Rhombohedral to Cubic Conversion of GeTe via MnTe alloying Leads

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Rhombohedral to Cubic Conversion of GeTe via MnTe alloying Leads to Ultralow Thermal Conductivity, Electronic Band Convergence and High Thermoelectric Performance Zheng Zheng, Xianli Su, Rigui Deng, Constantinos C. Stoumpos, Hongyao Xie, Wei Liu, Yonggao Yan, Shiqiang Hao, Ctirad Uher, Chris Wolverton, Mercouri G. Kanatzidis, and Xinfeng Tang J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.7b13611 • Publication Date (Web): 19 Jan 2018 Downloaded from http://pubs.acs.org on January 19, 2018

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Journal of the American Chemical Society

Rhombohedral to Cubic Conversion of GeTe via MnTe alloying Leads to Ultralow Thermal Conductivity, Electronic Band Convergence and High Thermoelectric Performance Zheng Zheng,a Xianli Su,a,b,* Rigui Deng,a Constantinos Stoumposb, Hongyao Xie,a Wei Liu,a Yonggao Yan,a Shiqiang Hao,c Ctirad Uher,d Chris Wolverton,c Mercouri G. Kanatzidisb,c,*, and Xinfeng Tang,a,* a

State Key Laboratory of Advanced Technology for Materials Synthesis and Processing, Wuhan University of Technology, Wuhan 430070, China b c

Department of Chemistry, Northwestern University, Evanston, Illinois 60208, USA

Department of Materials science and engineering, Northwestern University, Evanston, Illinois 60208, USA d

Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA

*Corresponding authors: X. Su ([email protected]), ([email protected]), X. Tang ([email protected]).

M.

G.

K.

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Abstract In this study, a series of Ge1-xMnxTe (x=0-0.21) compounds were prepared by melting-quenching-annealing process combined with Spark Plasma Sintering (SPS). The effect of alloying MnTe into GeTe on the structure and thermoelectric properties of Ge1-xMnxTe is profound. With increasing content of MnTe, the structure of the Ge1-xMnxTe compounds gradually changes from rhombohedral to cubic, and the known R3m to Fm-3m phase transition temperature of GeTe moves from 700 K closer to room temperature. First-principles density functional theory calculations show that alloying MnTe into GeTe decreases the energy difference between the light and heavy valence bands in both the R3m and the Fm-3m structures, enhancing a multi-band character of the valence band edge that increases the hole carrier effective mass. The effect of this band convergence is a significant enhancement in the carrier effective mass from 1.44 m0 (GeTe) to 6.15 m0 (Ge0.85Mn0.15Te). In addition, alloying with MnTe decreases the phonon relaxation time by enhancing alloy scattering, and reduces the phonon velocity, and increases Ge vacancies all of which result in an ultralow lattice thermal conductivity of 0.13 Wm-1K-1 at 823 K. Subsequent doping of the Ge0.9Mn0.1Te compositions with Sb lowers the typical very high hole carrier concentration

and brings it closer to its optimal value enhancing the power factor,

which combined with the ultralow thermal conductivity yield a maximum ZT value of 1.61 at 823 K (for Ge0.86Mn0.10Sb0.04Te). The average ZT value of the compound over the temperature range 400 K-800 K is 1.09, making it the best GeTe-based thermoelectric material.

2


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Introduction Thermoelectric solid state materials have attracted tremendous attention because they promise efficient and direct conversion of thermal energy to electricity and as such, can enable a distributed technology that can augment the efforts for energy efficiency, conservation and management.1-8 The conversion process is all electronic and offers an unprecedented reliability. The thermoelectric conversion efficiency is generally evaluated by the material’s figure of merit ZT, defined as = /

