Signature of topological Dirac fermion in magnetization measurements of three-dimensional topological insulator Bi1.5 Sb0.5 Te1.7 Se1.3 Prithwish Dutta, Arnab Pariari, and Prabhat Mandal∗
arXiv:1608.01203v1 [cond-mat.mtrl-sci] 3 Aug 2016
Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700 064, India (Dated: August 4, 2016) Here, we report the observation of anomalous cusp-like paramagnetic peak over the native diamagnetic background of three-dimensional topological insulator Bi1.5 Sb0.5 Te1.7 Se1.3 in magnetization measurements. Similar to that reported in Bi2 Se3 , Sb2 Te3 and Bi2 Te3 , the paramagnetic response, which is confined within a narrow magnetic field range, is robust against temperature and shows linear in field decay from the zero field value. The reconstruction of surface state with time is clearly reflected in the reduction of peak height with the age of the sample. This confirms that the paramagnetic response is originating from the singularity in electron spin orientation at the Dirac node of spin-momentum locked topological surface state. The interplay between metallic surface and insulating bulk is also visible in the electrical transport response. The field dependence of the resistivity shows weak anti-localization effect at low temperatures causes due to the two conducting surface channel.
Topological insulators have been a hot topic of research since it’s realization in Bi1−x Sbx , which is followed by Bi2 Se3 , Bi2 Te3 , Sb2 Te3 etc. This class of ‘insulators’, due to strong spin-orbit coupling, posses a metallic surface state with some unique properties[1]. The spin-momentum locked surface state, which is protected by time-reversal symmetry, makes the electronic transport robust against inelastic backscattering. Further theoretical investigation makes it a potential material for spintronics, quantum computation as well as host ground for Majorana fermions, Axion electrodynamics etc. However all of the above mentioned materials shows high bulk conductivity instead of the gap between the valance and conduction band. Hence the transport responses are metallic and dominated by bulk than surface. Thus the search for new topological insulators with higher surface to bulk transport ratio is still going on. It has been well stated in literatures[2, 3] that Crystal defects - namely antisite defect and selenium vacancies, are the main reason behind residual carriers in bulk. In order to achieve higher figure of merit, efforts were given to decrease the carrier density by using ternary and quarnary compounds of Bi, Sb, Te and Se instead of the undoped binary compounds. To understand the role of doping,it is necessary to note the mechanism of the crystal defects. The misplacement of Selenium from it’s lattice position causes increase in negative carrier concentration, which can be as high as ∼ 1019 , where as antisite defect of Bi,Sb/Te [3]produces holes. Thus replacing Bi by Sb and Se by Te in proper ratio can balance the opposite type of carriers and minimize the conductivity.Teramoto et al. show in their work [4] that in these group of compound with chemical formula Bi2−x Sbx Te3−y Sey , high resistivity can be achieved for a certain linear relation between x and y. Rein et al. [3] reinvestigated on that phase diagram and prescribed that Bi1.5 Sb0.5 Te1.7 Se1.3 (BSTS) has the
∗
[email protected]
highest resistivity in this range. Since then BSTS has been studied by various group through spectroscopy [7–11] and transport measurements [2, 5–8]. As angle resolved photo emission spectroscopy (ARPES) measurement directly looks into the band structure of the sample, it is one of the best tool to identify a TI state in a material. But, being a costly device and sophisticated experimental probe, it is not always easy to put a material under the scan of ARPES. Also, complex hybridization of the bulk and surface states at high lattice temperature is clearly observed by this high energy surface probe. Few indirect evidences of topological surface state like Weak Anti-localization effect(WAL) and low temperature saturation or drop in resistivity can be reflected in transport experiments. In this paper, we report a unusual paramagnetic response over diamagnetic background of the bulk near zero field in BSTS as a strong indirect evidences of its spinmomentum locked topologically nontrivial surface state. The results are consistent with recent reports