Đề tài Single Crystal Growth and Magnetic Properties of RRhIn5 Compounds ( R: Rare Earths )

4 Single Crystal Growth and Measurement Methods 4.1 Single crystal growth 1) Introduction to single crystal growth First, we mention about a meaning of ”crystal growth”. The word ”crystal” originates from the Greek which means coldness or ice. What are crystals? Crystals are ordered arrangements of atoms (or molecules). Materials in crystalline form has special opti- cal and electrical properties, in many cases improved properties over randomly arranged materials which are named amorphous or glass. What cause crystals to ”grow”? The driving force for crystallization comes from the lowering of the potential energy of the atoms or molecules when they form bonds to each other. The process of crystal growth starts with the nucleation stage. Several atoms or molecules in a supersaturated vapor or liquid start forming clusters: the bulk free energy of the cluster is less than that of the vapor or liquid. Crystal growth consists of two steps namedly nucleation and growth. If nucleation rates are slow and growth is rapid, large crystals will be grown. If nucleation is rapid, relative to growth, small crystals or even polycrystalline samples will be grown. There are three kinds of methods for single crystal growth: flux, czochralski and zone melting. - Czochralski method: A seed crystal is attached to a rod, which is rotated slowly. The seed crystal is dipped into a melt at a temperature slightly above the melting point. A temperature gradient is set up by cooling the rod and slowly withdrawing it from the melt (the surrounding atmosphere is cooler than the melt). Decreasing the speed, the crystal is pulled from the melt, increases the quality of the crystals (with fewer defects) but decreases the growth rate. The advantage of the Czochralski method is that large single crystals can be grown, thus it is used extensively in the semiconductor industry. - Zone melting: a polycrystalline specimen is prepared, typically in the shape of a cylinder and is placed into a crucible, with a seed crystal near the top of the crucible. The sample cylinder is placed in a furnace with a very narrow hot zone. The portion of the cylinder containing the seed crystal is heated up to the melting point, and the rest of the cylinder is slowly pulled through the hot zone. An advantage of the zone melting technique is that impurities tend to be concentrated in the melted portion of the sample. Consequently, this process sweeps them out of the sample and concentrates them at the end of the crystal ampoule, which is then cut off and discarded. Thus this method is sometimes used to purify semiconductor crystals and metals. As the same principle, a molybdenum crucible, which was supported by a tungsten bar, was heated up by the rf-water-cooled working coil. It is called the Brigman method which is used in a siliconit furnace at high heating temperature up to 1700 ◦C. 44 4.1. SINGLE CRYSTAL GROWTH 45 - Flux growth: slow cooling of the melt with congruently melting materials (those which maintain the same composition on melting). One simply melts a mixture of the desired compositions and then cools slowly through the melting point. More difficult with incongruently melting materials and knowledge of the phase diagram is needed. If the phase diagram is not known, consequently there is no guarantee that crystals will have the intended stoichiometry. Crystals grown in this way are not often rather large, and thus this method is frequently used in research, but no special technique is required. Therefore, it is an easy and economical method. The advantages of this technique are:67,68 1) Single crystal samples can be grown at the temperature which is lower their melting points. 2) The obtained crystal samples just have fewer defects and much less thermal strain. Flux metals can remove the impurity subtance because it make a clean environment in growing process. 3) Single crystal stoichiometry has an organic to drive back stoichiometric phenomina as oxidation or evaporation of material components. 4) This technique can be applied to the compounds with high evaporation pressure, since the crucible is sealed in the ampule and the flux prevents evaporation. There- fore, it is suitable for rare earth metals. 5) No special technique is required during the crystal growth and it can be done with the simple and inexpensive equipment. For this method, we must choose the flux metal which is better for each compounds. The flux metal often enters the crystal as an impurity. The excessive nucleation causes small crystals, which takes place either due to a too fast cooling rate, or supercooling of the melt by subsequent multiple nucleation. Another one more is quality of crucible because its reaction with materials occur at high temperatures. Finally, the ability to separate crystals from the flux at the end of growth needs special considerations. Rare earth metals often have a high vapor pressure at the melting points. Therefore, flux method is better to grow the single crystal for rare earth compounds. 