compounds containing alkali metals present promising thermoelectric ... K2Bi1-XSbXSe13 family of compounds [2] was done for the Sb- rich composition range ...
FTIR Reflectivity spectra of Thermoelectric K2Sb8Se13 crystals E. Hatzikraniotis1, Th. Hassapis1, Th. Kyratsi2,3, K. M. Paraskevopoulos1, M. G. Kanatzidis3 1 Physics Department – Aristotle University of Thessaloniki, 54124 Thessaloniki, GREECE 2 Department of Mechanical and Manufacturing Engineering – University of Cyprus 3 Department of Chemistry – Michigan State University - MI 48824 – USA Abstract Investigations of ternary and quaternary compounds, Bi and Sb chalcogenides, have shown that several multinary compounds containing alkali metals present promising thermoelectric properties. In this work, the FTIR reflectivity spectra of K2Sb8Se13 crystals are reported and analyzed for the first time. Optical investigations are conducted with or without polarization, in two directions, parallel and perpendicular to the needles. The observed distinct optical anisotropy investigated by means of polarization dependent reflectivity measurements is discussed with respect to the strong structural anisotropy.
The choice of alkali bismuth selenide in this work, is based on earlier investigations on ternary and quaternary chalcogenides compounds [3] which have shown that several members present promising thermoelectric properties. Alkali metals tend to create structural complexity in the crystal that can lead to complex electronic structure. Potassium members have a low symmetry monoclinic structure [4], which includes two different interconnected types of M/Se blocks. Interconnected blocks form tunnels where half of the K resides, while the other half, positionally and compositionally disordered with M lie on the block-connecting sites. These features seem to be responsible for the low thermal conductivity of this compound (~1.3 W/m.K at room temperature) [4]. In this work are reported for the first time the reflectivity spectra of K2Sb8Se13 crystals. The choice of the Sb-end member is due to its low carrier concentration, characteristic that is thus eliminating one of the main reasons that reflectivity may not have been widely used yet; the fact that optimized materials for best TE performance, are usually degenerate narrow gap semiconductors [5] and the high carrier concentration smears out the phonon signatures, making reflectivity analysis difficult. Optical investigations are conducted comparatively in two polarizations, parallel and perpendicular to the needles and the two most commonly used models for the description of the dielectric function are applied and extensively discussed.
Introduction Despite the fact that FTIR reflectivity is a non-destructive technique, widely used and valuably contributing in the understanding of semiconductors, limited attention has been drawn in the study of new promising thermoelectric (TE) compounds. Most of the work on new TE classes of materials is focused on structural, and electron-transport related properties, i.e. electrical conductivity (σ) - Seebeck coefficient (S) and thermal conductivity (κ), in the search of a higher ZT (ZT=σ.S2/κ). Fewer works appear which make use of reflectivity studies, as for example in the determination of rattler behavior. However, a better understanding in the vibrational modes may lead to a way for better tuning of the TE performance. One of the most commonly used ways in optimizing Experimental thermal conductivity and fine tuning the TE performance is The synthesis of K2Sb8Se13 was carried out by mixing through the scheme of solid solutions. In the analysis of potassium metal, antimony and selenium with 0.2wt% Se reflectivity data in the phonon spectral region, in the simplest Single crystals were well grown as highly oriented excess. of cases -one-mode behaviour-, as the concentration of dense ingots, using a modified vertical drop Bridgman components changes the optical phonon frequencies just move smoothly (almost linearly) between the two end- technique, with a diameter of 13 mm and a length of 20 mm. member structures [1]. A more complex, two-mode The highly anisotropic structure of K2M8Se13 family of behaviour, may also, arise from localized impurity modes in a compounds, results in needle-like morphology along the bcrystal being the resulting perturbation due to the substitution crystallographic axis [6], as shown in Figure 