in Bi2 Se3 , Bi2 Te3 , Sb2 Te3 [12] and ZrTe5 [13]. This signature being observable at very low field and room temperature, can serve as a easy tool to identify new topological insulators. Single crystals of BSTS were grown by melting the stoichiometric mixture of elements at 850 ◦ C for 48 h in a sealed vacuum quartz tube. The sample was cooled over a period of 48 h to 550 ◦ C and kept at this temperature for 96 h before cooling to room temperature. Large crystals of few mm size with shiny flat surface can easily be obtained from the ampule. Phase purity and the structural analysis of the samples were done by high resolution powder x-ray diffraction (XRD) technique (Rigaku, TTRAX II) using Cu-Kα radiation. The calculated values of lattice parameters are ahex = 4.284˚ A, chex = 29.87˚ A. The details are being discussed in the supplementary materials[14]. Due to tuning of the crystal defect, the bulk conductivity of BSTS becomes minimized and shows semicon-
ature up to the lowest measured temperature 2K, where the value becomes 31.64 Ω.Cm . The high temperature insulating behavior was fitted with Arrhenius equation and shown in Fig. 1(b).But, it does not fit well. A linear approximation at the high temp range(100K to 250K) provides activation energy ∆ ∼ 27meV. Hence we tried to fit the data with variable range hopping equation
5 0
4 0
ρ( Ω.C m )
3 0
2 0
ρ = ρ0 exp( 1 0
5 0
1 0 0
1 5 0
2 0 0
2 5 0
3 0 0
T (K )
2
2
0
0 0 .0
0 .1
0 .2
1 /T
ln ρ
4
ln ρ
4
0 .3
0 .4
T
(1)
,which has been shown in Fig. 1(c). In this case, we find a better fitting in the range 50 K to 190 K with T0 ∼ 106 K. So, it can be argued that though the bulk shows activation behavior at higher temperature, very soon it switches to variable range hopping due to lack of activated carrier in vicinity at lower temperature before being dominated by the surface contribution. The low temperature data, which is mainly dominated by the metallic response of the surface was fitted with the equation ρs = a + bT n and the value of the fitted parameters are a = 31.56 Ω.cm, b=1.29×10−3 Ω.cm.K−n and n= 2.03. The square dependence of temperature reveals the Fermi liquid nature of the surface carries at low temperature. To understand this interplay, we have assumed that the net resistivity is a parallel combination of surface resistivity and bulk resistivity. So, the equations are
0 0
T0 1/4 ) T
0 .5
-1 /4
FIG. 1. (a) The temperature dependence of resistivity of BSTS. The red line is the fit of the experimental data with equation 2. (b) Dependence of ln ρ with 1/T. The slope of this graph, which has been estimated by the red line, provides the band gap multiplied by 1/KB T . (c) Dependence of ln ρ with T−1/4 . The slope of this graph, which has been estimated by the red line, provides the value of T0 of equation 6.
ρtotal = [1/ρs + 1/ρb ]−1 The continuous red line fit to the experimental get a = 32.8Ω.cm, b = 5.17 × 10−6 Ω.cm and T0
ducting behavior due to its bulk band gap. The resistance increases with decrease in temperature following the Arrhenius’ equation ρ = ρ0 exp( kB∆T ). The surface due to it’s gapless Dirac cone in the band structure will be metallic in nature. Thus the surface resistance should follow the equation Rs = a + bT n . Due to it’s 2D nature, the cross section area of surface conduction is very small which leads to comparatively higher resistance value specially at high temperatures. Thus, at high temperature, the transport response is dominated by the bulk. It is to be noted that in a electrical transport measurement, the bulk and the surface contribute simultaneously and can be treated as parallel resistance. With decrease in temperature, the surface resistance decreases and the bulk resistance increases. At a certain point they cross each other giving a metal to insulator transition and then the low temperature conduction is dominated by the surface. The resistivity of BSTS increases with decrease in temperature, as shown in Fig. 1(a) down to 42 K. The value of resistivity is quite high compare to Sb2 Te3 as well other reported value[2, 5–8]. At room temperature ρ = 0.7Ω.Cm, which rises to 42 Ω.Cm at ∼ 42 K. There are very few TI, which shows such high resistivity with semiconducting behavior, essential for technological application. Below 42 K, the resistivity falls with temper-
(2)
in Fig. 1(a) is the theoretical data. From the fitting, we 8.28 × 10−3 Ω.cm.K −2 , Rb0 = = 4.7 × 106 K.