46CHAPTER 4. SINGLE CRYSTAL GROWTH AND MEASUREMENT METHODS 2) Single Crystal Growth of RRhIn5 Here are mentioned about the single crystal growth for RRhIn5 (R= rare earths). Figure 4.1 shows the growth process which was done in about 4 weeks, with 6 steps in O¯nuki Laboratory, Department of Physics, Graduate School of Science, Osaka University. The process with 6 steps follow: - STEP 1: Prepare the materials . Starting materials were 3N(99.9-pure)-R, 5N-Rh and 6N-In. . The R-metal block was cut by a spark cutter or diamond cutter and etched chemi- cally. An etching liquid for R is a 1:3 mixture of nitric acid and water. . We used the high-quality alumina crucible (Al2O3: 99.9%) as a container with outer diameter of 15.5mm, inner diameter of 11.5mm and length of 60mm. Since the crucible usually contains impurities, we cleaned the crucibles in alcohol, and bake them up to 1050 ◦C under high-vacuum (less than 1× 10−6 torr). - STEP 2: Seal materials in a quartz ampoule . Typically, we prepared the materials with the compositions of R :Rh : In=1 : 1: 10 or 15 ( offstoichiometry). . We put these materials into the alumina crucible and sealed in a quartz ampoule with 180mmHg Ar pressure, which is adjusted to reach at 1 atm at the highest temperature. - STEP 3: Furnace . We set the sealed ampoule in an electric furnace. The furnace possesses the tem- perature gradient naturally. As we know from our own experience, the better results are obtained when we put the ampoule where the temperature is more homogeneous. There- fore, we placed the ampoule at the highest and the flat temperature gradient position. Nevertheless, the temperature gradient is useful for growing some compounds. There are some reports of growing crystals by temperature gradient method.67 . The furnace is controlled by the PID temperature controller with Pt-PtRh13% (type-R) thermocouple. We obtained the temperature stability less than 0.1 ◦C. . We increased the temperature up to 1050 ◦C which is the maximum temperature of the electric furnace, as shown in Fig. 4.2. Then we kept it 48 hours. After that, the tem- perature was decreased slowly. The cooling rate was gradually increased with decreasing the temperature and the furnace was turned switch off at 650 ◦C follows the temperature 4.1. SINGLE CRYSTAL GROWTH 47 Single Crystal Growth Process 10mm 5mm 5mm 10mm 5mm 6 R Crucible Quart tube In Rh 1 2 4 3 Growing Process Furnace Heater quartz-ampoule alumina crucible Thermocouple 1 0 0 0 6 5 0 300 T e m p e ra tu r e ( O C ) Time ( hours ) Heat up 1050 C 0 InWoo l 5 1000 500 0 6004002000 High vacuum Fig. 4.1 Single crytal growing process in RRhIn5 compounds. 48CHAPTER 4. SINGLE CRYSTAL GROWTH AND MEASUREMENT METHODS 1000 500 0 T em p er at u re ( O C ) 6004002000 Time ( hours ) RRhIn5 Growing process YRhIn5 ( II ) ( I ) TmRhIn5 YbRhIn5 Fig. 4.2 Temperture profile for single crystal growth in RRhIn5. process. . Check the temperature dependence of the furnace every day and write it in a note- book. . The temperature of ampoules will cool down to room temperature after finishing growth process. STEP 4: New ampoule After taking out the ampoule from the furnace, we open ampoule and seal it again in a pyrex ampoule under high vacuum. The ampoule is heated up to 300 ◦C, which is sufficiently higher than the melting point of In, in the muffle furnace. STEP 5: Remove the remained indium metal Next ampoule is taken out quickly from the furnace and is set into the centrifuge. Finally we remove the flux from the crystals by spinning the ampule in the centrifuge. STEP 6: Single crystal sample The grown single crystal of RRhIn5 was a very large block of about 10×5×5mm3. Carefully seeing, we can find easily a cleared plane of the (001) plane, reflecting the layered crystal structure, as shown in Fig. 4.3. We succeeded in growing the single crystal samples of RRhIn5. We also succeeded in growing the single crystals of YRhIn5, TmRhIn5 and YbRhIn5 4.1. SINGLE CRYSTAL GROWTH 49 by the following process, as shown in Fig. 4.1. There are, however, some diffrences in the ratio of composition: . Starting materials were 3N(99.9% pure)-Yb or Tm, 5N-Rh and 6N-In with the ratio of 1 : 1: 30, and 1:1:10 for YRhIn5. . The temperature decrease of cooling process is faster than the other RRhIn5, taking about 17 days. The minium of temperature step is about 1 ◦C/1 hour, as shown in Fig. 4.2(II). . Typical sizes of YRhIn5, TmRhIn5 and YbRhIn5 are 5 x 5 x 3 (c-axis) mm 3, 5 x 3 x 0.3 (c-axis) mm3 and 2 x 1.5 x 0.2 (c-axis) mm3, respectively, as shown in Fig. 4.3(a), (k) and (j). 50CHAPTER 4. SINGLE CRYSTAL GROWTH AND MEASUREMENT METHODS LaRhIn5 5mm PrRhIn5 YbRhIn5 ( l ) ( b ) ( c ) SmRhIn5 GdRhIn5 5mm NdRhIn5( d ) ( e ) ( f ) 10mm TbRhIn5( h ) 10mm DyRhIn5 ( g ) 5mm HoRhIn5( i ) ErRhIn5( j ) TmRhIn5 ( k ) YRhIn 5( a ) Fig. 4.3 Single crystals of RRhIn5 obtained by the In-flux method. 4.2. MEASUREMENT METHODS 51 4.2 Measurement methods 4.2.1 Electrical resistivity 1) Introduction about the electrical resistivity Firstly, let’s mention about Matthiessen’s rule. There are four contributions in the electrical resistivity : the electron scattering due to impurities or defects ρ0, the electron- phonon scattering ρph, the electron-electron scattering ρe-e and the electron-magnon scat- tering ρmag: ρ = ρ0 + ρph + ρe-e + ρmag, (4.1) The ρ0-value, which originates from the electron scattering due to impurities and defects, is constant for a variation of the temperature. This value is important to know the quality of an obtained sample. If ρ0 is large, the sample contains many impurities or defects. A quality of a sample can be estimated by determining a so-called residual resistivity ratio (RRR = ρRT/ρ0), where ρRT is the resistivity at room temperature. Of course, a large value of RRR indicates that the quality of the sample is good. Next, a scattering lifetime τ0 and a mean free path l0 are also expressed in the view- point of the resistivity. ρ0 = m∗ ne · 1 τ0 , (4.2) where n is a density of carrier and e is an electric charge. Then τ0 and l0 values are τ0 = m∗ neρ0 , (4.3) l0 = vFτ0 = ~kF neρ0 , (4.4) The temperature dependence of ρph, which originates from the electron scattering by phonon, changes monotonously. ρph is proportional to T above the Debye temperature, while it is proportional to T 5 far below the Debye temperature, and ρph will be zero at T = 0. Here, the contribution of ρe-e, which can be expressed in terms of the reduction factor of the quasiparticle and the Umklapp process, is dominant at low temperatures for the strongly correlated electron system. Therefore, we can regard the total resistivity in non-magnetic compounds at low temperatures as follows: ρ(T ) = ρ0 + ρe-e(T ), (4.5) = ρ0 + AT 2, (4.6) where the coefficient √ A is proportional to the effective mass. Yamada and Yosida obtained the rigorous expression of ρe-e in the strongly correlated electron system on the 52CHAPTER 4. SINGLE CRYSTAL GROWTH AND MEASUREMENT METHODS basis of the Fermi liquid theory.27 According to their theory, ρe-e is proportional to the imaginary part of the f electron self-energy ∆k, and ∆k is written as ρe−e ∝ ∆k ' 4 3 (piT )2 ∑ k′,q piDfk−q(0)D f k′(0)D f k′+q(0) × { Γ↑↓2(k,k ′;k′ + q,k − q) + 1 2 Γ↑↑A 2 (k,k′;k′ + q,k − q) } , (4.7) where Γσσ is the four-point vertex, which means the renormalized scattering interac- tion process of k(σ)k′(σ) → k′ + q(σ)k − q(σ), ΓA↑↑ is denoted as Γ↑↑(k1,k2;k3,k4) − Γ↑↑(k1,k2;k4,k3), and D f k(0) is the true (perturbed) density of states of f electrons with mutual interaction in the Fermi level. This ∆k is proportional to the square of the enhancement factor and gives a large T 2-resistivity to the heavy Fermion system. Above the ordering temperature Tord, ρmag is given by ρmag = 3piNm∗ 2~e2εF |Jex|2(gJ − 1)2J(J + 1), (4.8) where Jex is the exchange integral for the direct interaction ρmag must be taken into a magnetic compound. This contribution describes scattering processes of conduction electrons due to disorder in the arrangement of the magnetic moments between the local moments and conduction electrons. When T = Tord, ρmag shows a pronounced kink, and when T < Tord, ρmag strongly decreases with decreasing temperature. The magnetic resistivities in the actinides, how- ever, are ascribed to strong scattering of the conduction electrons by the spin fluctuations of 5f electrons. This contribution to the resistivity at low temperatures is given by the square of the temperature, namely ρmag = A ′T 2. The coefficient A′ is extremely large in the heavy Fermion system. Therefore, ρmag and ρe-e are inseparable and ρmag can be considered to change to ρe-e. 2) Experimental method of the resistivity measurement The resistivity measurements have be done in our Laboratory by using a standard four-probe DC current method. We fixed sample on a plastic plate by an instant glue. The same is applied a DC current and measure the voltage via gold wire with 0.025mm and silver paste. We use 4He or 3He cryostat to measure the resistivity from 1.3 or 0.5K to the room temperature. The thermometers are a Cernox resistor for all temperature region or a combination between a RuO2 resistor at lower temperatures (below 20K) and a Diode resistor at higher temperatures. 4.2. MEASUREMENT METHODS 53 4.2.2 Specific heat 1) Introduction about the specific heat Next, we discribe about the specific heat measurement. At low temperatures, the specific heat consists of the electronic, lattice, magnetic and nuclear contributions: C = Ce + Cph + Cmag + Cnuc (4.9) = γT + βT 3 + Cmag + A T 2 , (4.10) where A, γ and β are constants with the characteristic of the compounds. When the magnetic and nuclear contributions can be neglected: C T = γ + βT 2, (4.11) We can estimate the electronic specific heat coefficient γ. Using the density of states D(εF), the coefficient γ can be expressed as γ = pi2 3 kB 2D(εF), (4.12) where kB is the Boltzmann constant. Since the density of states based on the free electron model is proportional to the electron mass, the coefficient γ possesses an extremely large value in the heavy Fermion system. According to the Debye T 3 law, for T ¿ ΘD: Cph ' 12pi 4NkB 5 ( T ΘD )3 ≡ βT 3, (4.13) where ΘD is the Debye temperature and N is the number of atoms. For the actual lattices the temperatures at which the T 3 approximation holds are quite low. It may be necessary to be below T = ΘD/50 to get a reasonably pure T 3 law. If the f energy level splits due to the crystalline electric field (CEF) in the paramag- netic state, the inner energy per one magnetic ion is given by ECEF = 〈Ei〉 = ∑ i niEi exp(−Ei/kBT )∑ i exp(−Ei/kBT ) , (4.14) where Ei and ni are the energy and the degenerate degree on the level i. Thus the magnetic contribution to the specific heat is given by CSch = ∂ECEF ∂T , (4.15) 54CHAPTER 4. SINGLE CRYSTAL GROWTH AND MEASUREMENT METHODS This contribution is called a Schottky specific heat. The entropy of the f electron S is defined in the viewpoint of the Schottky specific heat CSch: S = ∫ T 0 CSch T dT, (4.16) S = R lnW, (4.17) where W is a state number at temperature T . Therefore we acquire information about the CEF level. In the magnetic ordering state Cmag is: Cmag ∝ T 3/2 (ferromagnetic ordering) (4.18) ∝ T 3 (antiferromagnetic ordering). (4.19) When the antiferromagnetic magnon is accompanied with the energy gap ∆m, eq. (4.19) is modified to Cmag ∝ T 3 exp(−∆m/kBT ). 