University. We are thankful to Prof. Nygren and colleagues for their kind support. CR thanks Dr. H. Behrens and Dr. U. Bismayer for helpful discussions. CR and NR were supported in part by SFB 173 and BMFT 13N5738 7.
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158-166
SINGLE POLARON AND BIPOLARON ABSORPTION IN Ti407 ABOVE AND BELOW THE METAL-INSULATOR TRANSITION C. RUSCHER
and N. RUFFER
Institut ftir Mmeralogie der Universitiif Hannover, Welfengarten 1. 3000 Hannover, FRG
F.-J. SEIWERT
and R. GRUEHN
tnstitut ftir anorganische und analytische Chemie, Justus-Liebig-Umversitiit Glessen, Heinrich-Buff:Ring 58, 6300 Giessen, FRG Received 1 December 1989 Revised manuscript received
8 January
1990
The normal reflectivity of Ti,O, (T= 290 K) is measured in the range of 0.1 to 3 eV. The reflectivity shows a minimum which corresponds to the collective resonance of all Ti3+ valences. The optical absorption as calculated from the diffuse reflectivity of Ti407 is investigated in the spectral range of 0.5 to 2.5 eV and at temperatures between 10 and 300 K. A broad asymmetrical absorption band with a maximum at about 0.7 eV and an absorption sideband at about 1.5eV is observed. The overall line shape is explained by the polaronic absorption mechanism. From room temperature down to the metal-insulator transition at r, zz 150 K the total polaronic absorption cross section remains constant. It drops down in the intermediate phase (125-150 K) and then increases with decreasing temperature below the Verwey transition at T v z 125 K. The increasing absorption cross section below r, as well as the shift of the peak position to lower energy is in accordance with the gap opening at z 0.2 eV previously reported (Kaplan et al. [ 1 ] ) and is explained by the separation of single polaronic from bipolaronic energy levels. The magnetic susceptibility of Ti407 is discussed with respect to an exchange pairing mechanism above and below r,,,. It is suggested that a melting of the bipolaronic state during the metal-insulator transition results in a fluctuating valence bond system above T,.
1. Introduction The discovery of high temperature superconductivity has reinitiated the discussion of electronic ground states of crystals which cannot be understood by the common Wilson band formalism. For example, the description of pair correlation effects in the resonating valence bond (RVB) theory [ 2-41 is considered to explain the electronic properties above the superconducting transition temperature (T,) of the high-jr, systems Laz_,rSrXCu04 and YBa&u@_, [ 5,6]. However, in this theory electron-phonon coupling effects are fully ignored. It is well known that an electron can interact with phonons leading to a self trapping mechanism which is described in single polaron concepts (large polaron: [ 7-91; intermediate and small polaron: [ lo- 16 ] ) . For a random lattice, Anderson [ 171 has proposed an attractive Hubbard interaction energy of two electrons on the same lattice site. This bond was 0921-4534/90/$03.50 (North-Holland )
0 Elsevier Science Publishers
B.V.