The magnetoresistance(MR) of the sample is measured in the field range 0 to 9T at different temperature, as shown in Fig. 2(a). At the lowest temperature 2K and highest magnetic field 9T, the value of MR is 70%. The MR shows linear behavior in high field and low temperature. The nonlinearity at the low field is due to Weak Anti-localization(WAL) effect. With the application of initial field, the time reversal symmetry breaks down causing a gap in the gapless surface state. Hence, the rate of change of magnetoresistance, which is maximum at B → 0, decreases with increase in field strength following the HLN equation. 4 G(B) = −
αe2 1 Bφ Bφ [ψ( + ) − ln( )] πh 2 B B
(3)
Here α/2 is a parameter representing the number of chanh ¯ nels of conduction, Bφ = 4el 2 and lφ is the dephasing φ
length. The conventional magnetoresistance increases parabolically due to increase in the radius of gyration and hence scattering, which is expected from the bulk transport [6]. Consistence with the temperature dependence of ρ, the parabolic nature is almost absent at the low temperature in the present case and slowly comes into picture at higher temperature. The signature of the WAL effect, 2
which is solely a surface property, also diminishes with increase in temperature due to decrease in the dephasing length. These two observations re-establish the fact that the low temperature transport is dominated by surface than bulk, which gradually alters at higher temperature. To investigate deeply the WAL effect the magnetoconductance(MC) data is fitted with the HLN equation. The MC data at the low field fits well with the HLN equation confirming the WAL effect as shown in Fig 2(b). The derived values of lφ from the fit of equation 3, is plotted against temperature in Fig. 2(b)inset. Here, ∆G is approximately equals to ∆σtotal .t because ∆σbulk is negligible compared to ∆G/t in the equation ∆σtotal = ∆σbulk + ∆G/t at low temperature. According to literature, lφ ∝ T −0.5 , fitted satisfactorily to the experimental temperature dependence of lφ . The value of α was ∼ 1 at temperature 5 and 10 K, which indicates conduction through two non-trivial surface state. But it gradually drops down to 0.8 at 40 K and higher temperature. It is to be noted that the bulk conduction, being a 3D phenomenon, does not significantly contribute to WAL effect. At higher temperatures as the bulk conduction become more prominent and surface contribution decreases, the value of α decreases. Spin momentum locking is a integral characteristic of topological surface state of three-dimensional topological insulator. As a result of strong spin-orbit coupling and time reversal symmetry, the spin and the momentum wave vector of low energy quasi particle excitations are always perpendicular to each other. This leads to left-handed spin texture for the conduction bands and right-handed spin texture for the valence band on circular constant energy contour of Dirac cones. However at the Dirac node, where the two bands with opposite spin helicity touches with each other, the spins are free to orient in any direction due to singularity. As a consequence, it provides a paramagnetic contribution to the intrinsic magnetic moment of the system under application of magnetic field. It can be reflected as low-field paramagnetic peak in the susceptibility curve χ(H). Experimental discovery of this singular peak in χ has been reported for the family of three-dimensional topological insulators and identified as the fingerprint of the helical spin texture of the topological Dirac fermions on the surface state[12, 13, 15]. χ shows linear in field decay from the zero field value and completely suppresses within a few kOe field range [12, 13, 15]. This linear in field dependence of the topological response from 2D Dirac fermion on the surface, has also been established theoretically [12]. Considering T =0, this paramagnetic Dirac susceptibility has the form, (gµB )3 µ0 (gµB )2 χD ∼ = 4π 2[ h ¯ vF Λ − (¯ hvF )2 | B |] at the sample’s native chemical potential (µ = 0). Where g is the Land´e g-factor and Λ is the effective size of the momentum space, which is responsible for the singular response in χ. As the height of the susceptibility peak is determined by Λ, depending on the details of the bulk band, the peak height can vary from system to system within the family
6 0
5 K 1 0 K 2 0 K 3 0 K 4 0 K
5 0
M R (% )
4 0
3 0
2 0
1 0
0 0
2
4
6
8
B (T ) 0 .0
-0 .2
∆G ( e 2 / h )
-0 .4
-0 .6
5 0 0
-0 .8 l φ( n m )
4 0 0
-1 .0
3 0 0 2 0 0 0
1 0
2 0
3 0
4 0
T (K )
-1 .2 0 .0
0 .5
1 .0
1 .5
2 .0
B (T )
FIG. 2. (a) Dependence of magnetoresistance (MR) with magnetic field B at different temperature. (b)Field dependence of Magnetoconductance and the Fit with HLN equation. In the inset dephasing is plotted against temperature which damps following which is plotted by the red line.
of 3D topological insulators [12]. Whereas the nature of the topological response (i.e. cuspiness, linear-in-field decay, robustness against temperature, etc.,) is universal to the entire family of 3D topological insulators [12]. Fig. 3(a) shows the magnetization (M ) of a freshly cleaved single crystal of BSTS as a funtion of magnetic field (B), at several representative temperatures 2 to 300 K. Over the whole range of temperatures, it shows diamagnetic behaviour except at very low field, where a sharp paramagnetic upturn emerges in the magnetization with B. This can be more clearly visible from the susceptibility(χ) vs B plot, as shown in Fig. 3(b). The experimental value of χ has been obtained after taking the first order derivative of M (B) with respect to B. A cusp-like paramagnetic response in χ(B) sharply rises over the temperature dependent 3
6 4
2 K 2
2 0 K 5 0 K 1 0 0 K 3 0 0 K
4
e .m .u /g m )
0
6
(a ) 4
2 K 2
2 0 K 5 0 K 0 1 0 0 K -2 3 0 0 K 2 0
-4
(1 0
-3
-3
e .m .u /g m )
6
-2
-2
-6
M
M
(1 0
A g e
(a )
F re s h
-1 0
-4 -4
-6 -6 -1 0
0
-1 0
1 0
0
8
1 0
B (k O e )
B (k O e )
4 0
1 2
A g e 8
8
F re s h
(b )
8
0
(b )
e m u /g m -O e )
0
2 K -5
4
0
2 0 K 5 0 K 1 0 0 K 3 0 0 K 5
B (k O e )
2 0 K -8 5 0 K 1 0 0 K 3 0 0 K
-5
0
5
B (k O e ) 0
-1 0
-7
χ( 1 0
χ( 1 0
-7
0
2 K
4
e m u /g m -O e )
8
-4
-4
-4
-8 -1 0
-5
0
5
-1 0
1 0
0
1 0
B (k O e )
B (k O e ) FIG. 3. (a) Magnetization (M) vs B of freshly cleaved single crystal of BSTS at several representative temperatures from ) as a function of B, cal2 to 300 K, (b) Susceptibility (χ= dM dB culated by taking the first order derivative of magnetization.
FIG. 4. (a) Magnetization of the single crystal, which was kept for three days in air from the first measurements , (d) ) as a function of B at several represenSusceptibility (χ= dM dB tative temperatures.
diamagnetic background over a narrow field range ∼3 kOe. The height of the peak from the diamagnetic background almost remains same irrespective of temperature. This singular response shows no sign of thermal rounding up to the highest measuring temperature 300 K (∼27 meV), which is a significant fraction of the bulk energy gap of this type of materials. Similar to that observed in Bi2 Se3 , Sb2 Te3 , Bi2 Te3 [12, 15] and ZrTe5 [13], the singular paramagnetic response in the present system implies the helical spin texture of the 2D Dirac fermion on the surface. In this context, we would like to mention that the standard diamagnetic sample like bismuth can not have this singular paramagnetic response in χ (see supplementary information). Inset of figure 3(b) shows linear field decay of χ from zero field
value at a representative temperature 2 K, as predicted theoritically. The surface, when exposed to air for long time, gets effected due to doping from air, surface reconstruction and the formation of two-dimensional electron gas due to bending of bulk band at the surfaces [12]. As a result, the peak height has been observed to reduce with time [12, 13]. To probe the aging effect in BSTS, the magnetization measurements were done on the same piece of single crystal after 3 days of exposure to air and shown in figure 4(a). Although the nature of the paramagnetic peak in χ along with the diamagnetic background, as shown in figure 4(b), remains unchanged, a significant drop(∼ 25%) is observed in the peak height reflecting the expected aging effect in the present sample. Inset of figure 4(b) also shows linear in field decay of 4
nothing but the two surfaces of the sample. The anomalous singular paramagnetic peak over the diamagnetic floor, associated with the helical spin texture of 2D Dirac fermion on the surface of 3D topological insulators, has been observed in magnetization measurements, similar to that reported in Bi2 Se3 , Sb2 Te3 , Bi2 Te3 and ZrTe5 . The surface state reconstruction with time due to doping from air, known as aging effect for 3D topological insulators, is clearly reflected from the age dependent reduction of paramagnetic susceptibility peak.