2) Experimental method of the specific heat The quasi-adiabatic heat pulse method is used to measure the specific heat with dilution refrigerator, 3He and 4He cryostat at temperatures down to 0.1, 0.6 and 1.7K, respectively. The sample was put on the Cu-addenda. The RuO2 resister thermometer and two strain gage heaters were also put on the addenda. One of the strain gage heaters was used as a constant heater to compensate heat leak due to the heat radiation and/or the thermal conduction by the wire which was suspended on the addenda. We measured the temperature of the sample with constant heating, and the specific heat is thus deduced as follows: C = ∆Q ∆T = I · V ·∆t ∆T , (4.20) where ∆Q is the amount of heat, I and V are the current and the voltage flowing to the heater, respectively, ∆t is the duration of heating and ∆T is the change of temperature due to heating. This C includes both of the specific heat of the sample and that of the addenda. The relaxation method on a commercial Physical Property Measurement System (PPMS), produced by Quantum Design, is another methods to maasure the heat ca- pacity in the temperature from 2K to 300K and the magnetic field of 0 to 90 kOe. 4.2. MEASUREMENT METHODS 55 4.2.3 Magnetic susceptibility and magnetization 1) Introduction about the magnetic susceptibility and magnetization The magnetic susceptibily and magnetization was described on the basics of the crys- talline electric field scheme, as shown in Chap. 2. Here we agian summarize the important points. The 4f energy level in the Ce compounds with the non-cubic crystal structure splits into three doublets. Hamiltonian of this system is given by H = HCEF +HZeeman, (4.21) Here, HCEF in the tetragonal symmetry for Ce 3+ and R3+ is expressed as follows: HCEF = B 0 2O 0 2 +B 0 4O 0 4 +B 4 4O 4 4, (4.22) where Bml and O m l are the CEF parameters and the Stevens operators, respectively. Due to the CEF effect, the sixfold degenerate 4f -levels of the Ce ion split into three doublets. The CEF susceptibility is given by χiCEF = N(gµB) 2 1 Z ∑ m6=n |〈m|Ji|n〉|21− e −∆m,n kBT ∆m,n e − En kBT + 1 kBT ∑ n |〈n|Ji|n〉|2e− En kBT  (4.23) and Z = ∑ n e − En kBT , (4.24) where g is the Lande´ g-factor (6/7 for Ce3+ for example), J i is the component of the angular momentum and ∆m,n =En -Em . The magnetization can be also calculated by Mi = gµB ∑ n |〈n|Ji|n〉|e − En kBT Z , (4.25) The CEF susceptibility is also written: χCEF = lim H→0 dM dH , (4.26) The magnetic susceptibility including the molecular field contribution λi is given as follows: χ−1i = (χ i CEF) −1 − λi, (4.27) The eigenvalue En and eigenfunction |n〉 are determined by diagonalizing the total Hamiltonian: H = HCEF − gJµBJi(Hi + λiMi), (4.28) where the second term is the Zeeman term and the third one is a contribution from the molecular field. 56CHAPTER 4. SINGLE CRYSTAL GROWTH AND MEASUREMENT METHODS 2) Experimental method of the magnetic susceptibility and magnetization We measured the magnetic susceptibility and magnetization by a commercial SQUID magnet meter, produced by Quantum Design, at Low Temperature Center, Osaka Uni- versity and Advanced Science Center, Japan Atomic Energy Agency (JAEA) in Tokai. The magnetic susceptibility was usually measured under a small magnetic field of 1 kOe, and the magnetization was measured in the magnetic fields up to 70 kOe. 4.2. MEASUREMENT METHODS 57 4.2.4 High field magnetization 1) Introduction about the high field magnetization The signal of magnetization were detected under a pulse magnetic field to measure the induced electromotive force of a coil: V = −N dΦ dt , (4.29) where V is the induced electromotive force, Φ is the magnetic flux and N is the number of turns in a coil. Here, the coil is the coaxial compensated coil. If the A-coil (inside) and the B-coil (outside, reverse wound coil of the A-coil) are used: ΦA = BSA + µ0MSM , (4.30) and ΦB = BSB + µ0MSM , (4.31) Here, B is magnetic field. SA, SB and SM are the cross sections of A-coil, B-coil and a sample, respectively. The induced electromotive force is V = −d(NAΦA −NBΦB) dt , (4.32) because the coaxial compensated coil means that the induced electromotive force is can- celed without a sample: NASA = NBSB, (4.33) We get the equation of the induced electromotive force with a sample as follows: V = −µ0(NA −NB)SM dM dt , (4.34) By integrating this equation over the time, magnetization is obtained. 2) Experimental method of the high field magnetization The diagram of magnetization measurement is shown in Fig. 4.4. At first, the huge condenser bank was electrically charged. When we turned on the switch of a circuit, the charge flowed through the pulse magnet coil and generated high magnetic fields in the coil. The long-pulse magnet, which we used for the present research, generated the magnetic fields up to 500 kOe. The pulse width was 10msec. We used the uniaxial coil in order to compensate a background flux change due to a transient field. Ideally, we can compensate perfectly by A- and B-coils. However this scenario is ideal and in fact the background signals are not zero, containing linear and nonlinear components of the dM/dt. The linear component can be minimized by tuning a bridge balance circuit connected with a compensation coil-C. 58CHAPTER 4. SINGLE CRYSTAL GROWTH AND MEASUREMENT METHODS Pulse Generator Personal Computer X-Y Recorder Bridge Balance D.C. Amp. Digital Memory Liquid He Liquid N 2 Magnet Sample Field pick-up coil Pick-up coil Capacitor Bank Fig. 4.4 Diagram for the magnetization measurement in high pulse field. 