1. The purity of that is sufficiently strong, the vibrational states of host-crystal the crystals was confirmed by powder X-ray diffraction. Infrared spectra were recorded at nearly normal incidence phonon bands being split-off, producing non-propagating in the 70-500cm-1 spectral region, at room temperature, with a (and hence local) modes. Recent work from our group on K2Bi1-XSbXSe13 family of compounds [2] was done for the Sb- Bruker 113V FTIR spectrometer. The reflection coefficient rich composition range (x~6÷7). Preliminary analysis on was determined by typical sample-in-sample-out method with FTIR spectra of cold-pressed pellets has revealed a random a gold mirror as the reference. Polarized spectra of a single element isodisplacement –REI– model for the development of crystal were measured using a fixed-angle specular concecutive TO and LO modes; a mixed model behaviour, where one set reflectance accessory. For each spectrum 128 -1 . Crystals used scans were recorded with a resolution of 2cm of modes behaves like one-mode way and the other set in a for optical investigations were cleaved along the needles (btwo-mode way. It has been suggested that this model may axis) and then cut perpendicular to the b-axis, after which form a basis to explain the substantial suppression of the peak of lattice thermal conductivity in the above region of plates of 20 X 20 mm dimensions were polished within the xz-plane. stoichiometry. 1-4244-0811-3/06/$20.00 ©2006 IEEE 573 2006 International Conference on Thermoelectrics
Figure 1: SEM micrograph of the inside surface in a broken ingot. Results The IR reflectivity spectra are presented in Fig 2. The two polarizations (referred as “P” and “V”) correspond to parallel and vertical to the macroscopic needles. Solid lines in Figure 2 represents the best fit, as discussed later. As can be seen in Figure 2, IR reflectivity spectra present sharp narrow peaks in low wave numbers around 70cm-1, followed by two broader bands in the region of 100 and 150-160cm-1. A small shoulder, at about 200cm-1 is observed before the reflectivity minimum. At first the spectra were analyzed by the Kramers-Kronig method. Imaginary part of the corresponding dielectric function Im(ε), and the Im(-1/ε) are shown in figure 3. As can be seen, several discrete phonon modes can be identified, corresponding to characteristics observed in reflectivity spectra. In both polarizations, Im(ε) graphs seem similar and apart from any detailed difference in the texture and the peak height, the major feature in Figure 3a is the shift of the TO mode from 143cm-1 in the P, to 169cm-1 in the V polarization. Im(-1/ε) spectra (Figure 3b), also show similarity in the peak structure on both polarizations, and the main feature is the inversion of relative intensity in the peaks close to 195cm-1 and 206 cm-1. IR reflectivity spectra are fitted by classical dispersion theory; the complex dielectric constant is expressed in terms of the IR-active modes and in addition a Drude contribution, to encounter for the free carrier plasma contribution to reflectivity. Carrier contribution becomes evident, from the rising trend in the Im(ε) graph at low frequencies. For a successful fit for reflectivity data, 7 oscillators have been
ε (ω ) = ε ∞ + ∑ j
2 f j ⋅ ωTOj
ω
2 TOj
2 ε ∞ ⋅ ω PL 2 − ω − iγ jω ω − iγ PLω
−
2
(1)
where ε∞, refers to the high frequency contribution. The second term, Σ(...), describes the sum of all oscillatory contributions implying a restoring force (i.e. transverse optical TO modes, trapped polarons) each one with a characteristic frequency ωΤΟ,j, damping constant γTO,j and fj as the oscillator strength. The third term, refers to oscillations without restoring force such as the free-carrier contribution, which is expressed by the plasma frequency ωPL and the carrier damping constant γPL, which is related to the free carrier relaxation time (γPL~τ-1). 0.9
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considered with a 3-parameter model for the complex dielectric function ε(ω):
Reflectivity : P polarization
Spectra were analyzed by Kramers-Kronig method, and also fitted by classical dispersion theory. The complex dielectric constant is expressed in terms of a classical threeparameter model as well as the factorized (4-parameter) model. Kramers-Kronig (K-K) algorithm was optimized for increased accuracy in the high-end and low-end spectral regions.
