- Email: [email protected]
Magnetic phase transition in Heisenberg antiferromagnetic films with easy-axis single-ion anisotropy
Physica A 391 (2012) 1984–1990
Contents lists available at SciVerse ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Magnetic phase transition in Heisenberg antiferromagnetic films with easy-axis single-ion anisotropy Kok-Kwei Pan ∗ Physics Group, Center of General Education, Chang Gung University, No. 259, Wen-Hua 1st Road, Kwei-San, Tao-Yuan 33302, Taiwan, ROC
article
info
Article history: Received 14 August 2011 Received in revised form 5 November 2011 Available online 26 November 2011 Keywords: Heisenberg antiferromagnetic films Easy-axis single-ion anisotropy Critical behavior Shift exponent
abstract The staggered susceptibility of spin-1 and spin-3/2 Heisenberg antiferromagnet with easy-axis single-ion anisotropy on the cubic lattice films consisting of n = 2, 3, 4, 5 and 6 interacting square lattice layers is studied by high-temperature series expansions. Sixth order series in J /kB T have been obtained for free-surface boundary conditions. The dependence of the Néel temperature on film thickness n and easy-axis anisotropy D has been investigated. The shifts of the Néel temperature from the bulk value can be described by a power law n−λ with a shift exponent λ, where λ is the inverse of the bulk correlation length exponent. The effect of easy-axis single-ion anisotropy on shift exponent of antiferromagnetic films has been studied. A comparison is made with related works. The results obtained are qualitatively consistent with the predictions of finite-size scaling theory. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The study of the magnetic properties of thin films has attracted much attention in recent years [1]. Because of important progress in epitaxial techniques necessary for the precise preparation of thin films, our understanding of finite-size effects on the magnetic phase transition [2–4] in these quasi-two-dimensional systems has advanced considerably. The study of magnetic properties of magnets on a nanometer scale is in general not only of fundamental interest but also of technological importance. Of particular interest, from both the experimental and theoretical points of view, is the critical behavior of magnetic films, from which one can explore and test the universality hypothesis. In low-dimensional magnetic materials, single-ion anisotropy is important for establishing three-dimensional long-range order at a finite temperature [5] and plays a major role in determining the magnetic behavior of the system [6] with spin greater than 1/2. A number of theoretical works have been devoted over the years for investigating the magnetic properties and phase transition of the Ising and Heisenberg models in thin films [7–11] and superlattices [12]. However, systematic analytical studies of the influence of single-ion anisotropy on phase transition and critical behavior of magnetic thin films are still sparse, apart from theoretical studies on the magnetic ordering in Heisenberg thin films with single-ion anisotropy and dipolar interactions using the Green function theory [13] and the Monte Carlo simulation method [14]. In this work, we investigate the magnetic phase transition of the spin-1 and spin- 32 Heisenberg antiferromagnet with easy-axis single-ion anisotropy in cubic lattice films by the exact high-temperature series expansions [10,15]. We consider ∞ × ∞ × n simple cubic (sc) lattice which are infinite in two of its dimensions but of n finite layers in the third dimension. We impose free-surface boundary [7,8] condition, in which each surface spin lacks one nearest neighboring spin on the sc lattice. Restricting the size in one and more dimensions will affect the critical behavior of a system. If TN (n) denotes the Néel temperature for the n-layer film and TN (∞) is the three-dimensional (3D) bulk Néel temperature, the shifts
∗
Tel.: +886 3 2118800; fax: +886 3 2118033. E-mail address: [email protected].
0378-4371/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2011.11.048
K.-K. Pan / Physica A 391 (2012) 1984–1990
1985
in the Néel temperature TN (n) from the bulk value TN (∞) can be described by a finite-size scaling relation [7,8] that [TN (∞) − TN (n)]/TN (∞) varies with thickness n, as n−λ for large n with a shift exponent λ. The shift exponent λ is related to the bulk correlation length exponent ν by λ = 1/ν [2,16]. This paper is organized as follows. In Section 2, a description of the model and the formalism of the high-temperature series expansions (HTSE) method for the free energy and the staggered susceptibility of the Heisenberg antiferromagnet with easy-axis single-ion anisotropy on the sc lattice films are given. In Section 3, we present the analysis of the series and the results of the calculation. The conclusions are given in Section 4. 2. Model and method The Hamiltonian of the Heisenberg antiferromagnet model with easy-axis single-ion anisotropy is given by H =
−
Jij S⃗i · S⃗j − D
− − − (Sjz )2 − hs Siz + hs Sjz .