4.2. MEASUREMENT METHODS 59 4.2.5 de Haas-van Alphen effect 1) Introduction about the de Haas-van Alphen effect Next, we discribe the de Haas-Van Alphen effect measurement. The orbital motion of the conduction electron is quantized and forms Landau levels70 under a high magnetic field. Therefore various physical qualities shows a periodic variation with H−1 since increasing the field strength H causes a sharp change in the free energy of the electron system when Landaus level crosses the Fermi energy. In a three-dimensional system this sharp structure is observed at extremal areas in k-space, perpendicular to the field direction and enclosed by the Fermi energy because the density of state also becomes extremal. From the field and temperature dependence of various physical quantities, we can obtain the extremal area S, the cyclotron mass m∗c and the scattering lifetime τ for this cyclotron orbit. The magnetization or the magnetic susceptibility is the most common one of these physical quantities, and its periodic character is called the de Haas- van Alphen (dHvA) effect. It provides one of the best tools for the investigation of Fermi surfaces of metals. The theoretical expression for the oscillatory component of magnetization Mosc due to the conduction electrons was given by Lifshitz and Kosevich as follows: Mosc = ∑ r ∑ i (−1)r r3/2 Ai sin ( 2pirFi H + βi ) , (4.35a) Ai ∝ FH1/2 ∣∣∣∣ ∂2Si∂kH2 ∣∣∣∣−1/2 RTRDRS, (4.35b) RT = αrm∗ciT/H sinh(αrm∗ciT/H) , (4.35c) RD = exp(−αrm∗ciTD/H), (4.35d) RS = cos(pigirm ∗ ci/2m0), (4.35e) α = 2pi2kB e~ , (4.35f) Here the magnetization is periodic on 1/H and has a dHvA frequency Fi Fi = ~ 2pie Si = 1.05× 10−12 [T · cm2] · Si, (4.36) which is directly proportional to the i-th extremal (maximum or minimum) cross-sectional area Si (i = 1, . . . , n). The extremal area means a gray plane in Figure 4.5, where there is one extremal area in a spherical Fermi surface. The factor RT in the amplitude Ai is related to the thermal damping at a finite temperature T . The factor RD is also related to the Landau level broadening kBTD. TD is due to both the lifetime broadening and inhomogeneous broadening caused by impurities, crystalline imperfections or strains. 60CHAPTER 4. SINGLE CRYSTAL GROWTH AND MEASUREMENT METHODS Fermi surface dHvA frequency H 1/H 1/H H (a) (b) Fig. 4.5 Simulations of the cross-sectional area and its dHvA signal for a simple Fermi surface. There is one dHvA frequency in (a), while there are three different frequencies in (b). The factor TD is called the Dingle temperature and is given by TD = ~ 2pikB τ−1 = 1.22× 10−12 [K · sec] · τ−1, (4.37) RS factor is called the spin factor and related to the difference of phase between the Landau levels due to the Zeeman split. When gi = 2 (a free electron value) and m ∗ c = 0.5m0, this term becomes zero for r = 1. The fundamental oscillation vanishes for all values of the field. We call it is the zero spin splitting situation in which the up and down spin contributions to the oscillation cancelled out, and this can be useful for determining the value of gi. Note that in this second harmonics for r = 2 the dHvA oscillation should show full amplitude. The quantity |∂2S/∂kH2|−1/2 is called the curvature factor. The rapid change of cross-sectional area around the extremal area along the field direction diminishes the dHvA amplitude for this extremal area. The detectable conditions of dHvA effect are as follows: 1) The distance between the Landau levels ~ωc must be larger than the thermal broad- ening width kBT : ~ωc ¿ kBT (high fields, low temperatures). 2) At least one cyclotron motion must be performed during the scattering, namely ωcτ/2pi > 1 (high quality samples). In reality, however, it can be observed even if a cyclotron motion is about ten percent of one cycle. 4.2. MEASUREMENT METHODS 61 3) The fluctuation of the static magnetic field must be smaller than the field interval of one cycle of the dHvA oscillation (homogeneity of the magnetic field). 2) Shape of the Fermi surface The angular dependence of dHvA frequencies gives very important information about a shape of the Fermi surface is. As a value of Fermi surface corresponds to a carrier number, we can obtain the carrier number of a metal directly. We show the typical Fermi Fig. 4.6 Angular dependence of the dHvA frequency in three typical Fermi surfaces (a) sphere, (b) cylinder and (c) ellipsoid. surfaces and their angular dependences of dHvA frequencies in Figure 4.6. In a spherical Fermi surface, the dHvA frequency is constant for any field direction. On the other hand, 62CHAPTER 4. SINGLE CRYSTAL GROWTH AND MEASUREMENT METHODS in an ellipsoidal Fermi surface such as in Figure 4.6(b), it takes a minimum value for the field along the z-axis. These relatively simple shape Fermi surfaces can be determined only by the experiment. However, information from an energy band calculation is needed to determine a complicated one. 3) Cyclotron effective