0 400
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Figure 2: FIR reflectivity spectra at P (parallel) and V (perpendicular) to the needles polarizations. Spectra are shifted vertically for clarity. Solid line represents the best fit with the factorized model for the dielectric function. Plasma frequency is related to the concentration (n) and effective mass (m*), 2 = ω PL
n ⋅ e2 m * ε 0ε ∞
free
carrier
(2)
where e is the free electron charge, and ε0 the permitivity of the vacuum. Reflectivity, R(ω), was calculated by the Fresnel formula for near normal incidence: R (ω ) =
ε (ω ) − 1 ε (ω ) + 1
2
(3)
Each optical mode [transverse optical TO or longitudinal optical LO] is described by two parameters, frequency ωj and damping γj. The TO modes are the complex poles of ε(ω) while the LO modes are the complex roots of ε(ω)=0. in either case, the complex root or pole has the form of ΩL,T+iΓL,T, and
ω LO ,TO = Ω 2L ,T − 14 ΓL2,T
(4)
γ LO ,TO = − 12 ΓL ,T
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to encounter plasmon-phonon coupling. The other term, the factorized term in eq. 5, proposed by Kurosawa [10], is the extension of the Lyddane-Sachs-Teller (LST) relation to finite frequencies, and works well in numerous cases, including strongly damped soft modes [11]. In order to evaluate the more appropriate description for the dielectric function in K2Sb8Se13 crystals, reflectivity data were fitted by both models. Figure 4 shows an example of application for both models in the case of “P” polarization. In both models, the theoretical curves were extended to 0frequency limit, in order to examine the free carrier contribution. Non-linear least square fit (NLLS) was performed in Mathematica, ver.5. Best parameters, obtained from NLLS fit, are summarized in Table I. As can be seen in Figure 4, both models fit satisfactory the reflectivity data (Fig. 4a), however, the factorized model (eq. 5) fits far better the K-K calculations, especially in the ω>250cm-1 spectral region. Despite the quality of the fit, both models give the same values (with deviation less than 1cm-1) for the ωTO and ωLO modes, either as fit parameters in the factorized model (eq.5), or evaluated from the complex roots (eq.4) of the fitted 3-parameter model (eq.1). Discussion Space group symmetry calculations are not available neither for K2Sb8Se13 or any other member of the alkali-metal chalcogenide family of compounds. In polar vibrational modes, the electric field of the incident electromagnetic wave couples with the transverse TO optical phonon modes. The corresponding LO modes are shifted to higher frequencies higher than TO, due to Coulombic forces. In a single phonon case, the assignment of the modes is trivial. In a multi-phonon case, it is not trivial to assign a LO mode to the corresponding TO mode [12], namely, to the one with immediately lower frequency in the necessary sequence TO-LO-TO-LO on increasing frequency. 1
ω (cm-1)
As it is known [7, 8], in the case of broad reflection bands (large LO-TO splitting), it is more accurate to use a “fourparameter” or factorized model for the dielectric function,
ε (ω ) = ε ∞ ⋅ ε PL − LO ⋅ ε OSC ε PL − LO ε OSC
2 ω 2 − ω 2 − iγ L, Aω ωLB − ω 2 − iγ LBω = LA 2 ⋅ 2 − ω − iγ PLω ωT − ω 2 − iγ T ω
(5)
ω 2 , j − ω 2 − iγ L, jω = ∏ LO 2 2 j ωTO , j − ω − iγ T , jω
where, the first term (ε∞) is the high frequency contribution to the dielectric function, the second one (εPL-LO), proposed by Kukharskii [9] and used by several authors is added in order
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0.9 K2Sb8Se13 P polarization
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Figure 3: Imaginary part Im(ε) and Im(-1/ε) of the dielectric function for the two polarizations. Spectra are shifted vertically for clarity. Dots are from Kramers Kronig calculations and solid lines represent the best fit with the factorized model.
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ω (cm ) Figure 4: Best fit results for the two models presented comparatively for Reflectivity, Im(ε) and Im(-1/ε).
The LO modes, as the roots of eq.1, actually depend on the frequency and the oscillator strength of all the TO modes which are present. Furthermore, in the case of plasmon-LO phonon (PL-LO) coupling, it is known that two new peaks will appear in the Im(-1/ε) spectrum, with frequencies ωLA and ωLB. As shown by the work of Kukharskii [9], and verified by several authors [13-15], depending on the value of plasma frequency ωPL, the lower frequency mode ωLA will vary from zero (when ωPL=0) to ωTO (when ωPL»ωLO), and the higher frequency mode ωLB will vary from ωLO (when ωPL=0) to slightly higher than ωPL (when ωPL»ωLO). In all cases, the frequencies ωLA , ωLA , ωPL and ωLO are inter-related9 as:
ω LA ⋅ ω LB = ω PL ⋅ ωTO 2 2 2 2 ω LA + ω LB = ω PL + ω LO
(6)
In order to assign the LO modes which couple with free carriers in each one of the polarizations, the complex roots of eq.1 were evaluated in two limiting cases, one taking ωTO values and the corresponding oscillator strength f as obtained from fit (Table I) with ωPL, and without ωPL (ωPL=0). As stated earlier, ωLO frequencies, roots of eq.1, depend on the frequency and the oscillator strength of all TO modes and the Drude contribution. Therefore, the change in ωLO frequencies identifies the coupled PL-LO modes (Table I). As can be seen in Table I, for the V polarization, only the LO peak at 138.9cm-1 (ωPL≠0) is shifted to a lower value (135.9cm-1) when ωPL=0, which identifies that ωLB=138.9cm-1 and ωLA=33.5cm-1, as this mode vanishes when ωPL=0. Thus, the uncoupled LO mode is ωLO=135.9cm-1 and the corresponding TO mode (evaluated for the condition to ωLA≈ωTO when ωPL»ωLO) is ωTO=100.3cm-1. This mode (L3 in Table I) has the highest value of oscillator strength. For the rest modes, the ωTO ωLO couples are assigned on the basis of relative oscillator strength. For the P polarization, similar analysis shows that there are two LO modes (ωLB=122.3cm-1) and ωLB’=193.9cm-1) which are more significantly affected by the change ωPL≠0 to ωPL=0. The low frequency ωLA peak, ωLA=42.5cm-1, corresponds to the low frequency peak in figure 4b. The corresponding TO modes are ωTO=82.8cm-1 and ωTO=143.6cm1 (shown as L2 and L4 in Table Ι). Both modes are of high oscillator strength, with L2 mode stronger than L4. Electron coupling with two phonon modes (LO-PL-LO) is not in principle excludable, and has been observed in other materials, as, for example in the case of PbTe0.95S0.05 or A1– XMnXTe1–YSeY (A=Cd, Hg) alloys [16]. It is interesting that LO-PL-LO coupling is observed only in the P polarization (i.e. along the needles) and not in the V polarization. The case LO-PL-LO as due to alloying cannot be applied in K2Sb8Se13, however it is known that Potassium Bismuth-Antimony Selenides, tend to structural complexities especially in the metal sites which interconnect the two types of M/Se blocks, which is partially occupied by K, and this creates positional and compositional disorder. The two similar (in terms of coupling) L2 and L4 modes observed in the P polarization, may arise when the positional disorder is minimized leading to a -K-Sb-K-Sb- ordered sequence in the neighboring metal sites which interconnect the two types of M/Se blocks (local superstructure - band folding). From the remaining modes, two soft modes, namely L3 and L5, show slightly lower LO frequency than the corresponding TO (99.8cm-1 - 99.9cm-1 and 159.8cm-1 160.4cm-1 respectively). This unusual, are reported, case has been pointed out by Scott and Porto in the case of quartz [17], and also by Gervais and Piriou in quartz phenacite and rutile [18,19]. Physically, this situation occurs when a weak vibrational mode lies in the wing of strong mode, and may encountered in mixed crystals for a composition near a pole, or when a forbidden mode is activated. In the case of K2Sb8Se13, both weak L3 and L5 modes in the P polarization lie in the vicinity of the two strongest modes (82.8cm-1 and 146.3cm-1). Both weak L3 and L5 modes in the P polarization have almost the same frequency with the two strongest L3 and
L4 modes in the V polarization. It is believed that the weak L3 and L5 modes, actually forbidden, appear in the P polarization due to structural imperfections and local misalignment of the needles. Plasma frequency ωPL, in the in the view of PL-LO coupling gives ωPL=46.4cm-1 in the V polarization (eq.6), and ωPL=85.4cm-1 in the P polarization, taking the modified expression for plasmon-two phonon coupling [17]