⟨i,j⟩
i∈A
j
(1)
j∈B
∑
The first term is the Heisenberg antiferromagnetic exchange interaction between spins and the sum ⟨i,j⟩ is over all nearest neighbor bonds. The second term represents the easy-axis single-ion anisotropy with a positive anisotropy parameter D. hs in the Zeeman energy terms are staggered magnetic fields applied along the axis of magnetic ordering on two sublattices A and B for calculating the staggered susceptibility. The easy-axis anisotropy in the Hamiltonian [Eq. (1)] favors the Néel order along the z axis and the system shows an Ising-like critical behavior [17,18]. The Hamiltonian is divided into two parts, the mean-field Hamiltonian
− − 1 {D(Siz )2 + hm Siz } − {D(Siz )2 − hm Sjz } + NJz (M + )2
H0 = −
i∈A
2
j∈B
(2)
and the perturbation Hamiltonian H1 = J
− J − + − [Siz − M + ][Sjz + M + ] + [Si Sj + Si− Sj+ ] 2 ⟨i,j⟩
⟨i,j⟩
(3)
where hm = { JzM + + hs } and z is the number of nearest neighbors and N is the total number of spins in a thin film. M + is the sublattice magnetization which minimizes the free energy of the system. Here we give a brief description of the essentials of the HTSE method. According to the fluctuation–dissipation theorem [19], the high-temperature series expansions for the staggered susceptibility series per site for an n-layer film is obtained from the expansion of static two-spin correlation function by using the linked-cluster expansion [10,20]
β − − Tr {(Skz ) (−Slz ) e−β H } N k∈A l∈B Tr {e−β H } ∫ ∫ β 1 −− β = χ0s + dτk dτl ⟨T {[Skz (τk )][−Slz (τl )] S (β)}⟩c , nN2 β k∈A l∈B 0 0
χns =
(4)
where S (β) =
β
∫ ∞ − (−1)m m=1
m!
dτ1
β
∫
0
0
dτ2 · · ·
β
∫
dτm [H1 (τ1 )H1 (τ2 ) · · · H1 (τm )],
(5)
0
β = (kB T )−1 , T is the Dyson τ -ordering operator and the angular brackets represent the canonical thermal average over the unperturbed Hamiltonian H0 . The subscript c denotes that the cumulant part of the τ -ordered product or only the connected diagrams have to be considered. N2 is the number of lattice sites in each spin layer and χ0s is the staggered susceptibility term corresponding to the mean-field Hamiltonian H0 . The staggered susceptibility diagrams are therefore two-rooted diagrams with two spin operators Skz and Slz placed together on one of the sites of the free energy diagrams. The staggered susceptibility series expansion in [Eq. (4)] can be written in power of {β J } as
χ =χ + s n
s 0
∞ − (−1)m m=1
m!
−
W (cm )L(cm )I (cm ) {β J }m .
(6)
cm
The calculation of the coefficients in the mth-order term of the staggered susceptibility series expansion includes three parts: (a) The finding and enumerating of all two-rooted unique diagrams or graphs cm and the weight of each unique graph or the number of times the graph appears in the mth-order expansion W (cm ). (b) Calculating the lattice constant of each graph or the number of times each graph can be embedded in the sc lattice film system L(cm ). (c) Evaluating the multiple integrals containing τ -ordered cumulant products of spin operators for each graph I (cm ).
1986
K.-K. Pan / Physica A 391 (2012) 1984–1990
Table 1 The coefficients anl (x, y) from (Eq. (10)) in the text for spin-1 system are written in the form as anl (x, y) = (1 + 2x)−(l+1) αln,p,q up to order l = 2 are listed for n = 2–6 layers.