called bipolaron. Here the repulsive Coulomb energy is overcompensated by the introduced lattice distortion due to the electron-phonon coupling. Contradictory to the “Anderson bipolaron” there are also indications for next nearest neighbour (nnn) paired electrons as described for the insulating phase of VOz [ 181. The VO,-type pair is considered as “Heitler London” bond by Mott [ 19, p. 1941. A nnn-bipolaron (n-bip) theory is based on experimental observations on Ti,O, [ 20,2 11, where the Hubbard energy is assumed to be large and repulsive. T&O, shows a semiconductor to semiconductor transition at Tyz 125 K and a semiconductor to metal transition at T,,, N 150 K [ 221. Both transitions show a steep increase in the conductivity with increasing temperature, respectively. The magnetic susceptibility of Ti407 is small and temperature independent below T,.It shows an increase at T,,, and paramagnetic behaviour towards higher temperatures [ 231. ESR investigations, electrical resistivity, specific heat,
C. Riischer et al. /Single polaron and bipolaron absorption in Ti,O,
magnetic susceptibility and structural data on Ti407 [ 21,241 reveal an ordered bipolaronic state in the low temperature phase and a disordered bipolaronic state in the intermediate phase. The order-disorder transition has been discussed as a Verwey transition [ 251. It is still unclear whether there is a bipolaron liquid state in the intermediate phase of Ti407, whether the electronic system can be regarded as a “glass” state or if there occurs only a breaking of long range order into commensurate domains [ 25-27 1. Penson et al. [ 28 ] describe the stability ranges of the three Ti407 phases by order parameter theory. Using the number of pairs (n,) and the ordering of pairs (n) as corresponding order parameters the authors show that the free energy expression yields two first order phase transitions. The metallic phase corresponds in their theory to one-electron bonds. It is well known from the analysis of thermopower and magnetical susceptibility data, that there are certain deviations from the free electron approximation for the metallic phase of Ti407 [29]. Lakkis et al. [ 2 1 ] discussed the correlation enhancement of the Pauli susceptibility in terms of a Brinkman-Rice highly correlated electron gas. Penson et al. [ 28 ] showed that the concentration of bipolarons is not necessarily zero in the metallic state. In addition Riischer et al. [ 301 found polaronic like excitations by optical spectroscopy in the metallic phase of Ti407 (T=290 K) as observed in the quasi-metallic regimes of Nb02,5_-x (x>O.O2 [30]) and WO,_, (x>O.l [31]). It is the purpose of this study to investigate the metallic state of T&O,. For our analysis we have reinvestigated the temperature dependences of the electrical, magnetic and optical properties of Ti407. We present new spectroscopical results on the bipolaron state and on the metallic state of Ti407. We discuss the metallic state of Ti,O, in terms of Ti3+Ti3+ valence bond fluctuations in thermodynamical equilibrium with broken bonds instead of a free single electron approximation. This picture bases on the “van Vleck type analysis” of the magnetic susceptibility data, on our new spectroscopical results and on the knowledge of the bipolaron phases below the metal-insulator transition. The results are essential for the understanding of charge and spin fluctuations in “quasi metallic” d-metal compounds.
159
2. Experimental
Single crystals of circa 1x 1 x 1 mm3 and crystalline powders of T&O, were prepared by gaseous transport reactions with HCl in a temperature gradient. Details of the procedure are published by Seiwert and Gruehn [ 321. No impurity phases could be detected in the used samples by single crystal and powder X-ray diffraction techniques. The magnetic susceptibility and the DC conductivity were measured in the temperature range from 350 to 75 K. The experimental setups are published in detail elsewhere [ 33,341. The conductivity measurements were carried out using standard four-probe technique. The magnetic susceptibility measurements were done by the Faraday method with a field strength of Hc0.7 x lo6 A/m and calibrated by the standard method with CoHg(SCN),. Each measured point is an average of three switch on-switch off cycles of the magnetic field at a fixed temperature in the heating run. To study optical absorption we used the diffuse reflectance method [ 35 1. Spectra were taken in a single beam technique with 500 cm-’ step width. The chopped and monochromatic light beam (Zeiss M4QII) was directed onto the sample surface. The intensity of the diffuse reflected light was integrated by a white coated (MgO) shell. For temperature dependent measurements a helium closed cycle system (Leybold Heraeus) was used. The shell was placed with the sample holder on its bottom onto the top of the cold finger inside the vacuum chamber (2 X 10m6 mbar). The lOOohdiffuse standard reflection spectra was taken from MgO powder (Merck No. 5866). Each point of the reference spectra (lo) shows less than 1% deviation from the equivalent measurement as a function of temperature. Spectra in the range of 4 000 to 23 000 cm- I were measured at 100, 135,200 and 295 K. Closer temperature steps within the range of lo-300 K were taken at 6 000, 6 500, 9 000 and 11 000 cm-‘. The diffuse reflectance of Ti407 was measured from fine grained powder of the sample mixed with the white scatterer (I) relative to the pure MgO diffuse reflectivity: Rd = I/lo. The dilution was 1 mol T&O, to 300 mol MgO. The diffuse reflectivity (Rd) was recalibrated to
C. Riischer et al. /Single polaron and bipolaron absorption in Ti,O,
160
the absorption [35]:
using the Kubelka-Munk
formalism
F(&)=(l-&)2/2X&.