χ in the aged sample at a representative temperature 2 K. In conclusion, we have observed a transition from metallic to insulating behaviour in the temperature dependence of resistivity from 2 to 300 K in Bi1.5 Sb0.5 Te1.7 Se1.3 single crystal, which can be associated with the dominating surface and bulk state contribution to transport respectively. Prominent weak antilocalization effect at low temperature and low field in the magnetoconductance implies two 2D conducting channel protected by time-reversal symmetry, which are
[1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [2] A. A. Taskin, Z. Ren, S. Sasaki, K. Segawa, and Y. Ando, Phys. Rev. Lett. 107, 016801 (2011). [3] Z. Ren, A. A. Taskin, S. Sasaki, K. Segawa, and Y. Ando, Phys. Rev. B 84, 165311 (2011). [4] I. Teramoto and S. Takayanagi, J. Phys. Chem. Solids 19, 124 (1961). [5] K. Segawa, Z. Ren, S. Sasaki, T. Tsuda S. Kuwabata and Y. Ando Phys. Rev. B 86, 075306 (2012). [6] J. Lee, J. Park, J-H. Lee, J. S. Kim and H-J. Lee Phys. Rev. B 86, 245321 (2012). [7] E. Frantzeskakis, N. de Jong, B. Zwartsenberg, T. V. Bay, Y. K. Huang, S. V. Ramankutty, A. Tytarenko, D. Wu,Y. Pan, S. Hollanders, M. Radovic, N. C. Plumb, N. Xu, M. Shi, C. Lupulescu, T. Arion, R. Ovsyannikov, A. Varykhalov, W. Eberhardt, A. de Visser, E. van Heumen, and M. S. Golden Phys. Rev. B 91, 205134 (2015). [8] W. Ko, I. Jeon, H. W. Kim, H. Kwon, S-J. Kahng, J. Park, J. S. Kim, S. W. Hwang and H. Suh Science Reports 3, 2656 (2013). [9] K. W. Post, Y. S. Lee, B. C. Chapler, A. A. Schafgans, M. Novak, A. A. Taskin, Kouji Segawa, M. D. Goldflam, H. T. Stinson, Y. Ando, and D. N. Basov Phys. Rev. B 91, 165202 (2015). [10] S. Kim, S. Yoshizawa, Y. Ishida, K. Eto, K. Segawa, Y. Ando, S. Shin and F. Komori Phys. Rev. Lett. 112, 136802 (2014). [11] T. Arakane, T. Sato, S. Souma, K. Kosaka, K. Nakayama, M. Komatsu, T. Takahashi, Z. Ren,K. Segawa and Y. Ando Nat. Commun. 1639 3:636 (2012). [12] L. Zhao, H. Deng, I. Korzhovska, Z. Chen, M. Konczykowski, A. Hruban, V. Oganesyan, and L. KrusinElbaum, Nat. Mater. 13, 580 (2014). [13] Arnab Pariari, and Prabhat Mandal, arXiv:1603.05175. [14] Supplimentary material for “Signature of topological Dirac fermion in magnetization measurements of threedimensional topological insulator Bi1.5 Sb0.5 Te1.7 Se1.3 ”. [15] S.G. Buga, V.A. Kulbachinskii, V.G. Kytin, G.A. Kytin, I.A. Kruglov, N.A. Lvova, N.S. Perov, N.R. Serebryanaya, S.A. Tarelkin, and V.D. Blank, Chem. Phys. Lett. 631-632, 97 (2015). [16] Y. S. Hor, A. Richardella, P. Roushan, Y. Xia, J. G. Checkelsky, A. Yazdani, M. Z. Hasan, N. P. Ong, and R. J. Cava, Phys. Rev. B 79, 195208 (2009). [17] Y. S. Hor, P. Roushan, H. Beidenkopf, J. Seo, D. Qu, J. G. Checkelsky, L. A. Wray, D. Hsieh, Y. Xia, S.-Y. Xu,
D. Qian, M. Z. Hasan, N. P. Ong, A. Yazdani, and R. J. Cava, Phys. Rev. B 81, 195203 (2010). [18] Y. Onishi, Z. Ren, K. Segawa, W. Kaszub, M. Lorenc, Y. Ando and K. Tanaka Phys. Rev. B 91, 085306 (2015). [19] Y Pan, D Wu, J. R. Angevaare, H Luigjes, E Frantzeskakis, N de Jong, E van Heumen, T V Bay, B Zwartsenberg, Y K Huang, M Snelder,A Brinkman, M S Golden and A de Visser New J. Phys. 16 123035 (2014).