∑
p,q
αln,p,q xp y−q . The coefficients
l
p
q
αl2,p,q
αl3,p,q
αl4,p,q
αl5,p,q
αl6,p,q
0 1 2 2 2 2 2 2 2 2 2 2 2 2
1 2 0 1 1 1 2 2 2 2 3 3 3 3
0 0 3 0 1 3 0 1 2 3 0 1 2 3
2.00 20.00 5.00 1.67 −10.00 10.00 −6.67 −10.00 −10.00 −5.00 160.00 −10.00 −20.00 −10.00
2.00 21.33 5.33 1.78 −10.67 10.67 −7.11 −10.67 −10.67 −5.33 186.67 −10.67 −21.33 −10.67
2.00 22.00 5.50 1.83 −11.00 11.00 −7.33 −11.00 −11.00 −5.50 200.00 −11.00 −22.00 −11.00
2.00 22.40 5.60 1.87 −11.20 11.20 −7.47 −11.20 −11.20 −5.60 208.00 −11.20 −22.40 −11.20
2.00 22.67 5.67 1.89 −11.33 11.33 −7.56 −11.33 −11.33 −5.67 213.33 −11.33 −22.67 −11.33
All two-rooted connected graphs, the corresponding weight factors and the lattice constants are calculated by computer programs [10,18]. The following standard basis operators [18,20] are used in order to facilitate the calculation of the multiple integrals containing τ -ordered products of spin operators in series expansion: Lmn ≡ | ϵm ⟩⟨ ϵn |
(7)
where | ϵm ⟩, | ϵn ⟩ are eigenstates of mean-field Hamiltonian H0 . The spin operators are then rewritten in terms of the standard basis operators. For spin-1 system, S+ =
−
⟨ ϵm | S + | ϵn ⟩ Lmn =
m,n
S− =
−
⟨ ϵm | S − | ϵn ⟩ Lmn =
√
2(L12 + L23 ),
√
2(L21 + L32 ),
(8)
m,n
Sz =
−
⟨ ϵm | S z | ϵn ⟩ Lmn = (L11 − L33 )
m,n
and for spin-3/2 system, S+ =
−
⟨ ϵm | S + | ϵn ⟩ Lmn =
m,n
S− =
−
⟨ ϵm | S − | ϵn ⟩ Lmn =
√
√ 3L12 + 2L23 +
√
√ 3L21 + 2L32 +
3L34 , 3L43 ,
m,n
Sz =
−
⟨ ϵm | S z | ϵn ⟩ Lmn =
m,n
3 2
(9)
1
(L11 − L44 ) + (L22 − L33 ). 2
The multiple integrals containing τ -ordered products of spin operators in each connected diagrams of the staggered susceptibility series are calculated by using the multiple-site Wick reduction theorem and the standard basis operators [18,20]. The zero-field (hs = 0) high-temperature series for the staggered susceptibility series per site of n-layer lattice with free surface is obtained as
β −1 χns =
∞ −
anl (x, y){β J }l
(10)
l =0
where the coefficients anl (x, y) are polynomials in the variables x = eβ D and y = β D. We obtained the staggered susceptibility series up to sixth order for the sc lattice films. The whole coefficients anl are too long, so here we present these coefficients only to the second order in Tables 1 and 2 for spin-1 and spin-3/2, respectively. The full expressions of the staggered susceptibility series expansion of spin-1 and spin-3/2 systems are available upon request. The series obtained have been checked in the D = 0 limit. In the D = 0 limit, the results of the present calculation for the spin-1 and spin- 32 antiferromagnetic Heisenberg model on sc lattice films with n = 2–6 layers agree completely with previous results [10]. In addition, our results for the bulk value(n = ∞) of spin-1 system for finite D agree with the hightemperature series of spin-1 antiferromagnetic Heisenberg model with easy-axis single-ion anisotropy on the sc lattice [18].