(1)
This method was used before in optical absorption investigation on systems like MO’ _,WXOJ, W03_X, Nb02.s_, and Nb-W-oxides and was shown to give excellent results on localised electron effects [36,31,30,37]. The normal (regular) reflectivity was measured from polished crystal surfaces. For calibration an Almirror was used. The measurements were performed using a standard microscope technique (Bruker IFS 25 and IR-microscope A590: 600-5 000 cm-‘; Zeiss spectrometer, microscope: 5 000-22 000 cm-’ ).
3. Results 3.1. Conductivity and magnetic susceptibility For comparison with results previously reported by other authors [ 2 l-24,38,39] we have depicted our results of the conductivity and magnetic susceptibility in figs. 1 and 2, respectively. For the bipolaron order-disorder transition we find a hysteresis lag over a temperature range of 105-135 K, the main jump in the conductivity for this transition can be observed at 117- 120 K. We attribute deviations of the bipolaron order/disorder transition regime with other authors (Schlenker et al. [ 391, Schlenker and Marezio [ 241, Lakkis et al. [ 2 1 ] : 133- 142 K, Inglis et al. [38]: 125-139 K, Bartholomew et al. [22]: 126-140 K) to kinetical effects. We found different temperatures in TV measuring on the same crystal with different cooling/heating times. The onset of metallic behaviour (fig. 1) occurs at about 153 K in agreement with data reported previously (149-154 K: [21,22,24,38,39] ). Inglis et al. [ 381 observed anisotropic behaviour above T, with metallic or semiconducting behaviour for different crystallographic directions. We shall not go into further details about anisotropy effects, because they are not significant for our results - although they are of some importance for optical properties as well as for localisation effects of charge carriers [ 40,41 1. Systematical deviations of the specific conductivity with
results presented by others could be a result of uncertain geometrical form factors. Our magnetic susceptibility data of Ti407 are in good agreement with results previously reported [ 2 1,23,39]. The magnetic susceptibility of the semiconducting phases are related to van Vleck orbital paramagnetism [ 2 11, indicating the bipolaronic state below T,. At the metal-insulator transition temperature T,,, (see fig. 1) there is a jump in the susceptibility (see fig. 2 ) which indicates the increase of the number of Bohr’s magnetons due to the destabilization of singlet states of bipolarons [ 2 1,25 1. The high temperature phase susceptibility is attributed to a Pauli paramagnetism with a correlation enhancement of moments [ 2 11. We shall discuss below an order-disorder model for the magnetic susceptibility. 3.2. Rejlection measurements (T= 290 K) The specular or regular reflectance (fig. 3 ) reveals a broad minimum in the range 12 000- 16 000 cm- ‘. There is some soft structure near 10 000 cm- ’ and below 5 000 cm- ‘. In the inset of fig. 3 we show the theoretical dependence of the reflectivity for normal incidence as given by the Drude model: e’=e,(1-‘4$/(y2+w2)), t”=E,(yw;/(y*w+W3)), (2) R(w) 1+
(E’*+f”*)
= 1-t (t’2+e”*)
‘I*_
[2(E’*+en2)“2+2e’]“2 l/Z+ [2(e’*+e”*) ‘/2+2e’]‘/2
( wp = plasma frequency, y= damping constant, eoa= high frequency dielectric constant). Assuming a Drude free carrier model for the description of the reflectivity of Ti407 the observed minimum in the reflectivity corresponds to the plasma frequency of approximately all Ti3+ valences. On the other hand a single Drude model cannot explain the observed line profile. The low frequency side of the minimum resembles a highly damped Drude case (see fig. 3 ), whereas for the high frequency reflectivity a weakly damped Drude case has to be assumed. Furthermore taking into account the specific conductivity (a) of the high temperature phase of Ti,O, together with a charge carrier con-
C. R&her et al. /Single polaronand bipolaronabsorptionin T&O,
-5
I
I
3
I
I
5
I
1
7
I
I
9
l/T
,
11
161
I
1
13
(1O-3 K-')
Fig. I. Arrhenius plot of the conductivity of Ti,O, in the temperature range 290-75 K. Open circles represent the heating up measurement within the hysteresis. All data below the metal-insulator transition were taken within 1 h.
