Fe2VGa摻雜Ti及Si之電子結構跟熱電性質之研究 - 政大學術集成

全文

(1)國立政治大學理學院應用物理研究所碩士論文 Graduate Institute of Applied Physics, College of Science National Chengchi University Master Thesis. Fe2VGa 摻雜 Ti 及 Si 之電子結構跟熱電性質之研究治. 政. 大. 學. Ti and Si doped Fe2VGa. ‧. ‧ 國. 立 and thermoelectric properties of Electronic structure. n. Ch. er. io. 黃大頌. sit. y. Nat. al. i Un. v. Ta-Sung e n gHuang chi. 指導教授: 陳洋元博士 Advisor: Dr. Yang-Yuan Chen. 中華民國一 O 三年六月 June, 2014.

(2) 摘要熱電材料被視為其中一種可以解決能源問題的材料，其中具有高功率因子(power factor)的 Heusler 系統近年來被廣泛的研究。這篇論文中，我們利用取代效應探討鐵釩鎵(Fe2VGa) Heusler 系統的熱電性質以及磁性性質，其中包括鈦原子(Ti) 取代釩原子(V)跟矽原子(Si)取代鎵原子(Ga)。我們使用電弧熔煉法合成所有樣品，包括 Fe2V1-xTixGa (x = 0, 0.05, 0.1, 0.15, 0.2, 0.25) 和 Fe2VGa1-xSix (x = 0, 0.05, 0.1, 0.15, 0.2)。在 X 光繞射的分析中，我們展示了所有樣品都是 L21 的晶體結構還有. 政治大. 每個樣品的晶格常數；同時，我們利用能量分散式光譜儀揭露了樣品的化學計量. 立. 式。當取代濃度大於 0.1 時，兩個不同取代系統的功率因子(power factor=S2/ρ)皆. ‧ 國. 學. 會大幅度的提升，這現象可以歸功於能態密度中費米能階的移動。由能帶計算中我們得知鐵釩鎵系統的費米能階坐落在 pseudo gap 中，然而取代效應使費米能階. ‧. 移出 pseudo gap，進而跟能態密度有交錯，導致 Seebeck 常數上升，而功率因子. y. Nat. io. sit. 又與 Seebeck 常數成平方正比的關係，所以兩個不同取代系統的功率因子皆大幅. n. al. er. 度的提升。因為合金效應的關係，使所有有取代樣品的傳熱性都大幅度被壓抑，. Ch. i Un. v. 其中 Fe2VGa0.8Si0.2 的熱傳導性被抑制了兩倍。因為傳熱性的抑制以及同時功率. engchi. 因子的提升，使得 Fe2V0.8Ti0.2Ga 的熱電優值在 420 K 時較未被 Ti 取代之母材 Fe2VGa 提高了 10 倍。另外，我們也介由觀察樣品的磁化率以及磁化量探討了這些樣品的磁性性質。. i.

(3) Abstract Thermoelectric application has been considered as a possible solution for electric crises, and, recently, Heusler alloys have been studied for its large power factor near room temperature. In this thesis, we investigate the thermoelectric and magnetic properties of Ti-substituted (p-type) and Si-substituted (n-type) Heusler alloy Fe2VGa. All samples including Fe2V1-xTixGa (with x = 0, 0.05, 0.1, 0.15, 0.2, 0.25) and Fe2VGa1-xSix (with x = 0, 0.05, 0.1, 0.15, 0.2) are prepared through arc-melting. 政治大 lattice parameters, while the Energy-dispersive X-ray spectroscopy (EDX) reveals the 立. method. The X-ray refinement shows their L21 crystal structure and corresponding. ‧ 國. 學. stoichiometry. With proper substitution, with x > 0.1, the power factor of both systems is dramatically enhanced, which can be attributed to the Fermi level shifting.. ‧. According to the theoretical calculation for the density of state, the Fermi level of the. Nat. sit. y. un-doped sample is located within the pseudogap, while the Fermi level starts to move. n. al. er. io. out the pseudogap since the substituting effect applying, and it will consequently. i Un. v. intercept with the conduction or valence band. Due to the alloying effect, the thermal. Ch. engchi. conductivity of Fe2VGa0.8Si0.2 sample is significantly suppressed by a factor of 2. Therefore, we observed that the figure of merit (zT) in Fe2V0.8Ti0.2Ga sample is enhanced by 10 times at 420 K as compared with the parent compound Fe2VGa. Their magnetic properties are also investigated by means of susceptibility and magnetization measurements.. ii.

(4) Table of Contents 摘要................................................................................................................................. i Abstract ..........................................................................................................................ii Table of Contents ......................................................................................................... iii List of Figures ................................................................................................................ v Chapter 1 Introduction ................................................................................................... 1 Chapter 2 Basic Concepts .............................................................................................. 4. 政治大 2.1.1 Seebeck Effect ...................................................................................... 4 立. 2.1 Thermoelectric Effect ...................................................................................... 4. ‧ 國. 學. 2.1.2 Peltier Effect ......................................................................................... 6 2.1.3 Thomson Effect ..................................................................................... 7. ‧. 2.2 Electrical Conductivity .................................................................................... 9. Nat. sit. y. 2.3 Thermal conductivity ..................................................................................... 11. n. al. er. io. 2.3.1 Lattice Thermal Conductivity ............................................................. 12. i Un. v. 2.3.2 Electronic Thermal Conductivity ........................................................ 14. Ch. engchi. 2.3.3 Umklapp Scattering ............................................................................ 16 Chapter 3 Experimental Procedures............................................................................. 18 3.1 X-ray Diffraction (PANalytical X’pert Powder) ............................................ 18 3.2 Thermoelectric Properties Measurements ...................................................... 20 3.2.1 Thermal Diffusivity (LFA-457, NETZSCH) ...................................... 20 3.2.2 Seebeck Coefficient and Electrical Resistivity ................................... 23 3.3 Magnetic Susceptibility Measurements ......................................................... 24 Chapter 4 Results and Discussions .............................................................................. 27 4.1 Ti-doped Fe2VGa ........................................................................................... 28 iii.

(5) 4.1.1 Crystal Structure Analysis................................................................... 28 4.1.2 Electrical Resistivity ........................................................................... 31 4.1.3 Seebeck Coefficient ............................................................................ 34 4.1.4 Thermal Conductivity ......................................................................... 36 4.1.5 Power Factor ....................................................................................... 40 4.1.6 Figure of Merit (zT) ............................................................................ 41 4.1.7 Magnetic Properties ............................................................................ 42 4.2 Si-doped Fe2VGa ........................................................................................... 45 4.2.1 Crystal Structure Analysis................................................................... 45. 治政 4.2.2 Electrical Resistivity ........................................................................... 48 大立 4.2.3 Seebeck Coefficient ............................................................................ 49 ‧ 國. 學. 4.2.4 Thermal Conductivity ......................................................................... 51. ‧. 4.2.5 Power Factor ....................................................................................... 54. sit. y. Nat. 4.2.6 Figure of Merit (zT) ............................................................................ 55. io. al. er. Chapter 5 Conclusions ................................................................................................. 56. n. References .................................................................................................................... 58. Ch. engchi. iv. i Un. v.

(6) List of Figures Fig. 1.1 Heusler-type crystal structure ........................................................................... 2 Fig. 2.1 Seebeck effect ................................................................................................... 4 Fig. 2.2 Peltier effect ...................................................................................................... 6 Fig. 2.3 Positive Thomson effect ................................................................................... 7 Fig. 2.4 Negative Thomson effect .................................................................................. 8 Fig. 2.5 The heat propagation by phonons in a uniform temperature gradient ............ 12. 政治大 Fig. 3.1 XRD photo ...................................................................................................... 18 立 Fig. 2.6 The normal process and the Umklapp process ............................................... 17. ‧ 國. 學. Fig. 3.2 The Bragg’s diffraction ................................................................................... 19 Fig. 3.3 XRD schematic diagram ................................................................................. 19. ‧. Fig. 3.4 The configuration of LFA machine................................................................. 20. Nat. sit. y. Fig. 3.5 Sample holders of LFA machine .................................................................... 21. n. al. er. io. Fig. 3.6 The illustration of thermal diffusivity in LFA machine .................................. 22. i Un. v. Fig. 3.7 The illustration of ZEM-3 machine ................................................................ 23. Ch. engchi. Fig. 3.8 Components of the MPMS system ................................................................. 24 Fig. 3.9 Josephson junction .......................................................................................... 25 Fig. 3.10 the I-V curve and the -V curve of SQUID ................................................. 26 Fig. 4.1 X-ray diffraction patterns of Fe2V1-xTixGa samples ....................................... 28 Fig. 4.2 Enlargement of the XRD pattern in (2, 2, 0) phase ........................................ 29 Fig. 4.3 Lattice parameter of Fe2V1-xTixGa samples .................................................... 30 Fig. 4.4 Electrical resistivity of Fe2V1-xTixGa samples ................................................ 31 Fig. 4.5 DOS of Fe2V1-xTixGa with x = 0 and 0.05...................................................... 33 Fig. 4.6 DOS of Fe2V1-xTixGa with x = 0.1, 0.15, 0.2, and 0.25.................................. 33 v.