mass The cyclotron effective mass m∗c i can be determined from the temperature dependence of a dHvA amplitude data. Equation (4.35c) is transformed into: log { Ai [ 1− exp (−2αm∗ciT H )] /T } = −αm∗c i H T + const, (4.38) Therefore, from the slope of a plot of log{Ai[1 − exp(−2λm∗c iT/H)]/T} versus T at constant field H, the effective mass can be obtained. Let us consider the relation between the cyclotron mass and the electrical specific heat γ. Using a density of states D(EF), γ is written as γ = pi2 3 kB 2D(EF), (4.39) In the spherical Fermi surface, using EF = ~2kF2/2m∗c takes γ = pi2 3 kB 2 V 2pi2 ( 2m∗c ~2 )3/2 EF 1/2 = kB 2V 3~2 m∗ckF, (4.40) where V is molar volume and kF = (SF/pi) 1/2. We obtain from eq. (4.36) γ = kB 2m0 3~2 ( 2e ~ )1/2 V m∗c m0 F 1/2 = 2.87× 10−4 [(mJ/K2 ·mol)(mol/cm3)T−1/2] · V m ∗ c m0 F 1/2, (4.41) In the case of the cylindrical Fermi surface, γ = pi2 3 kB 2 V 2pi2~2 m∗ckz = kB 2V 6~2 m∗ckz, (4.42) Here, the Fermi wave number kz is parallel to an axial direction of the cylinder. If we regard simply the Fermi surfaces as sphere, ellipse or cylinder approximately and then we can calculate them. 4.2. MEASUREMENT METHODS 63 4) Dingle temperature Next, the Dingle temperature TD also be determined from measuring a field depen- dence of a dHvA amplitude data. Equations (4.35b)-(4.35d) yield log { AiH 1/2 [ 1− exp (−2λm∗c iT H )]} = −λm∗c i(T + TD) 1 H + const, (4.43) From the slope of a plot of log{AiH1/2[1−exp(−2λm∗c iT/H)]} versus 1/H at constant T , the Dingle temperature can be obtained. Here, the cyclotron effective mass must have been already obtained. We can estimate the mean free path l or the scattering life time τ from the Dingle temperature. The relation between an effective mass and lifetime takes the form ~kF = m∗vF, (4.44) l = vFτ, (4.45) Then eq. (4.37) is transformed into l = ~2kF 2pikBm∗cTD , (4.46) When the extremal area can be regarded as a circle approximately, using the eq. (4.36), the mean free path is expressed as l = ~2 2pikBm0 ( 2e ~c )1/2 F 1/2 ( m∗c m0 )−1 TD −1 = 77.6 [A˚ · T−1/2 ·K] · F 1/2 ( m∗c m0 )−1 TD −1, (4.47) 5) Field modulation method We use the usual ac-susceptibility field modulation method in dHvA effect measure- ment. A small ac-field h0 cosωt is varied on an external field H0 (H0 À h0) in order to obtain the periodic variation of the magnetic moment M . The sample is set up into a pair of balanced coils (pick up and compensation coils), as shown in Figure 4.7. An induced emf (electromotive force) V will be proportional to dM/dt: V = c dM dt = c dM dH dH dt = −ch0ω sinωt ∞∑ k=1 hk0 2k−1(k − 1)! ( dkM dHk ) H0 sin kωt, (4.48) 64CHAPTER 4. SINGLE CRYSTAL GROWTH AND MEASUREMENT METHODS 5φ 5 2φ Pick-up coil Compensation coil Sample Cotton Fig. 4.7 Detecting coil and the sample location. where c is constant which is fixed by the number of turns in the coil and so on, and the higher differential terms of the coefficient of sin kωt are neglected. Calculating the dkM/dHk it becomes V = −cωA ∞∑ k=1 1 2k−1(k − 1)! ( 2pih0 ∆H )k sin ( 2piF H + β − kpi 2 ) sin kωt, (4.49) Here, ∆H = H2/F . Considering h0 2 ¿ H02 the time dependence of magnetization M(t) is given by M(t) = A [ J0(λ) sin ( 2piF H0 + β ) + 2 ∞∑ k=1 kJk(λ) cos kωt sin ( 2piF H0 + β − kpi 2 )] , (4.50) -0.5 0 0.5 J k (λ ) 151050 λ k = 1 2 3 4 Fig. 4.8 Bessel function Jk(λ) of the first kind. 4.2. MEASUREMENT METHODS 65 1 2 3 5 4 1 2 3 4 5Lock-in Amp. Power Amp. Impedance Matching trance Degital Volt Meter Magnet Power Supply Oscillator Chart Recorder Computer Sample Superconducting magnet Pick-up coil Compensation coil Modulation coil 2ω ω ω Fig. 4.9 Block diagram for the dHvA measurement. where λ = 2piFh0 H0 2 . (4.51) Here, Jk is k-th Bessel function. Figure 4.8 shows the Bessel function of the first kind for the various order k. Finally we can obtain the output emf as follows: V = c ( dM dt ) = −2cωA ∞∑ k=1 kJk(λ) sin ( 2piF H0 + β − kpi 2 ) sin kωt, (4.52) We detect the signal at the second harmonic of the modulation frequency 2ω using a Lock-in Amplifier, since this condition may cut off the offset magnetization and then detect the component of the quantum oscillation only. The modulation field h0 to make the value of J2(λ) maximum, namely λ = 3.1. A modulation frequency of 3.5 Hz is used for dilution refrigerator.Figure 4.9 shows a block diagram for the dHvA measurement in the present study. 66CHAPTER 4. SINGLE CRYSTAL GROWTH AND MEASUREMENT METHODS 4.2.6 Neutron scattering 1) Introduction about the neutron Neutrons, which are first discovered in 1932, have become a quite useful probe which to study many important features of matter, especially condensed matter. Some basic properties of ”neutron” are shown in Table 4.I. The mass of the neutron is nearly that of the proton, and this relative large mass has several important consequences. From the production viewpoint, we can product the energetic neutron by fission of heavy nuclei or by collision with atoms of similar mass as hydrogen. There are some interesting properties of the neutron because of zero net charge: 1) It interacts very weakly with matter and penetrates deeply into sample. 2) It also easily transmits through sample enclosures used to control the evironment. 3) Zero net charge, there is no Coulomb barrier to overcome, so that the neutrons are oblivious of the electronic charge cloud and interact directly with the nuclei of atom. Another important character is that the neutron has a magnetic moment with spin 1/2, meaning that neutrons interact with the unpaired electrons in magnetic elements. Results of this interaction, neutron elastic scattering gives information about the spin configuration of electrons and the density distribution of unpaired electrons. Therefore, Table 4.I Some properties of the neutron72 Quantity Value Rest mass, mn 1.675 x 10−24 g Spin 12 Magnetic moment, µn 1.913 nuclear magnetons,µN Charge 0 regular periodic arrangements of atoms and magnetic moments in a crystalline sample can be detected. The differential scattering cross section is written as the sum of two terms, which represent respectively the nuclear and magnetic scattering:( dσ dΩ ) = N (2pi)3 V0 ∑ G [ δ(Q−G) ∣∣∣∣∣∑ i bie iQ·ri ∣∣∣∣∣ 2 + ( e2γn 2mc2 )2 δ(Q±K −G)f2(Q) ∣∣∣∣∣∑ i (µ0 sinαi)e iQ·ri ∣∣∣∣∣ 2 ] , (4.53) where N is the number of atoms, V0 is the unit cell volume, G is a reciprocal lattice vector, Q is a scattering vector, bi is the scattering length of the atom i, ri is the position 4.2. MEASUREMENT METHODS 67 of the atom, K is the propagation vector of the magnetic structure, f(Q) is the magnetic form factor, µ0 is the amplitude of the magnetic moment and αi is the angle between Q and the direction of the magnetic moment. The law of momentum and energy conservation governing all diffraction and scattering experiments are well known: Q = kf − ki|Q| = k2i + k2f − 2ki.kfcosθS~ω = Ei − Ef , (4.54) where, the wave-vector magnetitude k =2pi λ , with λ is the neutron wavelength of neutron beam, and the momentum transferred to the crystal is ~Q. The subscript i refers to the initial state (before the sample) and f the final state (after scattered by the sample). The angle between the incident and scattered beams is 2θS and the energy transfored to the sample is ~ω. Because of the finite mass of the neutron the dispersion realation for the neutron is: E = ~2k2 2mn ;E[meV ] = 2.072k2[A˚−2], (4.55) and the energy conservation law can be written as ~ω = ~2 2mn (k2i − k2f ), (4.56) ( a ) ( b )       q kf ki Q G o 2θS )       q Q G o ) ki kf 2θS Fig. 4.10 Vector diagram of inelastic scattering for (a) neutron energy loss (kf<ki), ~ω > 0 : energy is transferred from the incident neutron to the sample and an excitation is created; (b) neutron energy gain(kf>ki), ~ω < 0 : the sample gives up aquantum of energy to the neutron beam. O presents the origin of reciprocal space, G a recipocal- lattice, and q the momentum transfer a zone. We have two kind of scattering : elastic scattering (|ki|= |kj|=k) and inelastic scat- tering (|ki| 6= |kj|), as shown in Fig. 4.10. In the present work neutron elastic scattering is used for the magnetic structure study. The theory of the interaction between a neutron and the nuclei of an atom is very short range (about 10−13 cm = 1fm), which is much less 68CHAPTER 4. SINGLE CRYSTAL GROWTH AND MEASUREMENT METHODS than the wave length of thermal neutron, the interaction can be considered nearly point- like. Therefore, neutron-nucleus scattering contains only s-wave componients, which implies that the scattering is isotropic and can be characterized by the scattering length, b. The typical value of the scattering length for the elements is on the order 1×10−12cm, comparable to the nuclear radius. So the scattering cross section for a nucleus is equal to 4pib2 . The scattering length can be complex; however, the imaginary part only becomes significant near a nuclear absorption resonance. The real part is typically positive, but it can become negative at energies below a resonance; the sign of the scattering length is associated with the relative phase of the scattering neutron wave with respect to the incident wave. For the most elements, the scattering length in the thermal energy regime is essentially independent of energy. Contrast with electron and x-rays, the scattering does not depend upon the number of electrons and its strength varies somewhat ran- domly among the elements, and among isotopes of the same element. Because of the weakness of the interaction, the scattering amplitude from a sample is equal to the sum of the scattering amplitudes individual atoms. This results simplifies the interpretation of measurement. As described above, in spite of no net charge in the neutron, its internal structure of quarks and gluons gives the neutron a magnetic moment, 1.913 µB, interacts with the unpaired electron spins in magnetic atoms with a strength comparable to that of the nuclear interaction. The neutron is, therefore, one of the excellent probe of magnetic properties of solids and has contributed enormously to our understanding of magnetism. The effective magnetic scattering length, p for a magnetic atom with a moment of nµB Bohr magnetons is given by : p=0.27 x 10−12 µB cm. The constant is one-half of the classical radius of the electron multiplied by the gyro- magnetic ratio of the neutron. For nµB ≥ 1, it is apparent that p is comparable to the b values. It means that scattered intensity associated with magnetic effects is comparable to the scattering from the nuclei. The magnetic form factor f(Q) in eq. (5.27) is originated from the spatial distribution of the magnetic moment. The magnetic form factor is written as the Fourier transform of spatial distribution of magnetic moment σ(r): f(Q) = ∫ σ(r)e−iQ·rd3r, (4.57) From the eq.