l/&JO4
(emu/mol) 4
-1 -
50
100
150
200
250
300
350
Temperature (K)
Fig. 2. Inverse magnetical susceptibility of Ti,,O, as a function of temperature.
centration n x 10 x 102’ cmm3) which corresponds to the observed plasma frequency, we deduce mobilities (Jo) p=a/ne
(3)
of the order l-4 cm2/Vs (150-300 K). The same order of magnitude was found by Bartholomew and Frankl [ 22 J taking into account Hall effect measurements. These mobilities are intermediate be-
162
C. Riischer et al. /Single polaron and bipolar-on absorption in TilO,
1
F(Rd)
1.5 eV +
++
0.09
0
I
1
100
K
0 135 K , 200 K 0.08
A 290 K
+ 0.07
4+ ‘b+ lt
U.06
l
0.05
0.04
+
0’
0.03
Fig. 3. Reflectivity of Ti407 as a function of wavenumber. Two theoretical curves of a Drude free carrier model (see formula 2) for parameters t-=5.8, w,,= 12 760 cm-‘, y= 10 000 cm-’ (-) and e,=5.8. wP= 12 760 cm-r, y=3 000 cm-’ (---) are plotted in the inset.
tween those acceptable for a free electron model and small polaron picture [ 12,16 1. The free electron picture is, therefore, at the borderline of being applicable. 3.3. Optical absorption The Kubelka-Munk function of Ti407 is plotted in the spectral range of 4 000-16 000 cm-’ in fig. 4. The spectra of four different temperatures are shown. The overall feature is the appearance of a broad and asymmetric absorption band with a maximum in the range of 6 000-8 000 cm-’ and a second absorption structure at 12 000- 14 000 cm- ’ . The spectra measured at 135 K and 100 K indicate a small shift of the peak maximum towards lower energies as compared to the band maximum of the spectra taken above 150K. The temperature dependences are given in detail for 6000, 6 500, 9000 and 11000 cm-’ in fig. 5(a-d). Using the data shown in fig. 5 an increasing absorption in the low temperature phase of Ti407 is evident. In the intermediate phase between T, and T,the intensity drops whereas above T, the whole spectra remain constant with a scatter of the values on experimental reasons. Nearly no correlation to the phase transitions is observed in the high
0.02
L
I
t
I
I
1
t
4
6
8
10
12
14
Wavenumber
I lfi
(lo3 cm -')
Fig. 4. The Kubelka-Munk function of Ti407 at four different temperatures. Bars indicate position of peak maxima at 100 K ( 1) and 290 K (2) and sideband maxima (3). The arrow indicates the shift of peak maxima for decreasing temperature.
frequency tail of the peak, although there is a small increase below TV.The highest amount of increase in intensity is observed for the measurement at 6 000 cm-‘. Kaplan et al. [ 1 ] observed an optical gap of about 0.2 eV ( 1 613 cm- ’ ) below 140 K by measuring the transmission in the range of about 1 4002 500 cm-‘. Therefore, the absorption intensity somewhere below 5 000 cm-’ clearly decreases to lower intensity values crossing the high temperature curves.
4. Discussion The idea of the “next nearest neighbour” bipolaron state is well accommodated by structural details of T&O, [24,27,42,43]. The structure is built up of (Ti06)8octahedra in a rutile-type manner (fig. 6 (a) ). The oxygen array is approximately hexagonal close-packed and the Ti-ions occupy sites such that the octahedra share corners, edges and faces. The corner- and edge-shared octahedra form sheets or blocks which are truncated by the face shearing oc-
. . . . I
C. Riischeret al. /Single polaron and bipolaronabsorptionin Ti,O,
r(Rd) . l
a.
-I GO00 cm
l
. . .
“,*
.
0
. .
.
. .
0.07 L
JI_1
?O
(K)
260
60
-1
I,. 6500
Cl11
FW,,
I
0.00
F(R<,: d.
‘0 0.06
. .
.
.
.
. 0.05 J-l----_-L
20
60
r*‘, l*
-4
100
11000 cm-1
. a
.
0
II-1 180
l
.
220
’
.