(7) Fig. 4.7 Seebeck coefficient of Fe2V1-xTixGa samples ................................................ 34 Fig. 4.8 Thermal conductivity of Fe2V1-xTixGa samples ............................................. 36 Fig. 4.9 Comparisons between κ, κph, and κe of all Ti-substituted samples ................. 38 Fig. 4.10 temperature verse Ti substituting level ......................................................... 39 Fig. 4.11 Band-structure of Ti-substituted Fe2VGa samples ....................................... 39 Fig. 4.12 Power factor versus temperature in Fe2V1-xTixGa samples .......................... 40 Fig. 4.13 Figure of merit (zT) versus temperature of Fe2V1-xTixGa samples............... 41 Fig. 4.14 Magnetization versus magnetic field H in various temperatures ................. 42 Fig. 4.15 Enlargement of the M versus H at 2 K ......................................................... 43. 治政 Fig. 4.16 ZFC and FC dc susceptibility for Fe VGa.................................................... 44 大立 Fig. 4.17 ZFC and FC dc susceptibility for Fe V Ti Ga samples ............................. 44 2 2. 1-x. x. ‧ 國. 學. Fig. 4.18 X-ray diffraction patterns of Fe2VGa1-xSix samples ..................................... 45. ‧. Fig. 4.19 Enlargement of the XRD pattern at (2, 2, 0) direction ................................. 46. sit. y. Nat. Fig. 4.20 Lattice parameter of Fe2VGa1-xSix samples .................................................. 47. io. er. Fig. 4.21 Electrical resistivity of Fe2VGa1-xSix samples .............................................. 48. al. Fig. 4.22 Seebeck coefficient of Fe2VGa1-xSix samples............................................... 49. n. iv n C Fig. 4.23 Thermal conductivity of Fe Si samples ........................................... 51 h2eVGa n1-xg cx h i U Fig. 4.24 Comparisons between κ, κph, and κe of all Si-substituted samples ............... 52 Fig. 4.25 Temperature verse Si substituting level ........................................................ 53 Fig. 4.26 Power factor versus temperature of Fe2VGa1-xSix samples .......................... 54 Fig. 4.27 Figure of merit (zT) versus temperature of Fe2VGa1-xSix samples ............... 55. vi.

(8) Chapter 1 Introduction For decades, scientists have been eager to search alternative energies that have vast amounts of sources, and the thermoelectric application has long been considered as a solution due to its ability of directly converting waste heat into electricity. The conversion efficiency can be evaluated using the thermoelectric figure of merit zT=S2T/ρκ, where S is Seebeck coefficient, ρ is the electrical resistivity, T is the absolute temperature and κ is the thermal conductivity. Moreover, the power factor,. 政治大. PF=S2/ρ, is an important factor in the figure of merit, because it determines the power. 立. generation per unit time under a certain temperature difference. In other words,. ‧ 國. 學. although one material exhibits a high figure of merit compromised by a small power factor with a relatively small thermal conductivity, the energy generation per unit time. ‧. is still small. Since we have vast amounts of heat sources, we can rather search a large. y. Nat. n. al. er. io. figure of merit.. sit. power factor with a medium figure of merit than a small power factor having a high. Ch. engchi. i Un. v. In general, a Heusler alloy is a promising candidate for the thermoelectric power generation near the room temperature because of its high power factor value. The general form of the cubic L21 full-Heusler alloy is X2YZ (Fig. 1), where X and Y are two different transition metals and Z is usually for heavy element. It consists of four interpenetrating fcc sublattices and if one X atom is removed, then half-Heusler phase is presented. Seebeck coefficient and the electrical conductivity are both correlated with the carrier concentration, where the electrical conductivity is proportional to the carrier concentration and Seebeck coefficient is inversed, so both metal and insulator 1.

(9) are not good candidates for the thermoelectric material. Therefore, semiconductor or semimetal is a rather better choice because it combines a high Seebeck coefficient and a relatively high electrical conductivity. According to the theoretical calculation, semiconductor-like Heusler compounds can be identified by counting the number of valence electrons. If the total number of electrons is 18, it will construct a half-Heusler semiconductor; and if the total number of electrons is 24, it will form a full-Heusler semiconductor.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Fig. 1.1 Heusler-type crystal structure (http://en.wikipedia.org/wiki/Heusler_alloy#mediaviewer/File:Heusler_alloy_-_structure.png). Recently, a full-Heusler alloy, Fe2VAl, has been studied for its large power factor near room temperature.1-6 However, its thermoelectric performance is still not comparable to those of BiTe-based compounds.7 C.S. Lue and Y.K. Kuo et al.8 found that the 2.

(10) thermoelectric property of un-doped Fe2VGa could be better than un-doped Fe2VAl. Both of them possess similar Seebeck coefficient, but the electrical conductivity of Fe2VGa is one-order larger than Fe2VAl, which would exhibit a greater power factor, and the thermal conductivity of Fe2VGa is 20 % less than Fe2VAl. Moreover, Fe2VGa was theoretically predicted to be semi-metallic and nonmagnetic with a pseudogap in the density of states at the Fermi level.9 Recent NMR measurement consecutively confirms that Fe2VGa is a semimetal with a finite density of states at the Fermi level.10 However, the electrical resistivity data shows the metallic-like behavior.11. 治政 Besides, Fe VGa has been known as a weak and inhomogeneous ferromagnet, and its 大立 magnetic property is majorly due to local off-stoichiometry. However, an anomalous 2. ‧ 國. 學. GMR effect near 50% was discovered,12 and it has indicated another mechanism. ‧. which is caused by either for the specific-heat anomaly or the metal-insulator. sit. y. Nat. transition. The origin of GMR could be explained from the combination of the. io. er. semiconducting-like matrix and the magnetic clusters formed by anti-site magnetic Fe defects and local environment. Another group13 suggests that off-stoichiometric. al. n. iv n C Fe2VGa which exhibits the magnetic is the system with undercompensated h emoment ngchi U. Kondo effect due to atomic disorder. The Kondo coupling of the carriers and the magnetic clusters is not sufficient to screen the moments.. The aim of this study is to investigate the crystal structure, electrical transport properties, thermal transport properties, and the magnetic properties of Ti-substituted and Si-substituted Fe2VGa. We will also present the theoretical band-structure calculations in order to further explain the experimental results.. 3.

(11) Chapter 2 Basic Concepts 2.1 Thermoelectric Effect 2.1.1 Seebeck Effect In 1821, Thomas Johann Seebeck found that when two dissimilar metals form a circuit, with junctions at different temperatures, it would produce a voltage difference. 政治大 temperature gradient is called Seebeck emf (electromotive force). If the junctions are 立. that can drive an electric current in this circuit (Fig. 2.1). This voltage caused by the. ‧ 國. 學. held at different temperatures T1 and T2 (T1 > T2), the charge carriers would drift from high temperature to low temperature. This motion generates an electromotive force. ‧. (emf) given by ΔT = SAB(T1-T2) or SAB = ΔV/ΔT, where SAB is so called Seebeck. Nat. sit. y. coefficient between material A and material B. In general, the sign of Seebeck. n. al. er. io. coefficient can tell what the major charge carrier is. Positive sign is p-type material. i Un. v. which the major carrier is hole; and negative sign is n-type material which dominant carrier is electron.. Ch. engchi. Fig. 2.1 Seebeck effect 4.

(12) According to a theoretical calculation,14 we can deduce Seebeck coefficient from classic physics under certain assumptions. A conventional form of σ for the case of isotropic conductivity in a band is 1 3. s = -e2 ò t v 2 (. ¶f )N(E)dE ¶E. (1). , where N(E) is the density of state, f is the Fermi-Dirac function, and τ is the relaxation time. We can use the relationship. ¶f 1 = - f (1 - f ) / kBT and L2 = (t v)2 to ¶E 3. rewrite the electric conductivity. s =ò. 立. e2 L2 f (1- f )N(E)dE t k BT. 政治大. (2). Ss = -. 學. kB e2 L2 E - EF ( ) f (1- f )N(E)dE e ò t k BT kB T. io. sit. (4). n. er. Nat. p 2 kBT é 1 ¶s (E) ù S = -( ) 3e êë s (E) ¶E úû E=EF. y. When kBT << E-EF, we can deduce Seebeck coefficient as. al. (3). ‧. ‧ 國. A corresponding expression for Seebeck coefficient in the conduction band is. Ch. i Un. v. In order to preview Seebeck coefficient before doing the experiment, people usually. engchi. rewrite it as the function of the density of state, N(E), done by the band structure calculation.. p 2 k 2B é 1 ¶N(E) ù S=T 3e êë N(E) ¶E úû E=EF. 5. (5).

(13) 2.1.2 Peltier Effect The Peltier effect is the inverse of the Seebeck coefficient. That is to say, when a current flows through the junction of two different materials, heat would be either absorbed or evolved depending on the direction of the current (Fig. 2.2).. 立. 政治大. y. ‧. ‧ 國. 學. Nat. n. er. io. al. sit. Fig. 2.2 Peltier effect. Ch. i Un. v. The Peltier effect can be considered as the back-action counterpart to the Seebeck effect:. engchi. if a simple thermoelectric circuit is closed then the Seebeck effect will drive a current, which in turn (via the Peltier effect) will always transfer heat from the hot to the cold junction. The Peltier heat generated at the junction per unit time is equal to ·. Q = (P A - P B )I. (6). , where ΠA and ΠB are the Peltier coefficient of material A and B, respectively, and I is the electrical current from A to B. The Peltier coefficient represents how much heat is carried per unit time.. 6.