ω PL =
ω LA ⋅ ω LB ⋅ ω LB ' (1) ( 2) ωTO ⋅ ωTO
(7)
The observed difference in plasma frequency, parallel and perpendicular to the needles may be attributed to a difference in the carrier effective mass in the two directions. In fact, from eq. 2, we can estimate that m⊥/m||=2.78. Recent band structure calculations on the Bi-end member [20,21] have shown that the top-most valence band lies in the A point of the Brillouin zone(*), with high anisotropy in the λij parameters of the effective mass. Taking the λ values from ref. 20, the effective mass ratio for Bi-end member is m⊥/m||=2.40. The plasmon damping factor γPL, also shown in Table I, shows an anisotropy parallel and perpendicular to the needles. The observed anisotropy γPL(⊥)=8.6cm-1 and γPL(II)=56.8cm-1 indicates a mobility ratio µ||/µ⊥≈6.6, which suggests that electrical anisotropy along the two directions is only partially due to the anisotropy in the carrier effective mass; carrier mobility drops significantly when current flows perpendicular to the needles, not only due to a heavier carrier effective masses in this direction but also due to boundary scattering, caused by the needle misalignment, inter-needle space and/or micro-cracks, because of easy cleavage along the needle direction. The much larger electrical anisotropy than the effective mass anisotropy has been also observed in the Birich members [22] (µ||=16cm2/Vs and µ⊥≈3cm2/Vs for K2Bi8XSbXSe13, x=1.6). Conclusion This work reports the growth and the reflectivity investigations in the FIR region for K2Sb8Se13 single crystals. The IR reflectivity spectra were analyzed in two polarizations, parallel and perpendicular to the needles. Both the classical (3-parameter) and factorized form for the complex dielectric function has been successfully applied. Crystal vibrational modes were analyzed with seven discrete phonon modes and the ωTO – ωLO frequencies for each mode were assigned. Carrier contribution to dielectric function, and analysis has shown that plasmon is coupled with one principle LO mode in the direction perpendicular to the needles and with two LO modes along the needles. The plasma frequency ωPL, was decoupled from the corresponding LO-modes, and the analysis on plasma frequency ωPL, and has revealed the strong anisotropy in the carrier effective mass, in close agreement with the reported band structure calculations in other isostructural members of potassium bismuth (antimony) (*)
K2Sb8Se13 is p-type at room temperature, and thus the valence band characteristics are considered
selenides [20]. The discussion on the plasmon damping factor γPL shows that carrier mobility drops significantly when current flows perpendicular to the needles, not only due to a heavier carrier effective masses in this direction but also due to boundary scattering, micro-cracks or needle misalignment, and confirms earlier work on this family of compounds. FIR reflectivity analysis has shown to be an effective method in the understanding of TE material properties and performance. Acknowledgment Authors are grateful to Mr. O.G.Ziogos for his valuable assistance in the Kramers-Kronig calculations. This work is part of the THEMATA Project supported by the Cyprus Research Promotion Foundation. MGK thanks the Office of Naval Research for financial support.
mode LA L1 L2 L3 L4 L5 L6 L7 Drud e
Table I : Dispersion parameters for the best fit to the reflectivity data of K2Sb8Se13 at the two polarizations. Highlighted cells indicate the PL-LO coupled modes Parallel to the needles Perpendicular to the needles ωLO ωLO ωΤΟ ωLO ωLO ωΤΟ f f (cm-1) (cm-1) ωPL=0 (cm-1) (cm-1) ωPL=0 42.5 33.5 12. 73.0 76.6 76.6 3.6 66.0 69.6 69.4 3 20. 82.8 123.2 121.1 0.4 74.4 74.8 75.0 9 2 0.1 99.9 99.8 99.7 12. 100.3 138.9 136.0 9 2 4.5 143.6 193.9 188.0 1.3 163.2 174.2 173.7 7 4 0.2 160.4 159.8 159.5 0.3 179.7 191.4 189.8 7 9 0.0 202.3 203.5 200.9 0.1 199.9 207.3 205.2 3 8 0.5 241.7 243.2 243.2 0.1 289.0 292.9 292.7 8 9 from eq.7 ωPL=85.4 from eq.7 ωPL=46.4 ωPL=88.1 ωPL=50.0 from eq.1 from eq.1 γPL=8.6 γPL=56.8
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