260
(K)
Temperature
Fig. 5. The Kubelka-Munk function of Ti407 as a function of temperature at 6 000 (a), 6 500 (b), 9 000 (c) and I 1 000 cm-’ (d). Tv=125KandT,,,=150Kareindicatedbyarrows.
tahedra to give a fourfold modulation of the pseudorutile c-axis. Four crystallographical independent Ti sites, Ti ( 1) and Ti ( 2 ) inside the blocks and Ti ( 3 ) and Ti (4) at the end of the blocks can be outlined. The room temperature structure is assumed to consist of an average of Ti3.*+ valences (see fig. 6(b) )
163
distributed over the four Ti positions. In the low temperature phase (see fig. 6 (c) ) a separation into Ti3+ and Ti4+ can be attached to the Ti( 1 ), Ti( 3) and Ti (2), Ti (4) sites, respectively, as was concluded from changes in the Ti-0 and Ti-Ti distances. The peak profile of the room temperature absorption spectra was analysed earlier [ 301 in terms of Bryksins analytical expression for polaronic FranckCondon transitions. The superposed absorption structure in the high frequency wing ( 1.4- 1.7 eV, see fig. 4) was discussed [ 301 as a possible indication for the bipolaron state even in the metallic phase of Ti,O,. As is observed from the temperature dependence of the optical absorption the increase in the absorption intensity and the shift of the main peak maximum towards the region 0.3-0.6 eV correlates with the ordering of the Ti3+Ti3+ array. As there are no “drastical” changes in the line profile of the absorption spectra, we conclude that the main absorption mechanism remains unchanged in the metallic state as well as in the insulating phases. It is primarily the absorption cross section which changes, i.e. the probability of finding an empty nnn site. Kaplan et al. [ 1] termed transitions within the Ti( l)Ti( 3) slabs as intrinsic transitions, whereas transitions between the bipolaronic states and empty Ti( 2) and Ti (4) sites were denoted as extrinsic. They have measured only a small spectral range (0.18-0.32 eV ) and observed a small section of the low frequency part of the polaronic absorption band which they identified as extrinsic transitions. As the spectra are now known within the range of 0.18-2.8 eV as a function of temperature we explain the bandgap together with the increasing absorption intensity and the lowering of the position of the peak maximum by the separation of single polaronic levels from bipolaronic ground states. The 0.5 eV peak then has to be attributed to extrinsic transitions. Conclusions about the 1.5 eV sideband of the optical absorption of Ti,O, can be drawn from spectroscopical investigations on IWJO~.~_~[ 301 and NbOz [ 45 1. The line profiles of the optical absorption bands of Nb02.5_-x and Ti407 were found to follow the same nnn transition mechanism [ 301. For NbOz,s_, a crossover from mixed Nb5+Nb4+ to the “pure” Nb4+ valence state for NbOz can be followed by optical absorption spectroscopy. The mixed valence state shows both the 0.6 eV and the 1.5 eV ab-
164
C. R&her
et al. /Single palaron and bipolaron absorption in Ti407
Fig. 6. Structure principle of Ti407 replotted from Schlenker and Marezio [24] (a). The “valence disordered” state and the “valence ordered” insulating low temperature phase are outlined in (b) and (c), respectively (after Marezio et al. [43] ).