K.-K. Pan / Physica A 391 (2012) 1984–1990
1987
Table 2 The coefficients anl (x, y) from (Eq. (10)) in the text for spin- 32 system are written in the form as anl (x, y) = (1 + x2 )−(l+1) βln,p,q up to order l = 2 are listed for n = 2–6 layers.
∑
p,q
βln,p,q (x2 )p y−q . The coefficients
l
p
q
βl2,p,q
βl3,p,q
βl4,p,q
βl5,p,q
βl6,p,q
0 0 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
0 1 0 1 2 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3
0 0 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3
0.25 2.25 0.31 5.62 25.31 −0.10 1.52 0.70 −7.85 0.05 32.46 −14.30 −0.35 86.72 4.10 −17.11 7.85 227.81 3.16 −2.11 0.35
0.25 2.25 0.33 6.00 27.00 −0.08 1.62 0.75 −8.38 0.90 34.62 −15.25 −0.38 100.09 4.37 −18.25 8.38 265.78 3.38 −2.25 0.38
0.25 2.25 0.34 6.19 27.84 −0.07 1.68 0.77 −8.64 1.32 35.71 −15.73 −0.39 106.78 4.51 −18.82 8.64 284.77 3.48 −2.32 0.39
0.25 2.25 0.35 6.30 28.35 −0.06 1.71 0.79 −8.79 1.58 36.36 −16.01 −0.39 110.79 4.59 −19.16 8.79 296.16 3.54 −2.36 0.39
0.25 2.25 0.35 6.38 28.69 −0.05 1.73 0.80 −8.90 1.75 36.79 −16.20 −0.40 113.47 4.65 −19.39 8.90 303.75 3.59 −2.39 0.40
3. Results and discussions 3.1. Effect of anisotropy on the thickness dependence of the Néel temperature The series are extrapolated to study the crossover from three-dimensional (3D) critical behavior at high temperatures to two-dimensional (2D) critical behavior in critical temperature region. The Néel temperatures TN (n) for the n-layer films are estimated from the divergence of the paramagnetic staggered susceptibility series (Eq. (10)) by using the ratio method and Padé approximants [21]. We have applied [22] the ratio plot for (anl /anl−2 )1/2 versus {l(l − 1)}−1/2 to estimate the Néel temperatures TN (n) for the n-layer films. For each value of D/J, the Néel temperature TN (n) for the n-layer film is estimated self-consistently from the extrapolation given by kTN (n) J
= µn (l, m) =
√ √ [( l(l − 1) ) νln − ( m(m − 1)) νmn ] √ √ l(l − 1) − m(m − 1)
(11)
where νln = [anl (x, y)/anl−2 (x, y)]1/2 . In Figs. 1 and 2, we show the Néel temperature kTN /J as a function of single-ion anisotropy D/J of sc lattice films with n = 2, 3, 4, 5, 6, and ∞ layers for spin-1 and spin-3/2, respectively. They are plotted as a solid line which is estimated from the average of the sixth-order µn (6, 5) and µn (6, 4) extrapolations. The extrapolation shows good convergence in the sense that each extrapolation value differs from the average value by less than 1%. We have also used the Padé approximant analysis of the series to estimate the Néel temperature. The Néel temperature is estimated from the poles of direct Padé approximants to the series {χns }1/γ . The critical exponent for the 2D Ising [23], γ = 1.75, is assumed for all finite thickness films in order to obtain the best possible estimates of the Néel temperature. For the bulk (n = ∞), the critical temperature is obtained from the series with the 3D Ising [24] critical exponent γ = 1.25. The estimates of the Néel temperature obtained from the average of [2/3], [2/4] and [3/3] the Padé approximant for spin-1 and spin- 32 are also plotted in Figs. 1 and 2. The over-all results of the Padé approximants analysis of the series are in quantitative agreement with the ratio results with the uncertainties reaching 5% for the films of n = 5 and 6 layers. 3.2. The Néel temperature shifts and the shift exponent The shift of the reduced Néel temperature kTN (n)/J from the 3D bulk value kTN (∞)/J can be described by a power law characterized by a shift exponent λ according to the scaling theory [7,8] TN (∞) − TN (n) TN (∞)
≈ n−λ .