(14) 2.1.3 Thomson Effect The Thomson effect was predicted and subsequently observed by William Thomson in 1851. Seebeck coefficient is not constant in any temperature, and so a spatial gradient in temperature can result in a gradient in Seebeck coefficient. If the current density J is passed through a homogeneous conductor (Fig. 2.3), the heat production rate 𝑄̇ per unit volume is given by . Q  J 2  J xT. (7). 政治大 Thomson coefficient. The first term of the above equation is the Joule heating; while 立. , where ρ is the electrical resistivity, Ñ xT is the temperature gradient, and κ is the. ‧. ‧ 國. 學. the second term is so called the Thomson heating.. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. Fig. 2.3 Positive Thomson effect. 7. v.

(15) 政治大. 立. Fig. 2.4 Negative Thomson effect. ‧ 國. 學. Moreover, Thomson coefficient is different from other two thermoelectric coefficients. ‧. (Seebeck coefficient and Peltier coefficient), because it is the only one that can be. y. Nat. io. sit. measured directly in one material instead of the junction of dissimilar materials. In. n. al. er. 1854, Thomson found the relationships between the three coefficients, implying that. Ch. i Un. v. Seebeck, Peltier, and Thomson coefficients are different manifestations of one effect. The Thomson relations are. k=. engchi. dÕ - S and Õ = TS dT. (8). , where T is the absolute temperature, κ is Thomson coefficient, Π is Peltier coefficient, and S is Seebeck coefficient.. 8.

(16) 2.2 Electrical Conductivity From the theory of electromagnetism, if a particle of charge (-e) moves with velocity (v) in the presence of an electric field (E) and a magnetic field (B), then it will experience a force which is so called the Lorentz force (F). We can write down the Newton’s second law as F=m. dv = -e(E + v ´ B) . dt. (9). According to the fact that if we disregard the collision between electrons, an electron. 政治大 if we only apply the constant electric field to an electron gas which is centered at the 立. moving on the Fermi sphere with fixed energy would maintain its velocity. Therefore,. -eEt . m. (10). ‧. ‧ 國. v(t) - v(0) =. 學. origin when t = 0, we can rewrite the equation as. Moreover, we define the time (t) as the relaxation time (τ). Therefore, the current. y. Nat. e2t E. m. er. io. al. J = nqv = n. sit. density for n electrons per unit volume in the presence of the electric field is (11). n. iv n C The electric conductivity σ is defined so the electric conductivity can be h easnJg=cσE, hi U written as. ne 2   . m. (12). The quantum mechanism states that electrons in an atom cannot take on any energy value. Instead, there are fixed energy levels which the electrons can occupy, and it is impossible for the electrons to stand between levels. Moreover, the electrons seek to minimize the total energy in the material by going to lower levels. However, the Pauli exclusion principle tells us that they cannot all go to the lowest level. The electrons 9.

(17) instead fill up the band structure starting from the lowest level. The largest energy level to which the electrons have filled is called the Fermi level. Only electrons in energy levels near the Fermi level are free to jump among the partially occupied states in that region. Therefore, the position of the Fermi level with respect to the band structure is very important for the electrical conductivity.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 10. i Un. v.

(18) 2.3 Thermal conductivity In general, we can divide the thermal conductivity into the lattice part and the electronic part, which can be written down as κ = κL+κe. Unfortunately, it is hard to determine the lattice thermal conductivity directly as the mechanism inside the phonon scattering is much more complicated than the electronic thermal conductivity. Therefore, instead of directly deducing the lattice thermal conductivity, we use the Wiedemann-Franz law to determine the electronic thermal conductivity and indirectly get the lattice thermal conductivity by subtracting the electronic part from the total thermal conductivity.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 11. i Un. v.

(19) 2.3.1 Lattice Thermal Conductivity It is not an easy job to deduce the lattice thermal conductivity directly, so we would have to make some assumptions here. First, we deal with only monatomic Bravais Lattices, where the phonon spectrum has only acoustic branches. Furthermore, we also assume that the phonon dispersion relation be w = ck for all three branches in accordance with the Debye approximation.. 立. 政治大. ‧. ‧ 國. 學. io. sit. y. Nat. n. al. er. Fig. 2.5 The heat propagation by phonons in a uniform temperature gradient. Ch. engchi. i Un. v. Suppose a small temperature gradient is imposed along the x-direction (Fig.2.4), and those phonons emerging from collisions at position x are taken to contribute to the non-equilibrium energy density an amount proportional to the equilibrium energy density at temperature T (x) :. u(x) = ueq [T(x)].The net thermal current is the product. of the energy density and the x-velocity over all the places where the phonon’s last collision might have occurred. Assuming that the collision occurred a distance l = vτ from the point x0, in a direction making an angle θ to the x-axis we have. 12.

(20) j = ò vcosq u(x0 - l cosq ) 2p dq sinq = 1 ò 1 m d mvu(x0 - lm ). p. 4p. 0. 2. -1. (13). To linear order in the temperature gradient we then have j = -vl. ¶u 1 × ¶x 2. ò. æ ¶T ö ÷ è ¶x ø. m 2 d m or j = k ç -1 1. (14). , where the thermal conductivity κ is given by 1 3.   Cvl .. 立. (15). 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 13. i Un. v.

(21) 2.3.2 Electronic Thermal Conductivity Since the concept that we use to deduce the lattice thermal conductivity is much similar with the free electron gas approximation, we can directly use the relation 1 3.   Cvl and determine the electronic heat capacity Cel to obtain the electronic thermal conductivity. In order to evaluate the specific heat of a metal at temperatures small compared with TF we apply the Sommerfeld expansion to the electronic energy u and number densities n: u=. ò 0 eg(e )de +. n=. ò. m. m. 立. (16). p2 (kBT)2 g'(m )+O(T 4 ) , 6. (17). 學. ‧ 國. 0. g(e )de +. 政治大. p2 (kBT)2 [m g'(m )+ g(m )]+ O(T 4 ), 6. where g(ε) is the density of levels and μ is the chemical potential. m 0. H(e )de = ò H(e )de +(m - e )H(eF ), to rewrite the above eF. 0. ò. 0. eF 0. î. al. ü p2 (kBT )2 g'(e F )ý + (kBT )2 g(e F ) + O(T 4 ) 6 þ 6 v. p2. n e n g c hü i U (k T ) g'(e ) .. Ch. ì p2 g(e )de + í(m - e F )g(e F ) + 6 î. sit. ì. e g(e )de + e F í(m - e F )g(e F ) +. er. eF. n. n=. ò. io. u=. Nat. equations, and replace μ by εF in the terms of order T2, we find. y. ò. ‧. If we apply the expansion,. 2. B. F. (18). i. ý þ. (19). Since we are calculating the specific heat at constant density, n is independent of temperature, so the middle of the equation (18) can be determined as 0 = (m - e F )g(e F )+. p2 (kBT )2 g'(eF ) . 6. (20). Therefore, the thermal energy density is u  u0 . 2 6. (k BT ) 2 g ( F ) ,. where u0 is the energy density in the ground state. 14. (21).

(22) The specific heat Cel of the electron gas is 2 1 T  u   2 Cel     k B Tg ( F )   2 Nk B 2 TF  T  n 3. Use the relation  F  k BTF . (22). 1 me vF2 to rewrite the above equation, and we can get 2. the electronic thermal conductivity κe as 1 2    nk BT  2 2 1   v  l   nk BT . e   2 F 3  1 me v 2  3me  2k F  B  . (23).  e  2 nk B2T / 3me   2 k B2  . T    2   ne 2 / me  3e . 學. ‧ 國. 治政 We intentionally divide the electronic thermal conductivity 大 by the electrical 立 conductivity, and then we can find ‧. There we can define the Lorenz number L as. y. 2. n. er. io. al.  e 1  k B     T 3  e . sit. Nat. L. Ch. engchi. 15. (24). i Un. v. (25).

(23) 2.3.3 Umklapp Scattering Phonons can scatter through several mechanisms as they travel through the material. These scattering mechanisms are: phonon-phonon scattering, phonon-impurity scattering, phonon-electron scattering, and phonon-boundary scattering. All scattering processes can be taken into account using Matthiessen's rule. Then the combined relaxation time  C can be written as:. 1. C. . 1. U. 立. . 1. M. . 1. B. . 1.  phe. .. (26). 政治大 are due to Umklapp scattering, mass-difference. The parameters τU, τM, τB and τph-e. ‧ 國. 學. impurity scattering, boundary scattering and phonon-electron scattering, respectively. Umklapp scattering is a reflection or a translation of a wave vector to. ‧. another Brillouin zoneas a result of a scattering process. For example, an anharmonic. Nat. sit. y. phonon-phonon scattering process creates a phonon with a momentum k-vector. n. al. thermal conductivity in crystalline materials.. Ch. engchi. er. io. outside the first Brillouin zone. Umklapp scattering is one process limiting the. i Un. v. In figure 2.5, it shows the possible scattering processes of two incoming phonons with wave-vectors (k-vectors) k1 and k2 creating one outgoing phonon with a wave vector k3. As long as the sum of k1 and k2 stay inside the first Brillouin zone where k3 is the sum of the former two conserving phonon momentum, this process is called normal scattering. However, with increasing phonon momentum and thus the sum of the wave vector of k1 and k2 might point outside the Brillouin zone. In general, k-vector outside the first Brillouin zone are physically equivalent to vectors inside it and can be 16.

(24) mathematically transformed into each other by the addition of a reciprocal lattice vector G. These processes are called Umklapp scattering and change the total phonon momentum.. 立. 政治大. ‧ 國. 學. Fig. 2.6 The normal process and the Umklapp process (http://en.wikipedia.org/wiki/Umklapp_scattering#mediaviewer/File:Phonon_nu_process.png). ‧ y. Nat. sit. Furthermore, Umklapp scattering is the dominant process for thermal conductivity at. n. al. er. io. high temperatures for low defect crystals, since normal processes vary linearly with ω but Umklapp processes vary with ω2.. Ch. engchi. 17. i Un. v.

(25) Chapter 3 Experimental Procedures 3.1 X-ray Diffraction (PANalytical X’pert Powder) One of the ways to analyze the crystal structure of the material is the X-ray diffraction method. A typical machine consists of a source of radiation, a monochromator to choose the wavelength, slits to adjust to shape of the beam, and a detector to receive the reflected X-rays. (Fig. 3.1). 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. i n CFig. 3.1 XRD photo U hengchi. v. X-rays are used to produce the diffraction pattern because their wavelength λ is typically the same order of magnitude (1–100 angstroms) as the spacing d between planes in the crystal. The operation is followed by the Bragg’s law (Fig. 3.2) which is given by. 2d sin   n. (1). ,where d is the spacing between diffracting planes, λ is the wavelength of the indicated light (X-ray), and θ is the angle between the incoming X-rays and the outer 18.

(26) layer. Through the interference pattern, we can determine the crystal structure by comparing the signals and the reference.. 政治大 (http://physik2.uni-goettingen.de/research/2_hofs/methods/XRD) 立 Fig. 3.2 The Bragg’s diffraction. ‧ 國. 學. The source of the X-ray beams which we are using is Cu Kα, but it would generate. ‧. Kα1 (λ=1.54051 Å) and Kα2 (λ=1.54433 Å) simultaneously. However, the Kα2 signal. sit. y. Nat. can be eliminated by the software (HighScore Plus) analysis. To further understand. n. al. er. io. the details, the lattice constants of the Heusler compounds can be calculated from the. i Un. v. Rietveld refinement. (The instrument operational condition: voltage 45 KV, electrical. Ch. engchi. current 40 mA, the 2𝜃 range: 25。~85。).. Fig. 3.3 XRD schematic diagram 19.

(27) 3.2 Thermoelectric Properties Measurements 3.2.1 Thermal Diffusivity (LFA-457, NETZSCH) There are many methods to measure the thermal conductivity, such as the heat flow meter method, hot wire method, and the laser flash method. The reason why we choose the laser flash method is because it is cost-effective, easy-to-operate, and it covers a broad range of the temperature and the magnitude of the thermal conductivity. Therefore, we measure the thermal diffusivity instead of the thermal conductivity. 治政 directly, and the thermal conductivity (κ) can be determined 大 by 立   D C  d. (2). P. ‧ 國. 學. , where D is the thermal diffusivity, Cp is the specific heat, and d is the density of the. ‧. io. sit. y. Nat. n. al. er. compound.. Ch. engchi. i Un. v. Fig. 3.4 The configuration of LFA machine (http://www.netzsch-thermal-analysis.com/en/home.html) 20.

(28) The LFA-457 can be separated by three major parts (Fig. 3.4). First, the laser generator is at the downside of the machine, and the laser pulse is guided by a mirror to the sample carrier within the furnace. The middle part is the sample carriers covered by the forced-air-cooled high-temperature furnace. The furnace allows measurements between room temperature to 1100 0C , and the sample carriers can hold different samples for circular or square between 6 and 12.7mm (Fig. 3.5). Last, the detector is a highly sensitive MCT (Mercury Cadmium Telluride) IR-detector. It covers the temperature range from -125 to 1100 0C .. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Fig. 3.5 Sample holders of LFA machine (http://www.netzsch-thermal-analysis.com/en/home.html). 21.

(29) The measurement principle is as follows. The front side of a plane parallel solid sample is heated by a short laser pulse. The heat induced propagates through the sample and causes a temperature increase on the rear surface. This temperature rise is measured versus time using an infrared detector (Fig. 3.6). The thermal diffusivity is calculated by the Fourier equation for the heat diffusion, and it can be evaluated as the formula:. t2 D  0.1388  0.5. (3). , where τ0.5 is the half-time of the IR detector detecting the signal from minima to. 治政 maxima and t is the thickness of the sample. 大立. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Fig. 3.6 The illustration of thermal diffusivity in LFA machine. 22.

(30) 3.2.2 Seebeck Coefficient and Electrical Resistivity The ZEM-3 instrument is designed for measurement of Seebeck coefficient and the electrical resistivity of the sample simultaneously. The figure below shows the structure of the machine (Fig. 3.7).. 立. 政治大. ‧. ‧ 國. 學. Fig. 3.7 The illustration of ZEM-3 machine. Nat. n. al. er. io. sit. y. (http://www.ulvac.com/oem/technology.cfm?iid=22&tid=129&eid=78). i Un. v. The principle of the measurement is as follows. A rectangular or cylindrical sample is. Ch. engchi. set in a vertical position fixed by two current electrodes. The sample is heated by the furnace in helium atmosphere, while it is also heated by the heater which is at the lower block to create temperature gradient. Therefore, Seebeck coefficient is the ratio of the electromotive force ΔE measured by the thermocouples and the temperature difference measured by the same thermocouples. Simultaneously, the electric resistance is measured by the four-terminal method, in which an alternating current I is applied to both ends of the sample to measure the voltage drop dV between the same thermocouples.. 23.

(31) 3.3 Magnetic Susceptibility Measurements (Quantum Design MPMS SQUID Magnetometer) This instrument uses a superconducting quantum interference device (SQUID) magnetometer to monitor very small changes in magnetic flux and so discover the magnetic properties of samples. This instrument consists of the temperature control system which can precisely control the temperature in the range 2 K to 400 K, the magnet control system which can provide the magnetic field from zero to negative. 政治大 handling system, and the computer operating system. 立. and positive seven teslas, the superconducting SQUID amplifier system, the sample. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Fig. 3.8 Components of the MPMS system (http:/www.mrl.ucsb.edu/sites/default/files/mrl_docs/instruments/fundamentals.pdf). 24.

(32) Superconducting quantum interference device (SQUID) has two Josephson junctions in parallel in a superconducting loop. Figure 3.9 illustrates the structure of Josephson junction which is composed of two bulk superconductors separated by a thin insulating layer. It is possible for the cooper pairs to penetrate the energy barrier of the insulting layer if the insulating layer is thin enough. Moreover, this machine use two Josephson junctions in order to enhance the sensitivity of recognizing the magnetic field, which can enhance to 10-7~10-9 Gauss.. 立. 政治大. er. io. sit. y. ‧. ‧ 國. 學. Nat. Fig. 3.9 Josephson junction. al. If a small external magnetic field is applied to the super conducting loop, a screening. n. iv n C current Is will affect the critical current U as the current exceeds the h e nIgc, candh asi soon critical current of the Josephson junction, a voltage appears across the junction. The left side of the figure 3.10 shows the I-V curve under the external magnetic flux no and (n+1/2)o; and the right side of the figure shows the -V curve which behaves periodically. Before doing the experiment, we need to center the sample to the working area where the slope of the -V curve exhibits the greatest value. During the experiment, the working area would deviate from the position by the effect of the external magnetic field, but the negative feedback circuit will create an opposite. 25.

(33) magnetic flux to maintain the position of the working area. This feedback magnetic flux is the magnetic flux of the measuring sample.. 立. 政治大. ‧ 國. 學. Fig. 3.10 the I-V curve and the -V curve of SQUID (http://en.wikipedia.org/wiki/SQUID#mediaviewer/File:IV_curve.jpg). ‧. n. er. io. sit. y. Nat. al. Ch. engchi. 26. i Un. v.

(34) Chapter 4 Results and Discussions A good thermoelectric material should provide high figure of merit and also a large power factor. It implies that even one material processes a high figure of merit but a small power generation per unit time, we still cannot establish a good thermoelectric devise. In this chapter, the thermoelectric properties of Fe2V1-xTixGa and Fe2VGa1-xSix are performed. Besides, the crystal identifications and the magnetic properties of all samples are also showed.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 27. i Un. v.

(35) 4.1 Ti-doped Fe2VGa 4.1.1 Crystal Structure Analysis All polycrystalline samples presented here are prepared by arc melting method. The constituent elements including Fe (99.9%, Alfa Aesar), V (99.7%, Alfa Aesar), Ti (99.5%, ACROS Organics), and Ga (99.9999%, Alfa Aesar) are placing on a water cooled copper plate in a high purity argon atmosphere. Each sample is re-melted 10 times in order to create the homogeneity, but the losing weight of each sample upon. 治政 melting process is around 3%. After arc melting process, 大each sample is post annealed 立 in the vacuum-sealed (10 torr) quartz tube at 1000 C for 3 days followed by the 0. 學 ‧. 30. 40. y er. engchi. 50. 2. 60. i Un. v. 70. (422). (111). Ch. (400). n. Intensity(a.u.). io. sit. Nat. x=0 x = 0.05 x = 0.10 x = 0.15 x = 0.20 al x = 0.25. (220). furnace cooling.. ‧ 國. -6. 80. Fig. 4.1 X-ray diffraction patterns of Fe2V1-xTixGa samples. 28.

(36) Before discussing the thermoelectric properties, the examination of the crystal structure should be done first. Figure 4.1 shows the X-ray diffraction patterns of Fe2V1-xTixGa alloys, where x are 0, 0.05, 0.1, 0.15, 0.2, and 0.25, respectively. It is clear that all compounds exhibit the single-phase L21 crystal structure which is characteristic of the full-Heusler type crystal structure, and there is no second phase existed in all compounds.. In order to further examine the successful substitution of. Ti for V site, the enlargement of the (2, 2, 0) phase is showed in the figure 4.2. As we can see from the figure below, there is a trend of the peaks: the maximum peak is shifting toward small angle as the titanium substituting level is increasing. According. 治政 to the Bragg’s law: nλ=2dsinθ, the constructive diffraction 大 angle (θ) is decreasing as 立 the lattice spacing (d) is getting larger. Therefore, this is not conflicted with our ‧ 國. 學. intuition as the atomic radius of the titanium atom (147 pm) is larger than the. ‧. vanadium atom (134 pm).. Intensity (a.u.). Ch. 44.0. er. n. (220). io 43.2. sit. y. Nat. al. engchi. 2. i Un. v. x=0 x = 0.05 x = 0.10 x = 0.15 x = 0.20 x = 0.25. 44.8. Fig. 4.2 Enlargement of the XRD pattern in (2, 2, 0) phase 29. 45.6.

(37) Furthermore, the lattice parameter of Fe2V1-xTixGa compounds are calculated by the Rietveld refinement method under the software Highscore Plus and shown in figure 4.3. The lattice parameter of the un-doped sample, Fe2VGa, is around 5.7788 Å , and the lattice parameter is linearly increasing while the substituting level is increasing. Therefore, from the above evidences, the substitution in this series is successful.. Fe2V1-xTixGa. 5.795. 立. 政治大. ‧ 國. 學. 5.790 5.785. y 0.10. Ch. 0.15. x. n. al. 0.05. engchi. sit. 0.00. 0.20. er. io. 5.775. ‧. 5.780. Nat. Lattice Parameter (A). 5.800. i Un. v. Fig. 4.3 Lattice parameter of Fe2V1-xTixGa samples. 30. 0.25.

(38) 4.1.2 Electrical Resistivity The electrical resistivity of Fe2V1-xTixGa samples are measured and shown in the figure 4.4. All of them lie in the range between 0.8 μΩ m and 2.4 μΩ m from 300 K to 700 K. In this series, the vanadium atom is substituted by the titanium atom which has one more electron vacancy at the outer shell compared with the vanadium atom. Therefore, it is reasonable that Ti substitution will increase the charge carrier density concentration by accepting holes in the valence band, but it is also acceptable that the mobility of the charge carriers is decreased because the collisions between the charge. 治政 carriers is increasing. Our data shows that the electrical 大 resistivity of the un-doped 立 sample is around 1.40 μΩ m at the room temperature, and the electrical resistivity ‧ 國. 學. decreases as the substituting level increases, except x = 0.1 sample.. y. sit. al. er. 2.0. n. Resistivity ( m). ‧. io. 2.2. Nat. 2.4. 1.8. Ch. 1.6. engchi. i Un. v. x=0 x=0.05 x=0.1 x=0.15 x=0.2 x=0.25. 1.4 1.2 1.0 0.8 0.6 300. 400. 500. 600. 700. Temperature (K) Fig. 4.4 Electrical resistivity of Fe2V1-xTixGa samples. 31.

(39) For further discussing the behavior of the electrical resistivity, we can divide all different substituting levels into two groups: the first group includes x = 0 and x = 0.05, while the second group includes others. For the first group, both samples have upturns of the electrical resistivity near the room temperature, pointing out that both of them process semiconducting-like behaviors. However, for x > 0.1 samples, they process traditional metallic behavior. This phenomenon can be explained by theoretical calculation for the density of state of each sample, and the results are shown in figure 4.5 and figure 4.6. For x = 0 and x = 0.05, the position of their Fermi level lies in the pseudogap which is about 0.5 eV. However, while the substituting. 治政 level is increasing, Fermi level is moving out of the 大 pseudogap, and intercepts with 立 the valence band which creates more holes to transport the electricity. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 32. i Un. v.

(40) Fig. 4.5 DOS of Fe2V1-xTixGa with x = 0 and 0.05 (Calculated by S.W. Lin and G.Y. Guo). 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Fig. 4.6 DOS of Fe2V1-xTixGa with x = 0.1, 0.15, 0.2, and 0.25 (Calculated by S.W. Lin and G.Y. Guo). 33.

(41) 4.1.3 Seebeck Coefficient The results of the temperature dependent Seebeck coefficient of Fe2V1-xTixGa are plotted in the figure 4.7. As we known, the value of Seebeck coefficient is proportional to the value of the density of state and inversely proportional to the slope of the density of state at Fermi level, which is given by. S=-. p 2 k 2B é 1 ¶N(E) ù T . 3e êë N(E) ¶E úû E=EF. (1). Therefore, one of the ways to manipulate Seebeck coefficient is to substitute neighbor. 治政 atoms in order to optimize the position of the Fermi level. 大立 ‧ 國. -1. ‧ sit. io. 40. y. Nat. 50. n. al. er. Seebeck Coefficient (V K ). 學. 60. 30. Ch. engchi. i Un. v. x=0 x=0.05 x=0.1 x=0.15 x=0.2 x=0.25. 20. 10. 0 300. 400. 500. 600. Temperature (K) Fig. 4.7 Seebeck coefficient of Fe2V1-xTixGa samples. 34. 700.

(42) Seebeck coefficient of the un-doped sample, Fe2VGa, at the room temperature is 22 μV K-1, and it exhibits a maximum value 24 μV K-1 at 380 K, and then turns to decrease with higher temperature. This downturn of Seebeck coefficient is because of the bipolar effect. In other words, when the temperature increases, lots of the carriers will be thermally excited and jump across the energy gap to contribute the electron carriers. The total Seebeck coefficient can be determined by. S. ( S e e  S h h ) , e h. (2). so this offset will result in the decrease of Seebeck coefficient as the increase of the temperature.. 立. 政治大. ‧ 國. 學. The reason why we choose Ti to substitute for V site is because of the positive sign of Seebeck coefficient for the un-doped sample. Therefore, we try to give this system. ‧. one less electron to enhance Seebeck coefficient by the substitution of Ti for V site.. Nat. sit. y. For the substituting samples, the value of Seebeck coefficient increases initially with. n. al. er. io. the doping content, but tends to rebound with more Ti substitution onto V site. The. i Un. v. largest value of Seebeck coefficient is about 57 μV K-1 and is found in. Ch. engchi. Fe2V0.95Ti0.05Ga sample at the room temperature.. 35.

(43) 4.1.4 Thermal Conductivity The method we use to measure the temperature dependent thermal conductivity is the product of the experimental thermal diffusivity (α) data, the density (d), and the theoretical calculation of the specific heat (Cp). Each Sample is cut into a cylinder shape with 10 mm of the diameter and 2 mm of the height by the wire cutting method.. In order to evaluate the thermoelectric performance of the Ti-substituted Fe2VGa, the thermal conductivity was depicted in figure 4.8. The thermal conductivity of the. 治政 un-doped sample is around 24.3 W m K at 300 K, 大 and tends to decrease while the 立 temperature increases. The behaviors of all samples are same as the un-doped sample. -1. -1. ‧ 國. 學. Moreover, the reduction of the thermal conductivity by 40% in x = 0.25 sample at the. ‧. room temperature is found.. y. sit er. -1. al. n. -1. io. Thermal Conductivity (W m K ). Nat 26 24 22. Ch. engchi. i Un. v. 20 18. x=0 x=0.05 x=0.1 x=0.15 x=0.2 x=0.25. 16 14 12 10 300. 400. 500. 600. 700. Temperature (K) Fig. 4.8 Thermal conductivity of Fe2V1-xTixGa samples 36.