sorption bands. For NbOz only the 1.5 eV band becomes visible [ 45 1. NbOz has to be considered as a half tilled band case, where all sites are occupied by the “Nb4+ valence”. Therefore nnn Franck-Condon transitions are only possible into final states which are already filled. The 1.5 eV peak might be an indication for this situation in NbO*. We argue that the 1.5 eV absorption band observed in the Ti407 spectra (see fig. 4) indicates a nnn transition onto occupied sites, i.e. intrinsic transition. We shall now consider the magnetic behaviour of Ti407. From our experimental data we have concluded that there are no drastical changes in the electronic excitation mechanism above and below the metal-insulator transition. Therefore the electronic ground state configuration “as seen by light quanta” remains nearly unchanged and mainly the absorption intensity indicates the differences of the three Ti407 phases (see figs. 1, 5 (a-c) ). We now assume that the spins of two valence electrons of adjacent Ti-ions are coupled antiferromagnetically by the Heisenberg exchange effect. Within this picture only the exchange breaking mechanism can contribute to
the paramagnetic susceptibility. To analyse this effect as a possible explanation for the paramagnetic behaviour of Ti407 we have recalculated the measured magnetic susceptibility of fig. 2 in a “van Vleck” type manner. Thereby the square of the effective number of Bohr’s magnetons ( (Pan)*) is calculated from the measured molar susceptibility (xm) by:
<~r)~=xm3kTINa~~
(4)
(N, = Avogadro number, p = Bohr’s magneton ). The results shows a linear increase of ( ,u~~)* with increasing temperature together with the jump at T,. In order to show this effect compared to the contribution of all possible spins ( ( p,,,_ ) * ) as is given by the “spin only value” of Ti407, ( P”,X > =sJxs
(5)
(gz 2, gyromagnetical number; S= l/2; x=0.5, number of spins per Ti01.75) we have plotted in fig. 7 ((~Umax>2-(lleff)2)/(~,,,)2
(6)
C. R&her et al. /Single polaron and bipolaronabsorptionin T&O,
200
100
300
TEMPERATURE
165
400 (Kl
Fig. 7. Normalized square of the effective number of Bohrs magnetons ( + ) (see formulae (4-6) ) as a function of temperature. The explanation of T,, T2 and the solid and dashed lines are given in the text.
as a function of temperature. The observed temperature dependence of the normalized number of Bohr’s magnetons as calculated by eq. (6) obviously fits into a Landau order parameter behaviour (7) ( Q= order parameter) with two different transition temperatures T, and T2 above and below T,,,, respectively. From this observation we are motivated to denote formula (6) by Q2. From the linear extrapolation of Q2 down to Q2 = 0 (see arrows in fig. 7) the “virtual” transition temperatures T, and T2 are found to be 3 636 K and 1 350 K, respectively. The meaning of Q above T,,, could be the symmetry breaking of the local s= l/2 state, which is a constant local symmetry and orders into valence bonds already in the metallic state. Below T,,Q describes the single polaron/bipolaron ordering. It should be noted that a linear extrapolation of Q2 down to T= 0 K.points to 100% of bipolarons, whereas the “metallic pairs” would saturate below 100%. Lakkis et al. [ 2 1] have shown that the Pauli susceptibility data of Ti407 corresponds to a density of states of about 12 eV-’ per electron. Schlenker and Buder [29] calculated from the thermopower data
of the metallic state of Ti407 a density of states of about 20 eV-’ per electron within the free electron picture. In an ideal case of a free electron model for the metallic state of Ti407 both quantities for the density of states should be identical. Whether or not our bond model, where the nnn bonds are in thermodynamical equilibrium with broken bonds, can explain the above considered deviations from the free electron model or even reveals a description of the transport properties need further investigations.
5. Conclusion The average concentration of singlet paired states as a function of temperature can be read out of fig. 7. There are about 85O/oof pairs at 290 K and there is a jump from ca. 92% of “metallic pairs” in the metallic state to circa 98% bipolarons in the semiconducting state at 150 K. The picture that there is no drastic change in the electronic configuration has been demonstrated by optical absorption spectroscopy and is supported by the magnetic susceptibility data. As a result electron-phonon coupling is expected to be present above T,,although a complete
166
C. Riischer et al.
/Single polaron and bipolaron absorptron rn T&O,
decoupling cannot be ruled out by our experimental work. If there would be no electron-phonon coupling above T, then the valence bond state of Ti407 could resemble the “RVB”-liquid with the appearance of spinons and holons for charge and spin transport, respectively. If bipolarons are assumed to be stable above T,,, a “bipolaron liquid” or a “bipolaron gas” has to be considered for the metallic properties of T&O,. The jump of Q at 150 K is correlated with the lattice instability and the effective exchange interaction energy is increased by the separation of single polaronic levels from the bipolaronic one.
Acknowledgements The magnetic and conductivity measurements were done using the facilities of the Arrhenius Laboratory of Stockholm University. We are thankful to Prof. Nygren and colleagues for their kind support. CR thanks Dr. H. Behrens and Dr. U. Bismayer for helpful discussions. CR and NR were supported in part by SFB 173 and BMFT 13N5738 7.
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