(12)
A log–log plot of [TN (∞) − TN (n)]/TN (∞) versus n for spin-1 with various values of easy-axis anisotropy D/J is shown in Fig. 3. The inset in Fig. 3 shows a log–log plot of reduced Néel temperature shifts versus n for spin- 32 . The error bars represent the range of uncertainty in the estimate of the Néel temperature for each n-layer film. The uncertainties for the
1988
K.-K. Pan / Physica A 391 (2012) 1984–1990
Fig. 1. The Néel temperature as a function of easy-axis anisotropy of sc lattice films for spin-1 and n = 2, 3, 4, 5, 6, and ∞ layers. The solid line shows the results of the series expansion by using the ratio method. The triangles are the results of Padé approximants with estimated errors.
Fig. 2. The Néel temperature as a function of easy-axis anisotropy of sc lattice films for spin- 32 and n = 2, 3, 4, 5, 6, and ∞ layers. The solid line shows the results of the series expansion by using the ratio method. The triangles are the results of Padé approximants with estimated errors.
films of n = 5 and 6 are much larger. A good straight-line fit to various values of D/J for both spin-1 and spin- 23 is obtained. The estimated values of shift exponent λ for spin-1 and spin- 23 are λ ≃ 1.28 ± 0.3 and λ ≃ 1.25 ± 0.3, respectively. The uncertainties of the fits are determined by the lack of precision in the values of the Néel temperatures TN (n). In Fig. 4, we show [TN (∞) − TN (n)]/TN (∞) versus (1/n)λ for spin-1 with various values of easy-axis anisotropy D/J with λ ≃ 1.28 which corresponds to λ = 1/ν with ν ≃ 0.78. A good straight-line fit to various values of D/J is obtained. The corresponding plot for spin- 32 with λ ≃ 1.25(ν ≃ 0.80) is shown in the inset. These values are close to the critical exponent ν = 0.79 for Ising thin films by Monte Carlo simulation [11]. The results obtained are also consistent with the general universality principles [25] that the shift exponents as well as critical exponents depend only on spin and lattice dimensionality. 4. Conclusions In Section 4, we have studied the staggered susceptibilities of the spin-1 and spin-3/2 Heisenberg antiferromagnet with easy-axis single-ion anisotropy on a n-layer sc lattice films with free-surface boundary condition theoretically on the basis
K.-K. Pan / Physica A 391 (2012) 1984–1990
1989
Fig. 3. The Log–log plot of the shift of the reduced Néel temperature [TN (∞) − TN (n)]/TN (∞) versus layers n for spin-1 sc lattice films with various values of easy-axis anisotropy D/J. The error bars represent the range of uncertainty for each value of n. The straight line shows a linear fitting and the slope of the straight line yields the shift exponent λ. In the inset, we show a log–log plot of reduced Néel temperature shifts versus n for spin- 32 .
Fig. 4. The Néel temperature shift [TN (∞) − TN (n)]/TN (∞) versus (1/n)λ for spin-1 sc lattice films with various values of easy-axis anisotropy D/J with λ ≃ 1.28. The error bars represent the range of uncertainty for each value of n. In the inset, we show [TN (∞) − TN (n)]/TN (∞) versus (1/n)λ plot for spin- 32 with λ ≃ 1.25.