(44) For further discussing the mechanism of the thermal conductivity, we can divide it into two parts, the lattice thermal conductivity and the electronic thermal conductivity. Instead of directly deducing the lattice thermal conductivity, we use the Wiedemann-Franz law to determine the electronic thermal conductivity and indirectly get the lattice thermal conductivity by subtracting the electronic part from the total thermal conductivity. From the Wiedemann-Franz law: 𝜅𝑒 = 𝐿 ∙ 𝑇 ∙ 𝜎, we take 𝐿 = 2.45 × 10−8 𝑊 ∙ 𝛺 ∙ 𝐾 −2 , and apply the resistivity data to obtain the electronic thermal conductivity data.. 治政 Figure 4.9 shows the comparison between the lattice 大 and electronic thermal 立 conductivity. From the figures below, the lattice thermal conductivity of Fe VGa is 2. ‧ 國. 學. almost 4 times larger than the electronic part at the room temperature. However, it is. ‧. prone to decline as the temperature increases and the electronic part starts to take. sit. y. Nat. control of the thermal conductivity when the temperature is above 631 K. We. io. er. observed that when the doping content increases, the temperature which the electronic. al. part is dominated has a trend except x = 0.15 and we also plot this temperature versus. n. iv n C composition of all samples in the figure The temperature dependent behaviors of h e n4.10. gchi U the electronic thermal conductivity for x = 0.1, 0.15, 0.20, and 0.25 are similar, but the. value of x = 0.15 is relatively smaller than the rest of the samples. In order to explain this peculiar behavior, we should take a look at the theoretical band structure calculation which is simulated by VCA 3 atoms calculation. From figure 4.11, for the x = 0.15 sample, the sub-band at the X direction intercepts with Fermi level, which creates a second Fermi surface in this system. Logically, this Fermi surface would generate more carriers to conduct the electricity, but the electrical conductivity is. 37.

(45) decreasing. Therefore, we assume that this Fermi surface will change the electronic topological configuration and influence the electrical transportation.. x=0.15. e. 20. 15. 10. 5 600. 立. ‧ 國 400. e. 500. 600. e. 15. 10. 5. 700. Ch.  ph. 300. 400. i Un. engchi. v. 500. 600. Thermal Conductivity,  (W m-1 K-1). Thermal Conductivity,  (W m-1 K-1). 25.  ph. 20. e. 15. 10. 5 400. 500. 600. 700. 700. Temperature (K). x=0.1. 300. 700. 20. n. al. Temperature (K). 600. x=0.2.  ph. io. 300. 500. Temperature (K). ‧. 5. 400. x=0.05. Nat. 10. 300. 學. 15. 5. 700. 政治大. Temperature (K). 20. 10. y. 500. e 15. sit. 400.  ph. 20. er. 300. Thermal Conductivity,  (W m-1 K-1). Thermal Conductivity,  (W m-1 K-1).  ph. Thermal Conductivity,  (W m-1 K-1). Thermal Conductivity,  (W m-1 K-1). x=0 25. x=0.25. 20.  ph e. 15. 10. 5. 300. 400. 500. 600. Temperature (K). Temperature (K). Fig. 4.9 Comparisons between κ, κph, and κe of all Ti-substituted samples 38. 700.

(46) 650. Fe2V1-xTixGa. Temperature (K). 600 550 500 450 400 350 300 250 --. 0. 0.05. 0.1. 0.15. 0.2. 0.25. Composition, x Fig. 4.10 temperature verse Ti substituting level x=0. 立. 治政 x=0.05 大. ‧. ‧ 國. 學. n. al. er. io. sit. y. Nat x=0.15. x=0.1. Ch. e nx=0.2 gchi. i Un. v. x=0.25. Fig. 4.11 Band-structure of Ti-substituted Fe2VGa samples (Calculated by S.W. Lin and G.Y. Guo) 39.

(47) 4.1.5 Power Factor Although one material exhibits a high value of the figure of merit, we still cannot guarantee this material is good for the thermoelectric application. This is because the power generation per unit time is also a critical issue of the thermoelectric application. Since we have vast amounts of heat sources in our circumstance, we can rather search a large power factor with a mediate figure of merit than a small power factor having a high figure of merit. The power factor is defined as the ratio of the square of Seebeck coefficient to the electrical resistivity at a specific temperature. The temperature. 治政 dependent power factor of Ti-substituted Fe VGa samples 大 is plotted in figure 4.12. 立 The maximum value of the power factor for the un-doped sample is only 0.4 mW m 2. -1. ‧ 國. 學. K-2 at 340 K. However, when Ti is substituting for V site, the power factor is. ‧. dramatically enhanced by almost 10 times. We observed that the largest power factor. -1. -2. Power Factor (mW m K ). sit. n. al. er. io. 4. y. Nat. is around 4 mW m-1 K-2 and is found in Fe2V0.9Ti0.1Ga sample at room temperature.. Ch. 3. engchi U. v ni. x=0 x=0.05 x=0.1 x=0.15 x=0.2 x=0.25. 2. 1. 0 300. 400. 500. 600. 700. Temperature (K) Fig. 4.12 Power factor versus temperature in Fe2V1-xTixGa samples 40.

(48) 4.1.6 Figure of Merit (zT) The figure of merit is determined by the results of the power factor as well as the thermal conductivity from room temperature to 700K and is plotted in the figure 4.13. The un-doped sample exhibits low value of zT because of the combination of the small power factor and a high thermal conductivity. The largest value of zT is happened in Fe2V0.8Ti0.2Ga sample at 420 K and is about 0.067 which is almost 10 times larger than the un-doped sample. However, this value is relatively small compared with the conventional thermoelectric material. This is attributed to the huge. 治政 thermal conductivity which is almost one order larger 大than common thermoelectric 立 materials. Moreover, the maximum power factor of all samples is located at the low ‧ 國. 學. temperature region, but the thermal conductivity is decreasing while the temperature. ‧. is increasing. Although the zT of all samples is still low, we still can do further. y. sit. n. al. er. io. 0.07. Nat. research via multi-doping into this system.. 0.06 0.05. Ch. engchi. i Un. v. x=0 x=0.05 x=0.1 x=0.15 x=0.2 x=0.25. zT. 0.04 0.03 0.02 0.01 0.00 300. 400. 500. 600. 700. Temperature (K) Fig. 4.13 Figure of merit (zT) versus temperature of Fe2V1-xTixGa samples 41.

(49) 4.1.7 Magnetic Properties Except for the thermoelectric properties, we also investigate the magnetic properties of Ti-substituted Fe2VGa. Band-structure calculations15, 16 suggest that Fe2VGa is a nonmagnetic semimetal with a pseudogap at the Fermi level, but several experiments show the ferromagnetic property with a small hysteresis. Previous studies of this compound have attributed its weak ferromagnetic moment to antisite defects in the L21 structure. Moreover, another group17 suggests that Fe2VGa is characteristic of superparamagnetism and also points out that there must have two different magnetic. 學. ‧ 國. 治政 mechanisms. One of the mechanisms is the isolated Fe大 antisite defects, and another is 立 caused by the superparamagnetic moment which is three orders larger than the mechanism of the antisite defects.. y. sit er. n. M (emu). ‧. io. 0.2. 2K 20K 30K 50K al 180K. Nat. 0.4. 0.0. Ch. engchi. i Un. v. -0.2 -0.4 -8. -6. -4. -2. 0. 2. 4. 6. 8. H(T) Fig. 4.14 Magnetization versus magnetic field H in various temperatures. 42.

(50) The M vs H curves measured in various temperatures are displayed in figure 4.14. It is obvious that the behavior of the temperature 2K is different from the others, and this behavior is characteristic of the ferromagnetism. Moreover, if we enlarge the curve of the magnetization versus magnetic field, we can see a small hysteresis of 1000 Oe at T = 2 K, whereas magnetization does not show any hysteresis loops at T > 2 K. 0.4. 0.0. 政治大. 立. 學. -0.4 -0.4. -0.3. -0.2. -0.1. 0.0. 0.2. 0.3. 0.4. sit. Nat. H(T). 0.1. y. -0.2. ‧. ‧ 國. M (emu). 0.2. io. er. Fig. 4.15 Enlargement of the M versus H at 2 K. al. n. iv n C In figure 4.16, we display the zero-field (ZFC) and field cooling (FC) dc h e n gcooling chi U susceptibility for Fe2VGa sample. There is no obvious difference between the curves of ZFC and FC, and for T > 100 K the susceptibility obeys the Curie-Weiss law. Furthermore, we can calculate the Curie-Weiss temperature (θ) by fitting the χ-1 curve and we can get the paramagnetic Curie-Weiss temperature θ = -72 K. The negative paramagnetic Curie-Weiss temperature suggests the presence of Kondo-type interaction, and there is a relation between the Kondo temperature and the Currie-Weiss temperature if the sample’s magnetic moment is due to antisite defects.. 43.

(51) We also investigate the dc susceptibility of the doped samples, and we plot them in figure 4.17.. Fe2VGa ZFC Fe2VGa FC Fe2VGa ZFC. 300. 0.3 -1.  (mol*Oe*emu ). 250. -1. -1. dc(emu*mol *Oe ). 0.4. 150 100. -1. 0.2. 200. 0.1. 50. 立. Curie-Weiss temperature ~ -72 K. 政治大. 0. -100. -50. 0. 50. 100. 150. 200. 250. 300. Temperature (K) 50. 100. 150. 學. 0.0. 200. 250. T(K). 300. ‧. ‧ 國. 0.  (emu mol-1 Oe-1). sit. n. al. er. io. 0.4. y. Nat. Fig. 4.16 ZFC and FC dc susceptibility for Fe2VGa. 0.3. Ch. engchi. i Un. v. x=0 x=0.05 x=0.1 x=0.15 x=0.2 x=0.25. 0.2. 0.1. 0.0 0. 5. 10. 15. 20. 25. Temperature (K) Fig. 4.17 ZFC and FC dc susceptibility for Fe2V1-xTixGa samples 44. 30.

(52) 4.2 Si-doped Fe2VGa 4.2.1 Crystal Structure Analysis All polycrystalline samples presented here are prepared by arc-melting method. The constituent elements including Fe (99.9%, Alfa Aesar), V (99.7%, Alfa Aesar), Ga (99.9999%, Alfa Aesar), and Si (99.999%, CERAC) are placing on a water cooled copper plate in a high purity argon atmosphere. Each sample is re-melted 10 times in order to create the homogeneity, but the losing weight of each sample upon melting. 治政 process is around 3%. After arc-melting process, each大 sample is post-annealed in the 立 vacuum-sealed (10 torr) quartz tube at 1000 C for 3 days followed by the furnace -6. 0. y. sit er. (111). Ch. 30. 40. engchi. 50. 60. i Un. v. 70. (422). n. Intensity(a.u.). io. al. (400). Nat. x=0 x = 0.05 x = 0.10 x = 0.15 x = 0.20. (220). ‧. ‧ 國. 學. cooling.. 80. 2 Fig. 4.18 X-ray diffraction patterns of Fe2VGa1-xSix samples. 45.

(53) In figure 4.18, the X-ray diffraction patterns of Fe2VGa1-xSix alloys are presented, where x are 0, 0.05, 0.1, 0.15, and 0.2 respectively. All Si-substituted samples exhibit the full-Heusler type L21 crystal structure, which dominant constructive diffracting peak is located in (2, 2, 0) phase, and there is no second phase existed in all compounds. For further understanding the successful substitution of the silicon for the gallium site, the enlargement of the (2, 2, 0) phase is showed in the figure 4.19. There is a trend of the peaks: when the silicon substituting level is increasing, the constructive diffracting peak is shifting toward larger angle. This can be explained by Bragg’s law, nλ=2dsinθ, which implies that the constructive diffracting angle is. 治政 increasing as the lattice spacing is lowering. According 大 to the fact that the atomic 立 radius of the gallium atom (136 pm) is larger than the silicon atom (111 pm), the ‧ 國. 學. results is expected.. ‧. Intensity(a.u.). n 44.0. Ch. 44.2. engchi. 44.4. 2. sit er. io. (220). y. Nat. al. i Un. 44.6. x=0 x = 0.05 x = 0.10 x = 0.15 x = 0.20. v. 44.8. 45.0. Fig. 4.19 Enlargement of the XRD pattern at (2, 2, 0) direction. 46.

(54) Furthermore, the lattice parameter of Fe2VGa1-xSix compounds are calculated by the Rietveld refinement method under the software Highscore Plus and shown in figure 4.20. The lattice parameter of un-doped one, Fe2VGa, is around 5.7788 Å , and the lattice parameter is linearly decreasing while the doping content is increasing. Therefore, from the above evidences, the series of Si-substituted is successful.. 立. 5.780. 政治 Fe VGa 大 Si 2. 1-x. x. 5.775. ‧ 國. 學. 5.770 5.765. ‧. 0.05. 0.10. 0.15. 0.20. x. io. n. al. er. Nat 5.755. y. 5.760. sit. Lattice Parameter (A). 5.785. i Un. v. Fig. 4.20 Lattice parameter of Fe2VGa1-xSix samples. Ch. engchi. 47.

(55) 4.2.2 Electrical Resistivity In this case, we substitute the gallium atom on the silicon atom which has one more valence electron compared with the gallium atom in the outer shell. The electrical resistivity of Fe2VGa1-xSix samples are shown in figure 4.21, and all samples are located in the range between 1.40 μΩ m and 2.4 μΩ m from 300 K to 700 K. The resistivity of the un-doped sample is around 1.4 μΩ m at room temperature, and it process the semiconductor-like behavior in the temperature range 300 ~ 400 K. However, the electrical resistivity starts to act metallic behavior when the temperature. 治政 goes high. This is because electrons have more kinetic大 energy when the temperature is 立 high and this increases probability of the collision between electrons and atoms. The ‧ 國. 學. behavior of Fe2VGa0.95Si0.05 is much similar with the un-doped sample, but the. ‧. resistivity at the room temperature is 15% higher than the un-doped one. For x = 0.1,. sit. y. Nat. 0.15, and 0.2 samples, they have the same electrical behavior: they all have a broad. io. er. maximum in the temperature range 700 ~ 800K. We also found that the electrical. al. n. resistivity is increasing while silicon atoms are substituting gallium sites, which is. ni Ch against our purpose for a good thermoelectric material. U engchi 2.6. v. Resistivity ( m). 2.4 2.2 2.0. x=0 x=0.05 x=0.1 x=0.15 x=0.2. 1.8 1.6 1.4 300. 400. 500. 600. 700. 800. Temperature (K) Fig. 4.21 Electrical resistivity of Fe2VGa1-xSix samples 48.

(56) 4.2.3 Seebeck Coefficient The results of the temperature dependent Seebeck coefficient of Fe2VGa1-xSix are presented in the figure 4.22. As we known, the value of Seebeck coefficient is proportional to the value of the density of state and, simultaneously, inversely proportional to the slope of density of state at Fermi level, which is given by. S=-. p 2 k 2B é 1 ¶N(E) ù T . 3e êë N(E) ¶E úû E=EF. We have done the substitution of titanium atoms for vanadium sites for hole-type. 治政 substitution, but the results showed the maximum 大 value of Seebeck coefficient. 立 Therefore, we try to replace gallium atoms by silicon atoms to perform the ‧ 國. 學 ‧ y. sit. io. al. n. 0. er. 20. Nat. Seebeck Coefficient (V K-1). electron-substitution on Fe2VGa system.. -20. Ch. engchi. i Un. v. x=0 x=0.05 x=0.1 x=0.15 x=0.2. -40. -60 300. 400. 500. 600. 700. Temperature (K) Fig. 4.22 Seebeck coefficient of Fe2VGa1-xSix samples. 49. 800.

(57) Seebeck coefficients of Fe2VGa1-xSix samples are showed in figure 4.22, and a p-n transition is observed in this series. As we expected, the silicon atom has one more out-shell electron than the gallium atom, so the major carrier will become electron-type. Seebeck coefficient of the un-doped sample, Fe2VGa, at the room temperature is 22 μV K-1, and it exhibits a maximum value 24 μV K-1 at the 380 K, and then turns to a negative slope at the high temperature. The largest value of Seebeck coefficient is about -58 μV K-1 and is found in Fe2VGa0.85Si0.15 sample at room temperature.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 50. i Un. v.

(58) 4.2.4 Thermal Conductivity The thermal conductivity of Fe2VGa1-xSix samples is measured in the temperature range from 300 K to 820 K and is depicted in figure 4.23. The thermal conductivity of the un-doped sample is around 24.3 W m-1 K-1 at 300 K, and tends to decrease while the temperature increases. The behaviors of all samples are same as the un-doped sample. On the other hand, we found that the substitution of Si for Ga site results in the reduction of the thermal conductivity by 50% in x = 0.2 sample at room temperature.. 政治大. ‧ 國. 22. ‧. 20. io. 16. y. sit. 18. al. n 14. Ch. 12 10. 300. x=0 x=0.05 x=0.10 x=0.15 x=0.2. er. -1 -1. 學. 24. Nat. Thermal Conductivity (W m K ). 立. 400. engchi 500. i Un. 600. v. 700. 800. Temperature (K) Fig. 4.23 Thermal conductivity of Fe2VGa1-xSix samples. 51.

(59) For further discussing the mechanism of the thermal conductivity, we can divide it into two parts, the lattice thermal conductivity and the electronic thermal conductivity. Figure 4.24 shows the comparisons between the lattice and electronic thermal conductivity of all Si-substituted Fe2VGa compounds.. Thermal Conductivity,  (W m-1 K-1). Thermal Conductivity,  (W m-1 K-1). x=0 25.  ph e. 20. 15. 10. 5 300. 400. 12.  ph. 10. e. 8. 政治大. 立. 500. x=0.15. 14. 600. 6. 4. 700. 300. 400. 500. x=0.05. al. n 400. 500. 600. Temperature (K). 700. Ch. 800. 12.  ph. 10. e. 8. y. e. x=0.2. 6. sit. Thermal Conductivity,  (W m-1 K-1).  ph. io. 5. 300. Thermal Conductivity,  (W m-1 K-1). 14. 4. er. ‧ 國. 10. 2. engchi. 300. i Un. v. 400. 500. 600. 700. Temperature (K). x=0.1. 18 16.  ph. 14. e. 12 10 8 6 4 2 300. 400. 500. 600. 800. ‧. 15. 700. 學. 20. 600. Temperature (K). Nat. Thermal Conductivity,  (W m-1 K-1). Temperature (K). 700. 800. Temperature (K). Fig. 4.24 Comparisons between κ, κph, and κe of all Si-substituted samples. 52. 800.