( 5.21), nuclear Bragg peaks appear at Q = G, and magnetic Bragg peaks appear at Q = G±K. If the σ(r) is the delta function, f(Q)=1. Practically, the σ(r) has spatial distribution, f(Q) decreases rapidly with increasing Q. When there is significant orbital contribution to the magnetic moment, m = Sms +Lml, (4.58) The magnetic form factor can be written using spin and orbital contributions of magnetic form factor, fs(Q) and fl(Q), f(Q) = Sfs(Q) +Lfl(Q), (4.59) 4.2. MEASUREMENT METHODS 69 Table 4.II Some parametters of the neutron: wavelength, frequency, velocity and en- ery72 Quantity Relationship Value at E= 10 meV Energy [meV]=2.072 k2 [A˚−1] 10 meV Wavelength λ[A˚]=9.044/ √ E[meV ] 2.86 A˚ Wave vector k[A˚−1] = 2pi λ[A˚] 2.20 A˚−1 Frequency ν[THz] =0.2418 E [meV] 2.418 THz Wavenumber ν[cm−1] = ν[Hz]/(2.998x 1010 cm/s) 80.65 cm−1 Velocity v [km/s] =0.6302 k[A˚−1] 1.38 km/s Temperature T [K= 11.605 E [meV] 116.05 K Confinement radius R 0.7fm Beta-decay lifetime τ 885.9 ±0.9 sec Quark structure udd 1 up, 2 down Within a dipole approximation, the individual form factors are then expanded in the basis of Bessel functions, fs(Q) ', (4.60) fl(Q) ' + , (4.61) Concerning the magnetic form factor of f -electrons:73 rare earths and actinides, where the spin-orbit coupling is large, spin and orbit couple together to give a total angular momentum. J = S +L, (4.62) In such case, J is a good quantum number and the magnetic moment can be expressed as m = gJJ , (4.63) Thus, the simplified magnetic form factor is expressed as follows, f(Q) = +C2 , (4.64) where, C2 = 2 gJ − 1 = J(J + 1) + L(L+ 1)− S(S + 1) 3J(J + 1)− L(L+ 1) + S(S + 1) , (4.65) The neutron used in a scattering experiment can be obtained from a nuclear reactor, where the neutron arise from the spontaneous fission of 235U , or from a parallel source where the neutrons are produce by bombarding a heavy target as U, W, Ta, Pb or Hg with high-energy protons. There are amny kinds of reactors : High Flux Isotope Re- actor (HFIR), High Flux Reactor (HFR), High Flux Beam Reactor (HFBR). Efficient use of the neutron-beam time from a spallation source compemsates for the lower time- adveraged flux. JRR-3M is a reactor which located in Tokai, Japanese Atomic Energy Agency (JAEA) with 20MW from 1990. Japan is one of the second power econony nation on the world which uses nuclear resource for the target of peace. 70CHAPTER 4. SINGLE CRYSTAL GROWTH AND MEASUREMENT METHODS 2) Experimental method of the neutron scattering experiment Neutron scattering from a single crystal sample was measured using a thermal neutron triple-axis spectrometer. Fig. 4.11 and Fig. 4.12 show the schematic view of the triple- axis spectrometer. The three axes are those of the monochromator, sample table and analyzer. Continuous neutrons from the nuclear reactor must be monochromatized to a desired wavelength. The continuous neutrons are incident on a pyrolytic graphite (PG) monochromator crystal and only the desired neutrons, which satisfy the Bragg’s law λ = 2dM sin θM (dM: (002) plane spacing in PG, θM: half of scattering angle), go through the 1st and 2nd collimators. The monitor detects the intensity of incident neutron beam. The neutron beam prepared in this way falls on the sample to be investigated. Then the scattered neutrons with the scattering angle of 2Φ pass through the 3rd collimator, and the neutrons are scattered by the (002) plane of PG analyzer crystal. Finally the detector counts only the neutrons which satisfy the Bragg’s law λ = 2dM sin θA. We can measure the intensity as a function of energy (wave length) by changing the angle 2θA. Neutron scattering data of RRhIn5 was measured at JRR-3M in Japan Atomic Energy Agency (JAEA). Magnetic scatterings with the single crystal were measured by a triple- axis spectrometer TAS-1 installed at reactor hall, and TAS-2 and LTAS installed at guide hall. PG monochromator and analyzer were used to get monochromatic beam with λ= 2.36 and 4.19 A˚. Harmonic contamination was removed via a 20 cm thick Be polycrystalline block cooled down to 10 K and a PG filter as thick as 12 cm for cold (LTAS) and thermal (TAS-1 and TAS-2) neutron experiments, respectively. Powder diffraction data were measured by using a high resolution powder diffractome- ter, HRPD. The incident beam was monochromatized at λ= 1.823 A˚ with a Ge (5,3,3) monochromator. The angle resolution of about ∼0.1◦ was obtained with the collimation of 6’-12’-6’. 4.2. MEASUREMENT METHODS 71 Fig. 4.11 The neutron three-axis spectometer (TAS) is the most versatile and useful instrument for use in inelastic scattering because it all allows one to probe nearly any coordinates in energy and momentum space in a precisely controlled manner. The three axis correpond to the axes of rotation of the monochromator, the sample and the analyzer. The monochromator defines the direction and magnitude of the momentum of the incident and the analyzer peform a similar function for the scattered or final beam. A typical set- up of a TAS: the incident and scattering neutron wave vectors, ki and kf are selected by Bragg diffraction on the monocromator and analyzer crystals. 72CHAPTER 4. SINGLE CRYSTAL GROWTH AND MEASUREMENT METHODS Reactor PG (002) monochrometer 1st collimator 2nd collimator PG filter Monitor Sample table 3rd collimator 4th collimator PG (002) analyser Detector Fig. 4.12 Schematic view of the triple axis spectrometer.