of exact high-temperature series expansions. The effect of easy-axis single-ion anisotropy on the thickness dependence of the Néel temperature TN and shift exponent λ of antiferromagnetic films has been studied. The shift of the Néel temperature TN for various values of easy-axis anisotropy D/J is described by a scaling relation with a shift exponent of λ ≃ 1.28 ± 0.3 for spin-1 and λ ≃ 1.25 ± 0.3 for spin- 32 . Our estimates of the shift exponent for spin-1 and spin- 32 are in agreement with
1990
K.-K. Pan / Physica A 391 (2012) 1984–1990
Monte Carlo simulations results. They are also consistent with the finite-size scaling prediction for the 3D Ising universality class and the general universality principles of the spin independent shift exponents. The lack of precision in the values of Néel temperatures and the limited number of layers due to the shortness of the series lead to uncertainties in the estimates of the shift exponent λ. Longer series are essential to obtain a more precise estimate of the shift exponent. Acknowledgments This research was supported by the Chang Gung University of Republic of China under Grant No. UMRPD590081. We are grateful to the National Center for High-performance Computing (NCHC) for computer time and facilities. References [1] C. Lu, S. Bader, in: L.M. Falicov, F. Mejia-Lira, J.L. Moran-Lopez (Eds.), Magnetic Properties of Low Dimensional System II, Springer Verlag, Berlin, 1990. [2] M.N. Barber, in: C. Domb, J.L. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, vol. 8, Academic, New York, 1983. [3] S. Andrieu, C. Chatelain, M. Lemine, B. Berche, Ph. Bauer, Phys. Rev. Lett. 86 (2001) 3883; Malte Henkel, Stéphane Andrieu, Philippe Bauer, Michel Piecuch, Phys. Rev. Lett. 80 (1998) 4783. [4] T. Ambrose, C.L. Chen, Phys. Rev. Lett. 76 (1996) 1743; Yi Li, K. Baberschke, Phys. Rev. Lett. 68 (1992) 1208. [5] W. Knafo, C. Meingast, K. Grube, S. Drobnik, P. Popovich, P. Schweiss, P. Adelmann, Th. Wolf, H.v. Löhneysen, Phys. Rev. Lett. 99 (2007) 137206. [6] J.E. Crow, R.P. Gruertin, and T.W. Mihalisin (Eds.), Crystalline Electric Field and Structural Effect in f -Electron Systems, Plenum, New York, 1980. [7] G.A.T. Allan, Phys. Rev. B 1 (1970) 352; T.W. Capehart, M.E. Fisher, Phys. Rev. B 13 (1976) 5021. [8] D.S. Ritchie, M.E. Fisher, Phys. Rev. B 7 (480) (1973) 480. [9] H.T. Diep, Phys. Rev. B 43 (1991) 8509. [10] K.K. Pan, Phys. Rev. B 71 (2005) 134524; K.K. Pan, Phys. Rev. B 64 (2001) 224401. [11] P. Cossio, J. Mazo-Zuluaga, J. Restrepo, Physica B 384 (2006) 227; J. Mazo-Zuluaga, J. Restrepo, Physica B 384 (2006) 224. [12] H. Bakrim, M. Hamedoun, A. Hourmatallah, J. Magn. Magn. Mater. 261 (2003) 415; S. Zhang, G. Zhang, J. Appl. Phys. 75 (1994) 6685; A.S. Carriço, R.E. Camley, Phys. Rev. B 45 (1992) 13117. [13] P. Fröbrich, P.J. Kuntz, Phys. Rep. 432 (2006) 223. [14] J.J. Alonso, J.F. Fernández, Phys. Rev. B 74 (2006) 184416. [15] M. Wortis, in: C. Domb, M.S. Green (Eds.), Phase Transitions and Critical Phenomena, 3, Academic, New York, 1974. [16] M.E. Fisher, in: C. Domb, M.S. Green (Eds.), Critical Phenomena, Academic, New York, 1971. [17] J. Oitmaa, C.J. Hamer, Phys. Rev. B 77 (2008) 224435. [18] K.K. Pan, Phys. Rev. B 79 (2009) 134414, and references therein. [19] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Clarendon Press, Oxford, 1971. [20] C. Wentworth, Y.L. Wang, Phys. Rev. B 36 (1987) 8687; Y.L. Wang, C. Wentworth, B. Westwanski, Phys. Rev. B 32 (1985) 1805. [21] D.S. Gaunt, A.J. Guttman, in: C. Domb, M.S. Green (Eds.), Phase Transitions and Critical Phenomena, 3, Academic, New York, 1974. [22] K. Binder, P.C. Hohenberg, Phys. Rev. B 9 (1974) 2194; K. Binder, P.C. Hohenberg, Phys. Rev. B 6 (1972) 3461. [23] J.H. Chen, M.E. Fisher, B.G. