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Size Dependence of Magnetostructural Transition in MnBi Nanorods H. M. Lu and X. K. Meng* Department of Materials Science and Engineering, National Laboratory of Solid State Microstructures, Nanjing UniVersity, Nanjing 210093, People’s Republic of China ReceiVed: December 22, 2009
In terms of the size-dependent second-order Curie temperature model originally proposed for spherical nanoparticles, we develop an analytical model to describe size dependence of magnetostructural transition temperature Tt(D,L) in MnBi cylindrical nanorods through considering the effects of both diameter D and length L. It is found that the size-dependent magnetostructural transition temperature decreases with declining diameter and length where the diameter effect is the principle factor while the length effect is the secondary one. Moreover, the size dependence of Tt(D,L) of nanorods is weaker than that of zero-dimensional spherical nanoparticles while is stronger than that of one-dimensional cylindrical nanowires. The accuracy of the developed model is verified by the available experimental results.
[ (
Introduction
TC ) T0 1 + β
Nanocrystals are under considerable investigation worldwide because of their wide scientific and technological interest. Due to the unique properties of nanocrystals, the fabrication of nanostructural materials and nanodevices with desirable properties in atomic scale has become an emerging interdisciplinary field involving solid-state physics, materials science, chemistry, and biology.1 In this case, it is important and necessary to understand and predict the thermodynamics of nanocrystals for fabricating the materials for practical applications. Because of its high uniaxial magnetic anisotropy (∼2 × 107 erg/cm3 at 500 K) along its c axis and record magneto-optical Kerr rotation of 1.25° at room temperature, MnBi has been considered for potential applications in high-temperature permanent magnets, data storage, and magneto-optical media.2-6 At room temperature MnBi is ferromagnetic with the hexagonal NiAs structure (low-temperature phase or “FLTP”), and it undergoes a bulk first-order magnetostructural transition at Tt ) 633 K to the paramagnetic “stuffed” Ni2In structure (hightemperature phase or “PHTP”).7-10 Magnetostructural transitions comprise simultaneous magnetic and structural phase changes of an abrupt and hysteretic nature, which are different from the canonical ferromagnetic transition of second-order thermodynamic character. Guillaud’s first report of the magnetostructural transition of MnBi in 1951 was followed by a number of publications with phenomenological derivations and expressions to explain the characteristics of the transition.11-15 All explanations were derived from the concept of a temperature-dependent orbital overlap in the exchange integral and the dependence of the magnetic exchange on unit cell volume V0 is termed volume exchange striction.15 Bean and Rodbell investigated this type of transition in MnAs by assuming that the interaction may be approximated by the molecular field model13 * To whom correspondence should be addressed. Tel.: +86-025-83685585. Fax: +86-025-8359-5535. E-mail:
[email protected].
V - V0 V
)]
(1)
where TC is the Curie temperature while T0 is the Curie temperature if the lattice is not compressible. V is the volume and V0 is the volume in the absence of exchange interactions. β is the slope of the dependence of TC on volume. Although the β value of MnAs had been determined through fitting experimental data,13 there is no report on β value of MnBi. As a result, the existence of unknown fitted parameter β makes it difficult to directly predict TC in terms of eq 1. Moreover, to the best of our knowledge, there is no other analytic expression proposed to describe the size dependence of magnetostructural transition in MnBi nanocrystals. Recently, in terms of the size-dependent melting temperature model, we have established an analytic expression to predict the size-dependent second-order Curie temperature of Ni and FePt zero-dimensional nanoparticles.16,17 In this work, this analytic expression is extended to describe the size-dependent magnetostructural transition in MnBi nanorods through considering the influences of both the diameter and the length. Methodology In terms of the bond-order-length-strength (BOLS) correlation mechanism and the Ising premise,16-19 the second-order Curie transition temperature TC is determined by the exchange interaction energy Eexc and these two parameters have the same size dependence16,17
(
TC(D) Eexc(D) -2Svib 1 ) ) exp TC(∞) Eexc(∞) 3R D/D0 - 1
)
(2)
where ∞ denotes the bulk and R is the ideal gas constant. D0 is a critical diameter at which all atoms are located on its surface. D0 depends on their dimension d and the nearest atomic distance h between Mn and Bi through D0 ) 2(3 - d)h, where d ) 0 for nanoparticles, d ) 1 for nanowires, and d ) 2 for thin films.16,20 Svib is the vibrational component of the melting entropy Sm at the bulk melting temperature. For MnBi alloys, the elec-
10.1021/jp912081f 2010 American Chemical Society Published on Web 02/01/2010
Magnetostructural Transition in MnBi Nanorods
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tronic entropy can be neglected since the nature of chemical bonds does not vary during the transition, namely, Svib ≈ Sm Spos with Spos being the positional entropy. Similarly to the second-order Curie transition in Ni and FePt, the first-order magnetostructural transition in MnBi nanocrystals also transforms from the ferromagnetic phase to paramagnetic one and this transition is also controlled by the magnetic exchange interaction. Considering that magnetic exchange interaction energy is essential in describing the size-dependent magnetic transition, we assume that eq 2 can be extended to predict the magnetostructural transition temperature of MnBi nanocrystals. Simultaneously, the shape of nanocrystals also affects the phase transition temperature.21-23 It was reported that the differences in shape were important for the Curie temperature calculations, since the shape determined the number of surface atoms whose reduced coordination number would affect the exchange energy locally.21 Nanorods exist within the transitional regime between zero-dimensional nanoparticles and onedimensional nanowires and exhibit both diameter- and lengthdependent properties.24,25 Combining eq 2 and the consideration of the shape effect on the ratio of surface atoms to the total atoms, a unified model to describe size- and shape-dependent magnetostructural transition temperature Tt(D,λ) can be proposed as
(
Tt(D, λ) -2λSvib 1 ) exp Tt(∞) 3R D/D0 - 1
)
(3)
(4a)
where VC and Va denote the volumes of nanocrystals and atoms, respectively. Similarly, the number of surface atoms n of the nanocrystals can be calculated as
n ) ηSAC /Aa
(4b)
where AC and Aa are the surface areas of nanocrystals and atoms. ηS denotes the surface packing density, which describes the ratio of the surface area occupied by atoms to the total surface area. In terms of eqs 4a and 4b, δ can be described as
δ ) n/N ) (ηS /ηL)(ACVa /AaVC)
shapes, ηS and ηL can be assumed to be shape-independent as a first-order approximation. In this case, λ can be written as
λ ) δ2 /δ1 ) (AC2 /AC1)(VC1 /VC2)
λ, describing the shape effect on the ratio of surface atoms to the total atoms δ, can be determined as the ratio of δ between nanocrystals with other shape and those with basal shape (e.g., spherical nanoparticles and cylindrical nanowires for zero- and one-dimensional nanocrystals, respectively). To calculate δ and λ values, the numbers of surface atoms and the total atoms should be determined first. To calculate the total number of the atoms N of the nanocrystals, the effect of lattice packing density ηL, which denotes the ratio of the volume of crystal occupied by atoms to the total volume, should be taken into account. In this case, N can be determined as
N ) ηLVC /Va
Figure 1. Tt(D) function of MnBi nanorods with L ) 30 nm where the solid line is plotted based on eq 3 while the closed circular and triangular symbols denote the experimental data on heating and cooling,6,9 respectively. Note that the composition of the sample is MnxBix with x ) 0.05,6,9 Sm(MnBi) ≈ xSm(Mn) + (1 - x)Sm(Bi) ≈ 19.45 J/(mol K) and Spos(MnBi) ≈ -R[x lnx + (1 - x) ln(1 - x)] ≈ 1.66 J/(mol K), Svib(MnBi) ≈ Sm - Spos ) 17.79 J/(mol K), and h ≈ 0.263 nm.26-28
(5)
Assuming that the nanocrystals with other shapes take the same surface and lattice packing modes as those with basal
(6)
where the subscripts 1 and 2 denote the nanocrystals with basal shapes and other shapes. According to the definition of λ, λ is equal to 1 for spherical nanoparticles and cylindrical nanowires. Note that Va and Aa terms disappear in eq 6 since atomic diameter remains constant for the same nanocrystals with different shapes. Moreover, although the ηL term also disappears in eq 6, it is important to determine the grain size in terms of eq 4a. Combining with eqs 3 and 6, size- and shape-dependent magnetostructural transition temperature Tt(D,λ) of MnBi nanocrystals can be determined. Results and Discussion Recently, Kang et al. synthesize isolated FLTP MnBi nanorods with average diameter and length of 10 and 30 nm.6,9,10 Due to the longer length, the third confinement dimension in MnBi nanorods becomes fully relaxed so that the nanorods acquire the two-dimension confinement of nanowires, namely, D0 ) 4h with d ) 1 in this case. It is evident that AC ) 2π(D/ 2)2 + πDL and VC ) π(D/2)2L for nanorods where L denotes the length of nanorods. While for nanowires, the surface areas of the top and bottom of nanowire are negligible as a firstorder approximation due to the length L′ . D, namely, AC ≈ πDL′ and VC ) π(D/2)2L′. In terms of eq 6, the shape factor λ for nanorods can be determined as
λ ≈ (D/2 + L)/L
(7)
In terms of eqs 3 and 7, Figure 1 shows the calculated Tt(D) functions of Mn0.05Bi0.95 nanorods with L ) 30 nm on heating (the red solid line) and cooling (the blue solid line) processes. It is evident that Tt(D) decreases with declining D and the drop becomes dramatic at D < 8 nm. As a comparison, available experimental results (red circular and blue triangular symbols for heating and cooling processes, respectively) are also listed. Due to the hysteretic nature of magnetostructural transition in
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Lu and Meng with λ ) 1, it is obvious that the size dependence of magnetostructural transition temperature for nanorods is stronger than that for cylindrical nanowires in terms of eq 3. However, although λ is also equal to 1 for spherical nanoparticles, the size dependence of Tt(D) for nanorods is weaker than that of spherical nanoparticles since D0(nanoparticles) ) 6h is 1.5 times larger than D0(nanorods) ) 4h. Namely, Tt(D, cylindrical nanowires) > Tt(D, nanorods) > Tt(D, spherical nanoparticles) at the same D value. Conclusions
Figure 2. Tt(D,L)/Tt(∞) function of MnBi nanorods described by eq 3 with λ ≈ (D/2 + L)/L.
MnBi nanocrystals, both Tt(D ) 10 nm, L ) 30 nm) and Tt(∞) values (525 and 633 K) on heating are different from those (490 and 603) on cooling.6,9 However, Tt(D ) 10 nm, L ) 30 nm)/ Tt(∞) ) 525/633 ≈ 0.83 on heating is very close to that (490/ 603 ≈ 0.81) on cooling, which indicates that size dependence of magnetostructural transition temperature on heating is nearly the same as that on cooling. As shown in Figure 1, the model predictions correspond well to the corresponding experimental results on heating and cooling. In contrast to the Mn0.05Bi0.95 results, Kang et al. found that the second-order transition temperatures of Mn0.1Bi0.9 nanoparticles with grain size of 50-100 nm are comparable to the respective bulk values on heating and cooling.6 On the basis of eq 3 with D0 ) 6h, λ ) 1, and Svib(Mn0.1Bi0.9) ≈ 16.2 J/(mol K), the depression of second-order Curie temperature of Mn0.1Bi0.9 nanoparticles is only 2-4%, which indicates that its size dependence is indeed negligible in this case. Together with previous work on size-dependent Curie temperatures of Ni and FePt nanoparticles,16,17 the work here indicates that eq 2 or eq 3 can not only describe the second-order magnetic transition temperature but also determine the first-order magnetostructural transition temperature, which also confirms the necessity of magnetic exchange interaction energy in describing the effect of size on magnetic transition. Note that exp(-x) ≈ 1 - x when x is small enough (e.g., x < 0.1), eq 3 with λ ≈ (D/2 + L)/L for nanorods can be simplified as