( + ), where α is the Seebeck coefficient, σ is the electrical conductivity, T is the absolute temperature, κe is the electronic thermal conductivity and κL is the lattice

thermal conductivity.9-13 The performance of a thermoelectric material can be improved in a synergistic way by enhancing the electronic properties through doping,14-16 band structure modifications,17,18 introducing resonant states in the vicinity of the Fermi level,19,20 and achieving band convergence.21,22 Coupled with a successful suppression of the thermal conductivity by forming solid solutions or introducing nano-structural features that enhance phonon scattering,23-26 a superior thermoelectric material can be designed.27,28 As the Group 14 tellurides PbTe,29-32 SnTe33-36 and GeTe37,38 are narrow band gap semiconductors with high thermoelectric performance that can be utilized for power generation in the intermediate temperature region (~500-800 K). PbTe, crystalizing with a rock salt structure, has been engineered to possess excellent thermoelectric properties. For example, Tan et al.31 synthesized Na-doped Pb0.98Na0.02Te-x%SrTe alloys by non-equilibrium methods and obtained a ZT value of 2.5 at 923 K. 3



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Compared to PbTe, GeTe is a far less studied p-type narrow band gap semiconductor.39 Because of the high concentration of intrinsic Ge vacancies, the hole carrier concentration is quite high,40 on the order of 1021 cm-3, which leads to excessively high electrical conductivity (~8500 S/cm), low Seebeck coefficient (~34 µV/K), and high thermal conductivity (~8 W/mK) at room temperature.40 Therefore, the thermoelectric properties of pristine GeTe are rather poor, with the maximum ZT of less than 0.8 at 720 K. The optimization of thermoelectric properties of GeTe has historically focused mostly on suppressing the high hole carrier concentration, and on enhancing alloy scattering to lower the thermal conductivity. Considerable progress has been achieved in recent years. For instance, Perumal et al.41 reported that Sb-doped

GeTe

(rhombohedral

structure),

exhibits

decreased

hole

carrier

concentration and improved valence band convergence (decreased the energy separation between the light and heavy valance bands) while Sb doping point defect phonon scattering, reducing the lattice thermal conductivity and leading to a ZT of 1.85 at 725K. Wu et al.42 reported a large enhancement of ZT from 0.8 to 1.9 in GeTe via alloying with PbTe, which also led to improved valence band convergence, suppressed the concentration of Ge vacancies and lowered the lattice thermal conductivity. GeTe exhibits a ferroelectric phase transition at 700 K from the low temperature polar rhombohedral structure (R3m) to the high temperature cubic rock-salt structure (Fm-3m).43 For thermoelectric materials intended for high temperature power generation, phase transformations are problematic as they could speed up the 4


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deterioration in the thermoelectric performance and even lead to a mechanical failure of the device due to cracking and discontinuity in the thermal expansion coefficient. Therefore, for the long-term thermal stability and reliability of operation it is important to lower the ferroelectric phase transition temperature or even to suppress it altogether. In this work, we prepared a series of Ge1-xMnxTe (x=0-0.21) compounds by a melting-quenching technique followed by annealing and combined with spark plasma sintering (SPS), and we investigated the influence of MnTe content in GeTe on the structure and thermoelectric properties of Ge1-xMnxTe. With increasing content of Mn in the structure, Ge1-xMnxTe gradually transforms from the rhombohedral phase to the cubic phase and the transformation takes place at a progressively lower temperature. Moreover, alloying with MnTe promotes a convergence of the light and heavy valence bands of GeTe, leading to an increase in the hole effective mass. The convergence in this case occurs via two different mechanisms, first the Mn substitution lowers the maximum of light hole band relative to that of the heavy one and second the increase in crystal symmetry from rhombohedral to cubic increases the degeneracy of the valence band extrema by making several points on the Brillouin zone more equivalent. The effect of this band convergence is a large enhancement in the carrier effective mass from 1.44 m0 (GeTe) to 6.15 m0 (Ge0.85Mn0.15Te) which is reflected in an enhanced Seebeck coefficient. Alloying with MnTe also decreases the phonon relaxation time by enhancing alloying point defect scattering, and lowers the phonon velocity. This leads to an ultralow lattice thermal conductivity of 0.13 Wm-1K-1 at 823 5



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K. ZT values as high as 1.61 at 823 K were achieved for the Ge0.86Mn0.10Sb0.04Te alloy when optimized via Sb doping.