(60) From the figures above, the lattice thermal conductivity of Fe2VGa is almost 4 times larger than the electronic part at room temperature. However, it is prone to decline with the temperature increasing and the electronic part starts to take control of the thermal conductivity when the temperature is above 631 K. We observed that when the substituting level increases, the temperature which the electronic part is dominated has a trend except x = 0.15 sample and we also plot this temperature versus composition of all samples in the figure 4.25. We also observed a strange electrical transportation on x = 0.15 sample, and we think the reason is same as the Ti-substituted series. The intersection of the DOS and Fermi level would result in the. 治政 change of the electrical topological configuration, and大 further influence the electrical 立 transportation. ‧. ‧ 國. 學. al. er. sit. Fe2VGa1-xSix. n. Temperature (K). io 600. y. Nat. 650. Ch. engchi. i Un. v. 550. 500. 450 0. 0.05. 0.1. 0.15. x Fig. 4.25 Temperature verse Si substituting level. 53. 0.2.

(61) 4.2.5 Power Factor The temperature dependent power factor of Si-substituted Fe2VGa samples is depicted in figure 4.26. The maximum value of the power factor for the un-doped sample is only 0.4 mW m-1 K-2 at 340 K. However, when silicon atoms are going into this system, the power factor will decrease first, and then dramatically increase as the doping level increases. The reason why Fe2VGa0.95Si0.05 shows a smaller power factor is probably because the major carrier changes from hole-type to electron-type, and the power factor is proportional to the square of Seebeck coefficient. Therefore, the. 治政 power factor would exhibit a small value during this 大transformation. The largest 立 power factor is 2.1 mW m K and is found in Fe VGa Si at room temperature. -1. -2. 2. 0.8. 0.2. ‧ 國. 學. Compared with the Ti-substituted samples, although the power factor is smaller than. ‧. Ti-substituted samples, the largest power factor is happened in Fe2VGa0.8Si0.2, and it. sit. y. Nat. seems to increase with the substituting level increased. Therefore, we will try to. io Power Factor (mW m-1 K-2). n. al. er. substitute more silicon atoms for further research.. 2.0. Ch. engchi. i Un. v x=0. 1.5. x=0.05 x=0.1 x=0.15 x=0.2. 1.0. 0.5. 0.0 300. 400. 500. 600. 700. 800. Temperature (K) Fig. 4.26 Power factor versus temperature of Fe2VGa1-xSix samples. 54.

(62) 4.2.6 Figure of Merit (zT) We use the results of the power factor as well as the thermal conductivity to determine the figure of merit from room temperature to 820 K and plot it in the figure 4.23. The un-doped sample exhibits low value of zT because of the combination of the small power factor and a huge thermal conductivity. The largest value of zT happens in Fe2VGa0.8Si0.2 sample at room temperature and is about 0.052. The figure of merit is proportional to the ratio of the power factor over the thermal conductivity, and the power factor seems to increase as the doping level is increased, and, simultaneously,. 治政 the thermal conductivity seems to decrease. Therefore,大 we will try to substitute more 立 silicon atoms for further research. ‧. ‧ 國. 學. 0.06. sit. n. al. er. io 0.04. y. Nat. 0.05. Ch. zT. 0.03. x=0 x=0.05 x=0.1 x=0.15 x=0.2. engchi. i Un. v. 0.02 0.01 0.00 300. 400. 500. 600. 700. 800. Temperature (K) Fig. 4.27 Figure of merit (zT) versus temperature of Fe2VGa1-xSix samples. 55.

(63) Chapter 5 Conclusions This thesis reports the thermoelectric properties of Ti-substituted (p-type) and Si-substituted (n-type) Heusler alloy Fe2VGa. All samples including Fe2V1-xTixGa (with x = 0, 0.05, 0.1, 0.15, 0.2, 0.25) and Fe2VGa1-xSix (with x = 0, 0.05, 0.1, 0.15, 0.2) are prepared through arc-melting method. Each sample was carefully examined by x-ray diffraction analysis and found to be single-phase L21 Heusler-type structure. Upon Ti substitution, Seebeck coefficients are positive in temperature range 300 ~. 政治大 the enhancement of Seebeck coefficients are observed, which can be attributed to 立 700 K, signifying that hole carriers dominate the electrical transportation. Moreover,. ‧ 國. 學. Fermi level shifting toward lower energy level. The electrical resistivity data reveals that the substituted samples are more conducting than un-doped sample. This can also. ‧. be explained by Fermi level shifting toward lower energy level in band-structure. Nat. sit. y. calculation. Therefore, the increase of Seebeck coefficient and the decrease of the. n. al. er. io. electrical resistivity give rise to the dramatic enhancement of the power factor. With. i Un. v. proper substitution (Fe2V0.9Ti0.1Ga), the power factor is enhanced by 11 times at the. Ch. engchi. room temperature. Due to the alloying effect, the thermal conductivities of substituted samples are significantly suppressed, around 40% decreases. The largest figure of merit happens in Fe2V0.8Ti0.2Ga sample at 420 K and is about 0.067 which is almost 10 times larger than the un-doped sample. In the beginning, we want to observe the superparamagnetic behavior in our sample, so we measure the ac susceptibility in different frequencies. However, the signal of the ac susceptibility is small and rough, and we didn’t find the frequency dependent behavior in our samples. Nevertheless, we observe a small hysteresis of 1000 Oe at T = 2 K in magnetization measurement, whereas magnetization does not show any hysteresis loops at T > 2 K. 56.

(64) On the other hand, we also investigate the thermoelectric properties of Fe2VGa1-xSix samples for n-type material. Upon Si substitution, an enhancement of absolute value accompanied with the p-n transition is found. This phenomenon can be attributed to the shift of Fermi level toward higher energy level. However, the electrical resistivity is increasing while silicon atoms are substituting gallium sites, which is against our purpose for a good thermoelectric material. The largest power factor happens in Fe2VGa0.8Si0.2 sample at room temperature and is about 2.1 mW m-1 K-2 which is almost 6 times larger than the un-doped sample. Although the power factor is smaller than the Ti-substituted samples, the largest power factor happens in Fe2VGa0.8Si0.2,. 治政 and it seems to increase as the doping level increasing. 大 Therefore, we will try to 立 substitute more silicon atoms for further research. The largest zT happens in ‧ 國. 學. Fe2VGa0.8Si0.2 sample at room temperature and is about 0.052. Although we have. ‧. found a huge enhancement of the thermoelectric performance, it still low compared. n. al. er. io. sit. y. Nat. with state-of-the-art thermoelectric materials.. Ch. engchi. 57. i Un. v.

(65) References 1.. C.S. Lue, C.F. Chen, J.Y. Lin, Y.T. Yu, and Y.K. Kuo, Phys. Rev. B 75, 064204 (2007). 2. 3. 4. 5.. Y. Nishino, and Y. Tamada, J. Appl. Phys. 115, 123707 (2014) M. Vasundhara, V. Srinivas, and V.V. Rao, Phys. Rev. B 77, 224415 (2008) Y. Nishino, S. Deguchi, and U. Mizutani, Phys. Rev. B 74, 115115(2006) K. Renard, A. Mori, Y. Yamada, S. Tanaka, H. Miyazaki, and Y. Nishino, J. Appl. Phys. 115, 033707 (2014). 6. 7.. H. Kato, M. Kato, Y. Nishino, U. Mizutani, and S. Asano, J. Jpn. Inst. Met. 65, 652 (2001) W. Xie, S. Wang, S. Zhu, J. He, X. Tang, Q. Zhang, and T.M. Tritt, J. Mater. Sci. 48, 2745–2760 (2013). 立. C.S. Lue, and Y.K. Kuo, Phys. Rev. B 66, 085121 (2002) A. Bansil, S. Kaprzyk, P. E. Mijnarends, and J. Toboła, Phys. Rev. B 60, 13396 (1999). ‧ 國. 學. 8. 9.. 政治大. 10. C. S. Lue and J. H. Ross, Jr., Phys. Rev. B 63, 054420 (2001). ‧. sit. n. al. er. Jpn. 66, 1257 (1997). A. Slebarski, and J. Goraus, Phys. Rev. B 80, 235121 (2009) Melvin Cutler et al. Physical Review 181, 1336 (1969) L.S. Hsu, Y.K. Wang, G.Y. Guo, and C.S. Lue, Phys. Rev. B 66, 205203 (2002) C.S. Lue, W.J. Lai, C.C. Chen and Y.K. Kuo, J. Phys.: Condens. Matter 16, 4283 (4283). io. 13. 14. 15. 16.. y. Nat. 11. N. Kawamiya, Y. Nishino, M. Matsuo, and S. Asano, Phys. Rev. B 44, 12406 (1991) 12. K. Endo, H. Matsuda, K. Ooiwa, M. Iijima, K. Ito, T. Goto, A. Ono, J. Phys. Soc.. Ch. engchi. i Un. v. 17. C.S. Lue, Joseph H. Ross Jr, K.D.D. Rathnayaka, D.G. Naugle, S.Y. Wu, and W.H. Li, J. Phys.: Condens. Matter 13, 1585 (2001). 58.

(66)