Tt(D, L) 4hSvib 1 2 ≈1+ Tt(∞) 3R L D
(
)
(8)
It is evident that Tt(D,L) drops with decreasing D and L. Note that if L > D (or 1/L < 1/D) usually for nanorods and the coefficient of 1/L is 1 while that of 1/D is 2, it can thus be claimed that the diameter effect is the principle factor in the depression of Tt(D,L) while the length effect is the secondary one. The above conclusion can also be found in Figure 2 where the effects of both the diameter and the length on Tt(D,L) functions are displayed in terms of eq 3. For example, when D is fixed at 10 nm, Tt(10 < L < 40 nm)/Tt(∞) ranges from 0.78 to 0.83, while at L ) 10 nm, Tt(2 < D < 10 nm)/Tt(∞) is in the range of 0.16-0.78. Considering that λ ≈ (D/2 + L)/L and D < L for nanorods, there is 1 < λ < 1.5. In comparison with cylindrical nanowires
A unified thermodynamical model has been developed to describe the diameter and length dependences of magnetostructural transition temperature Tt(D,L) of MnBi nanorods. It can be found that Tt(D,L) decreases with decreasing D and L where the diameter effect is the principle factor while the length effect is the secondary one. Reasonable agreement between the model and corresponding experimental results can be found for the Tt(D,L) function. Considering the effects of size and dimension, the depression tendency of Tt(D) function takes the following sequence: ∆Tt(D, spherical nanoparticles) > ∆Tt(D, nanorods) > ∆Tt(D, cylindrical nanowires) at the same diameter with ∆Tt(D) ) Tt(∞) - Tt(D) being the difference. Moreover, together with previous work,16,17 the work here suggests that our model can determine the size dependences of both the second-order magnetic transition temperature and the first-order magnetostructural transition temperature, which results from the fact that our model correctly describes the size-dependent magnetic exchange interaction energy. Acknowledgment. Financial support from the State Key Program for Basic Research of China (2004CB619305, 2010CB631004) and the Postdoctoral Science Foundation of China (200801370) is acknowledged. References and Notes (1) Gleiter, H. Acta Mater. 2000, 48, 1. (2) Chen, D.; Aagard, R. L. J. Appl. Phys. 1970, 41, 2530. (3) Yang, J. B.; Kamaraju, K.; Yelon, W. B.; James, W. J. Appl. Phys. Lett. 2001, 79, 1846. (4) Saha, S.; Obermyer, R. T.; Zande, B. J.; Chandhok, V. K.; Simizu, S.; Sankar, S. G.; Horton, J. A. J. Appl. Phys. 2002, 91, 8525. (5) Srajer, G.; et al. J. Magn. Magn. Mater. 2006, 307, 1. (6) Kang, K.; Moodenbaugh, A. R.; Lewis, L. H. Appl. Phys. Lett. 2007, 90, 153112. (7) Heikes, R. R. Phys. ReV. 1955, 99, 446. (8) Yoshida, H.; Shima, T.; Takahashi, T.; Fujimori, H.; Abe, S.; Kaneko, T.; Kanomata, T.; Suzuki, T. J. Alloys Compd. 2001, 317-318, 297. (9) Kang, K.; Lewis, L. H.; Moodenbaugh, A. R. Appl. Phys. Lett. 2005, 87, 062505. (10) Kang, K.; Yoon, W. S.; Park, S.; Moodenbaugh, A. R.; Lewis, L. H. AdV. Funct. Mater. 2009, 19, 1100. (11) Guillaud, C. J. Phys. Radium 1951, 12, 223. (12) Smart, J. S. Phys. ReV. 1953, 90, 55. (13) Bean, C. P.; Rodbell, D. S. Phys. ReV. 1962, 126, 104. (14) Goodenough, J. B.; Kafalas, J. A. Phys. ReV. 1967, 157, 389. (15) Morosin, B. Phys. ReV. B 1970, 1, 236. (16) Lu, H. M.; Cao, Z. H.; Zhao, C. L.; Li, P. Y.; Meng, X. K. J. Appl. Phys. 2008, 103, 123526. (17) Lu, H. M.; Li, P. Y.; Huang, Y. N.; Meng, X. K.; Zhang, X. Y.; Liu, Q. J. Appl. Phys. 2009, 105, 023516. (18) Sun, C. Q.; Zhong, W. H.; Li, S.; Tay, B. K.; Bai, H. L.; Jiang, E. Y. J. Phys. Chem. B 2004, 108, 1080. (19) Sun, C. Q. Prog. Mater. Sci. 2009, 54, 179. (20) Jiang, Q.; Tong, H. Y.; Hsu, D. T.; Okuyama, K.; Shi, F. G. Thin Solid Films 1998, 312, 357. (21) Evans, R.; Nowak, U.; Dorfbauer, F.; Shrefl, T.; Mryasov, O.; Chantrell, R. W.; Grochola, G. J. Appl. Phys. 2006, 99, 08G703.
Magnetostructural Transition in MnBi Nanorods (22) Guisbiers, G.; Kazan, M.; Van Overschelde, O.; Wautelet, M.; Pereira, S. J. Phys. Chem. C 2008, 112, 4097. (23) Lu, H. M.; Li, P. Y.; Cao, Z. H.; Meng, X. K. J. Phys. Chem. C 2009, 113, 7598, and references therein. (24) Li, L. S.; Hu, J. T.; Yang, W. D.; Alivisatos, A. P. Nano Lett. 2001, 1, 349.
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