Experimental Section Compounds with the nominal composition of Ge1-xMnxTe (x=0-0.21) and Ge0.9-yMn0.1SbyTe (y=0-0.10) were synthesized by vacuum melting combined with the SPS process. High purity Ge (bulk, 99.999%), Te (bulk, 99.999%), Mn (pellet, 99.99%), and Sb (bulk, 99.99%) were weighed and mixed in stoichiometric proportions to achieve the desired composition (5 g). The mixtures were sealed in evacuated quartz tubes (diameter of 15 mm) and heated to 1373 K in 10 hours, kept at this temperature for 24 h, quenched in supersaturated salt water and then annealing at 773 K for 3 d. The obtained ingots were ground into fine powders, which were vacuum sintered using a Spark Plasma Sintering (SPS) apparatus under a pressure of 50 MPa at 773 K for 5 min to obtain fully dense bulk samples. Powder XRD analysis (PANalytical–Empyrean; Cu Kα) was used to identify phase composition of the samples. The Rietveld refinements of the XRD patterns were performed using JANA software. The morphology of the bulk samples was studied using Electron Probe Microanalysis (EPMA, JEOL JXA-8230) and High-resolution transmission electron microscopy (HRTEM, JEM-2100F, JEOL). The chemical valence of elements was determined using X-ray photoelectron spectroscopy (XPS, VG Multilab 2000; Thermo Electron Corporation). The phase transition temperature of the samples was measured by using a differential scanning calorimeter (DSC Q20; 6


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TA instrument) shown in Supplementary Figure S1. The electrical conductivity and the Seebeck coefficient were measured using a ZEM-3 apparatus (Ulvac Riko, Inc.) under a helium atmosphere from 300 to 823 K. The thermal conductivity was calculated from = , where λ is the thermal diffusivity measured in an argon atmosphere by the laser flash diffusivity method (LFA 457; Netzsch), the specific heat capacity (Cp) was calculated by Dulong-Petit law and the density (ρ) of the samples was determined by the Archimedes method and was summarized in Supplementary Table S1 which is ranged from 5.86 gcm-3 to 6.15 gcm-3 with the relative density above 95% at room temperature. The electrical conductivity (σ), the Hall coefficient (RH), and the low-temperature heat capacity (Cp) were measured using a Physical Property Measurement System (PPMS-9: Quantum Design). The carrier concentration (n) and the carrier mobility (µH) were calculated from n=1/eRH and µH= σ /ne. Electronic Band Structure Calculations: The total energies and relaxed geometries of GeTe and Ge1-xMnxTe were calculated by density functional theory (DFT)

within

the

generalized

gradient

approximation

(GGA)

of

Perdew-Burke-Ernzerhof with Projector Augmented Wave potentials.44 We use periodic boundary conditions and a plane wave basis set as implemented in the Vienna ab initio simulation package.45 The total energies were numerically converged to approximately 3 meV/cation using a basis set energy cutoff of 500 eV and dense k-meshes corresponding to 4000 k-points per reciprocal atom in the Brillouin zone. We consider up to 2 Mn atoms in GeTe 54 atom cell. For Ge25Mn2Te27, we consider multiple two Mn substitution Ge configurations and adopt the most favorable one 7



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with 2 nearest neighbor MnGe for electronic band structure calculations. Even though Mn has been explored as multivalent species (+2, +3, +4, etc.), we consider Mn in GeTe as an isovalent doping with Mn substituting for Ge.46 The scalar relativistic spin polarization effect has been considered with an initial magnetic moment of 5 µB for the substituted Mn. The substitution defects in GeTe completely change the symmetry of the original primitive cell. Thus, for the purposes of a more direct comparison with GeTe we transformed the eigenstates for defect structures into a so-called effective band structure in the primitive Brillouin zone of the parent compound GeTe using a spectral decomposition method.47

Results and discussion 1. Effect of alloying with MnTe on the structure, phase composition, band structure and thermoelectric properties of Ge1-xMnxTe 1.1 Phase composition and microstructure Figure 1(a) and (b) are the XRD patterns of Ge1-xMnxTe powders before and after SPS at room temperature. Before SPS, all samples were single phase materials within the in-house XRD detection limit. After SPS, the samples with x ≤ 0.18 were single phases having the rhombohedral structure. In samples with x > 0.18 a small amount of second phase MnTe2 was detected. The XRD of the samples with x > 0.21 after annealing is shown in Supplementary Figure S2. Typically, the presence of double peaks in the 2θ range of 23o to 27o and 41o to 45o is a characteristic feature of the rhombohedral phase. With increasing content of Mn, the double peaks gradually merge and become wide single peaks, indicating R3m results from the rock-salt 8