5
Supplementary information for “Signature of topological Dirac fermion in magnetization measurements of three-dimensional topological insulator Bi1.5 Sb0.5 Te1.7 Se1.3 ”
B.
We have measured field response of magnetic moment of standard bismuth and palladium samples in SQUIDVSM [MPMS 3, Quantum Design] prior to BSTS single crystal. FIG.6 (a) shows that linear diamagnetic moment of bismuth at 2 and 100 K passes through the origin. This is more clear from FIG.6 (b), which shows magnetic field dependence of differential susceptibility (χ= dM dB ). Absence of any paramagnetic peak around zero field implies that singular paramagnetic susceptibility in BSTS single crystal is not due to any spurious response in our system. FIG.7 (a) shows the expected magnetic behaviour of
Characterization
A.
Magnetic measurements of standard samples
The phase purity of the samples was checked using powder x-ray diffraction(XRD) method with CuKα radiation in a Rigaku x-ray diffractometer(TTRAX II).The XRD pattern can be fitted well with a hexagonal structure with space group R3m for both the sample. The evaluated lattice parameters are ahex = 4.284˚ A, chex = 29.87˚ A for BSTS. Presence of the peaks(1 0 7) and (0 0 12) are very important for ordering of chalcogen layer. The presence of only a preferential direction peaks in the XRD graph of a single flake, indirectly confirm the crystal formation of the sample[17].
0 .4
2 K 1 0 0 K
0 .2 0 .0 -0 .2
M
( ×1 0
-3
e .m .u ./g m )
(a )
-0 .4
-2
-1
0
1
2
B (k O e ) 1 2 0 0 0 0
(a )
In te n s ity ( a r b .)
8 0 0 0 0
(1 0 7 )
6 0 0 0 0
3 0
(0 0 1 2 )
4 0
2 q (d e g re e )
4 0 0 0 0 2 0 0 0 0 0 4 0
6 0
8 0
2 θ(d e g re e ) 1 0 0 0 0 0 0
(0 0 1 5 )
-2 0 -2
(b )
8 0 0 0 0 0
-1 6
−7
2 0
2 K 1 0 0 K
(b )
2 0 0 0 0
χ (×10 e . m . u . / g m - O e )
In te n s ity (a r b .)
-1 2
1 0 0 0 0 0
-1
0
1
2
In te n s ity (a r b .)
B (k O e ) 6 0 0 0 0 0
(0 0 1 8 )
4 0 0 0 0 0
(0 0 6 ) 2 0 0 0 0 0
(0 0 9 )
(0 0 1 2 )
(0 0 2 1 )
FIG. 6. (Color online) (a) Magnetization measured at several representative temperatures for standard diamagnetic bismuth sample. (b) Differential susceptibility (χ= dM ) obdB tained after taking numerical derivative of the magnetization with respect to external magnetic field.
0
2 0
4 0
2 q (d e g re e )
6 0
8 0
FIG. 5. (Color online) (a) X-ray diffraction pattern of powdered single crystals of BSTS. Red open circles are experimental data and red continuous line is the calculated pattern. The peaks(1 0 7) and (0 0 12) are shown in the inset. (b) XRD data of single flake of BSTS which contains peaks of a preferential directionwhich can be seen from the indek mentioned.
paramagnetic palladium sample, provided by the Quantum Design. FIG.7 (b) shows the low field susceptibility of palladium at 2 K and room temperature. The nonlinear behaviour of χ at low field and a weak zero field peak at 2 K are completely suppressed at room temperature. This is entirely different from the singular, robust and linear low field paramagnetic response from the topological surface state in BSTS. 6
0 .0 4
P d
(a )
0 .0 0
2 K 3 0 0 K
M
(e m u /g m )
0 .0 2
-0 .0 2
-0 .0 4
-2
-1
0
1
2 0
B (k O e ) -1 -2 -3
1 2
-4
2 K 3 0 0 K 9
-5
-6
e m u /g m -O e )
P d
(b )
1 5
χ ( ×1 0
0 .0
3
6
-2
-1
0
1
2
B (k O e )
FIG. 7. (Color online) (a) Magnetization of standard paramagnetic palladium sample at 2 and 300 K, (b) Differential susceptibility (χ= dM ) obtained after taking numerical derivadB tive of the magnetization with respect to external magnetic field,
7
0 .5
1 .0