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Fm-3m because of the strong tendency of Ge2+ ion to move off-center from the octahedral site in order to lower the energy of the 4s2 lone pair. As Mn, which does not have a stereoactive lone pair, substitutes on the Ge site in the R3m structure, it attempts to approach the center of the position, which essentially characterizes the cubic rock salt structure. Beyond the regular XRD analysis, SEM and back scattered electron (BSE) images (see Figure 2) were carried out by EPMA for Ge1-xMnxTe (x=0-0.15). No significant difference between the bright and dark contrast was observed in any of the samples. The distribution of elements in all samples was homogeneous on a micro-scale, with no detection of any secondary phases, further confirming that when x ≤ 0.18, the samples are single phase materials, consistent with the XRD results. Figure 3 shows the typical rhombohedral and cubic structure of GeTe. The difference between the structures is that in the rhombohedral motif the Ge atom sits off-center of the octahedron formed by six Te atoms in the rhombohedral structure while in the rock-salt motif it is located in the center of the octahedron. Due to the off-center position of the Ge atom, the octahedra in the rhombohedral structure are asymmetric with six Ge-Te bonds split into three shorter bonds (red stick, 2.844 Å) and three longer bonds (green spring, 3.156 Å), while in the cubic structure all Ge-Te bonds are of equal length, as shown in Figure 3. In the rhombohedral structure, the Ge atoms form a strongly distorted octahedral (GeTe6) local structure, as shown in Figure 3(a). Due to the bond distance difference, the bond angle between the shorter Ge-Te bonds and longer Ge-Te bonds deviate from 180 or 90 deg. During the phase 9



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transition, on heating, the Ge atom shifts on average from the off-center position (1/2-x, 1/2-x, 1/2-x) to the center position (1/2, 1/2, 1/2),48 resulting in equal Ge-Te bond distances of the cubic structure, as shown in Figure 3(b). We used DSC analysis to characterize the phase transition temperature of the Ge1-xMnxTe (x=0-0.45) samples, and the results are summarized in Figure 4. As shown in Figure 4(a), the phase transition temperature from the rhombohedral to cubic structure decreases gradually from 665 K to 416 K as the content of Mn increases in the single phase region. When the content of Mn exceeds 0.18, a secondary phase is detected. The JANA software was employed for the structure refinement. Detailed information is presented in Supplementary Figure S3. With increasing content of Mn, the lattice parameter along the c-axis decreases from 10.67 Å for GeTe to 10.31 Å for Ge0.82Mn0.18Te. In contrast, the lattice parameter along the ab plane increases as the content of Mn increases. The overall effect is a decrease in the unit cell volume with increasing content of Mn because of the large decrease of the lattice parameter along the c-axis, as shown in Figure 4(b). Moreover, alloying with MnTe shortens the long Ge-Te bond distances and lengthens the short bond distances, with a concomitant increase of the Te-Ge-Te bond angle approaching the 180o of the cubic structure, see Figure 4(c). To assess the microstructure by TEM characterization, we have selected Ge0.86Mn0.10Sb0.04Te as a typical example in the series. In Figures 5(a) and 5(b) we observe a number of characteristic herringbone structures with their fish skeleton-like rows of short slanted parallel lines alternating row by row, which is a typical 10


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microtwinned structure in the GeTe system.49 The presence of herringbone structures with alternating dark and bright stripes typically implies that each domain possesses a different crystal axis, crystalizing as a merohedral twin crystal. In Figure 5(c) we can resolve the interplanar spacing of 0.34 nm, corresponding to the (0, 2, 1) plane of the rhombohedral structure of GeTe. In Figure 5(d), the region 1 is basically free of defects, and its inverse Fourier image is shown in Figure 5(e). The lattice stripes are arranged regularly, with few defects, and in the corresponding Fourier images, the diffraction spots are along straight lines. Area 2 in Figure 5(d) is rich with defects. The diffraction spots of the Fourier image, the inset in Figure 5(f), lie on a circle, and many defects, such as dislocations and stacking faults are observed in the inverse Fourier image. It is the presence of these defects that are believed to enhance phonon scattering, resulting in very low lattice thermal conductivity. Figure 6 shows X-ray photoemission spectra of Ge, Te and Mn in Ge1-xMnxTe (x=0-0.21) samples. The binding energy of the Ge-3d core state is 32.43 eV, corresponding to the 3d orbital of the Ge atom, Figure 6(a). We can confirm that Ge is in the Ge2+ state in Ge1-xMnxTe. Figure 6(b) is the photoemission spectrum of the Te-3d core state, including Te-3d3/2 and Te-3d5/2. The binding energy of Te-3d5/2 and Te-3d3/2 are 572.90 eV and 583.29 eV, respectively. The Te-3d region has well separated spin-orbit components (Δmetal =10.4 eV). In addition, the shape of the peak is approximately symmetric, and the intensity ratio of the two peaks is 3:2, indicating that the valence of Te is -2 in Ge1-xMnxTe (x=0-0.21). Figure 6(c) shows the photoemission spectrum of the Mn-2p core state, including Mn-2p1/2 and Mn-2p3/2. 11



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The binding energy of Mn-2p3/2 and Mn-2p1/2 are 641.62 eV and 653.74 eV, respectively. The Mn-2p spin-orbit components are well separated (Δmetal =11.2 eV), and the shape of the peaks is asymmetrical. We also note the presence of a satellite peak of Mn-2p3/2 observed at about 647 eV, which is the characteristic peak of Mn2+. Thus, Mn is in the Mn2+ state rather than Mn4+/3+ state in Ge1-xMnxTe. Regardless of Mn content, no chemical shift is observed in the peak position of each element, indicating that the chemical environment of Mn2+ atoms has not changed at room temperature. The evolution towards the cubic structure with increasing MnTe fraction in GeTe and its influence on the transport properties is discussed in detail below.

1.2 Thermoelectric Performance of Ge1-xMnxTe 1.2.1 Band structure of Ge1-xMnxTe Before we delve into the experimental measurements of thermoelectric transport, we present the results of DFT calculations of the electronic structure which provide useful insight on how the valence band structure is modified with the introduction of Mn in GeTe. The experimental results are then discussed in the context of the changes in the electronic structure. In order to better understand the impact of alloying of GeTe with MnTe on the electronic band structure, we calculated the band structure of Ge27-xMnxTe27 (x=0, 1, 2) compounds in both the low temperature rhombohedral structure (Figures 7(a)-7(c)) and the high temperature cubic structure (Figures 7(e)-7(g)). Brillouin zones of GeTe with low temperature rhombohedral structure and high temperature cubic structure are shown in Figure 7(d) and Figure 7(h), 12


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respectively for comparison. The band calculations show that the conduction bands are formed mainly from Ge 4p states and the valence bands mainly from Te 5p states for both the rhombohedral and rock-salt structures. In the rhombohedral Ge1-xMnxTe structure, the energy difference between the first and second valence band maximum is almost unchanged from 0.15 eV for pure GeTe to 0.14 eV in Ge26MnTe27 but decreases to 0.04 eV for Ge25Mn2Te27, indicating that the band convergence of the two valence bands is occurring. In addition, alloying with MnTe introduces new states in the band gap regardless of crystal structure, and they arise from Mn 3d states, shown in red dots in Figure 7. The Fermi level is right in the middle of the gap, however, it is well know that the presence of a large amount of Ge vacancies in GeTe shifts the Fermi level into the valence band resulting in a p-type highly degenerate semiconductor.50 At high temperatures, the stable phase is the rock-salt cubic structure. Figures 7(d)-7(f) show the band structure of the Ge27-xMnxTe27 (x=0, 1, 2) compounds in their high temperature cubic phase. The main contribution to the hole transport comes from the valence band maximum at the L and Σ points of the Brillouin zone. In pure GeTe with cubic structure, the energy difference between the valence band maxima at L and Σ points is 0.21 eV. As the content of MnTe increases, the energy difference decreases to 0.05 eV and 0.01 eV for Ge26MnTe27 and Ge25Mn2Te27, respectively. Clearly, alloying with MnTe promotes merging of the two valence bands regardless whether Ge1-xMnxTe is in the rhombohedral or cubic phase. Moreover, alloying with MnTe gives rise to extra states in the gap in both the low temperature rhombohedral phase 13



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and the high temperature cubic phase. 1.2.2 Electronic transport properties To shed light on the influence of the modified band structure on the charge transport, it is useful to compare the room temperature plots of the Seebeck coefficient S versus the carrier concentration n, in the so-called Pisarenko relation for the Ge1-xMnxTe compounds. The black dash line is calculated by using a two valence band model.37 The blue solid trend lines are calculated using a single parabolic band model and assuming that scattering is dominated by acoustic phonons, as expressed in Equation 1. =

/

∗

(1)

Here, S is the Seebeck coefficient, kB is the Boltzmann constant, h is the Planck constant, e is the electron charge, n is the carrier concentration, and m* is the effective mass. Figure 8(a) shows the Pisarenko plot for Ge1-xMnxTe compounds together with those previously reported of Sb-doped Ge1-xSbxTe and undoped GeTe. For undoped GeTe and Sb-doped Ge1-xSbxTe, the trend line of the Pisarenko plot indicates an effective mass of 1.44 m0, while for the Ge1-xMnxTe compounds, the effective mass increases dramatically with the increasing content of MnTe in the structure and the Seebeck coefficient is much larger than that predicted by two valence band model.37 For instance, the trend line indicates that the effective mass of charge carriers in Ge0.85Mn0.15Te is closer to 4.69 m0, considerably larger than the effective mass in pure GeTe. The enhanced effective mass is also experimentally verified by the low temperature heat capacity measurements which derive the Sommerfeld constant γ 14


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which is proportional to the density of states effective mass. The value of γ increases dramatically from 3.27 mJmol-1K-1 for pure GeTe to 24.38 mJmol-1K-1 for Ge0.82Mn0.18Te, so the corresponding density of states effective mass increases from 2.88 m0 for pure GeTe to 42.1 m0 for Ge0.82Mn0.18Te. Such a huge change in density of states effective mass m* indicates the participant of heavy valence band for charge transport after alloying with MnTe (based on equation 15). This value is significantly greater than the one obtained from the Seebeck coefficient analysis which gives 4.01 m0 because of the contribution of the localized magnetic moments of Mn2+ ions from the d5 orbital manifold as indicated by the band structure calculations in Figure 7. The unpaired electrons of the Mn2+ ions do not contribute to the charge transport in the materials but are contributing to the heat capacity as reflected in the high Sommerfeld parameter γ. In addition, for the Ge0.90-xMn0.10SbxTe compounds, the trend line of the Pisarenko plot indicates the effective mass of 3.47 m0, which is larger than that of the pure GeTe and the Sb doped Ge1-xSbxTe. The reason for such a rapid increase in the effective mass is the gradual convergence of the two valence band edges as the content of Mn increases, and the increasing role the heavy valence band in the carrier transport. Room temperature transport parameters for the Ge1-xMnxTe samples, including the electrical conductivity, Seebeck coefficient, carrier concentration, carrier mobility, effective mass, and the lattice thermal conductivity are shown in Table 1. At room temperature, the hole carrier concentration increases with the increasing MnTe content. Figure 8(b) shows the constituency ratio measured by using Electron Probe 15



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Microanalysis and the carrier concentration in Ge1-xMnxTe compounds as a function of the content of Mn. The overall atomic ratio between cations (Ge+Mn) and anions (Te) decreases as the Mn content increases, which means that the concentration of Ge vacancies increases, resulting in an increase in the carrier concentration from 7.84× 1020 cm-3 for pure GeTe to 4.78×1021 cm-3 for the Ge0.85Mn0.15Te compound. Figure 9(a) shows the temperature dependence of the electrical conductivity for Ge1-xMnxTe. Since compounds with MnTe content above 0.18 contain MnTe2 impurity phase, we did not measure their electronic transport properties. The electrical conductivity of all compounds decreases monotonically with increasing temperature, behaving as highly degenerate semiconductors. At room temperature, the decrease in the electrical conductivity with increasing content of Mn is attributed to increased scattering and, hence, reduced carrier mobility. The Seebeck coefficient of all compounds is positive exhibiting a p-type character with holes as the dominant charge carrier, see Figure 9(b). We should mention a slight upturn in the Seebeck coefficient near the phase transition, it is particularly evident in pure GeTe. With the increasing content of Mn, the Seebeck coefficient gradually decreases. Although the band degeneracy increases as discussed above, the significantly increased carrier concentration leads to an overall lower Seebeck coefficient. Figure 9(c) displays the power factor of the Ge1-xMnxTe compounds as a function of temperature. After alloying with MnTe, the carrier concentration becomes greater than the optimal carrier concentration (1-3×1020 cm−3)51 and charge carrier scattering is enhanced, resulting in a significant decrease in the mobility of holes. 16


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1.2.3 Thermal Conductivity Figure 10 (a) depicts the temperature dependence of the total thermal conductivity of the Ge1-xMnxTe (x=0-0.18) compounds. Pure GeTe shows a rapidly decreasing thermal conductivity with the increasing temperature until it reaches the phase transformation temperature just above 700 K. At that point, the trend is interrupted and the conductivity starts to increase. The Ge1-xMnxTe composition with x = 0.03 has a substantially weaker temperature dependence of the thermal conductivity.

The

compositions with greater content of Mn actually have a thermal conductivity that initially increases, reaches a peak value in the 500 K to 600 K range, and then decreases. The peaks fall within the temperature range where the structures undergo the phase transformation. The lattice thermal conductivity can be estimated by subtracting the electronic thermal conductivity from the measured total thermal conductivity, +

!

= − = − #

(2)

where κL is the lattice thermal conductivity, κamb is the bipolar thermal conductivity (here likely negligible since no hint of intrinsic excitations has been detected in measurements of any one of the transport parameters within the covered temperature range), κ is the total thermal conductivity, and κe is the thermal conductivity contributed by charge carriers, respectively. The latter can be estimated by the Wiedemann– Franz law: = #

(3) 17



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where σ is the electrical conductivity and L is the Lorenz number. Assuming a single parabolic band model, the Lorenz number L is calculated from52

#= $

'

%& (

%& (

+ (,) )* . (,) )*

+

%&(

−/

)*

%& (

)*

- (,)

0 1

. (,)

(4)

The reduced Fermi level η can be calculated from the temperature dependent Seebeck coefficient: =±

+

%& (

3

6 (4 ) = 7A

)*

%& (

@

89

)*

- (,) . (,)

:& ;?

− 45

(5) (6)

where kB is the Boltzmann constant, e is the elemental electron charge, s is the Seebeck coefficient, r is the scattering factor. We assume that acoustic phonon scattering as the predominant scattering mechanism, thus r = –1/2. Figure 10(b) displays the temperature dependence of the lattice thermal conductivity. The lattice thermal conductivity decreases with the increasing temperature and decreases with the increasing content of Mn due to the enhanced alloy phonon scattering. Thus, while pure GeTe at room temperature has the lattice thermal conductivity of ~3.37 Wm-1K-1, the lattice thermal conductivity of Mn0.15Ge0.85Te is only about half of that value, ~1.64 Wm-1K-1, at room temperature. At 800 K, the lattice thermal conductivities of the samples alloying with MnTe become very low and fall in the range ~0.25-0.5 Wm-1K-1. This is comparable or even lower than an estimate of the minimum lattice thermal conductivity for GeTe of about 0.5 Wm-1K-1calculated on the basis of the glass limit model of Cahill et al.,53 see Figure 10(b). The low values of the lattice thermal conductivity of Ge1-xMnxTe 18


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compounds in comparison to pure GeTe reflect the strongly enhanced alloy scattering. To further analyze the influence of introducing Mn atoms in the GeTe lattice on the thermal conductivity, we make use of the Callaway model.54,55 Assuming that the grain size is similar in all Ge1-xMnxTe samples, we only consider the influence of Umklapp scattering and point defect scattering on the lattice thermal conductivity. According to Callaway, the lattice thermal conductivities of Ge1-xMnxTe (κL) and that of pure GeTe ( ) are related via:

B

BC

=

F =

D

Rhombohedral to Cubic Conversion of GeTe via MnTe alloying Leads

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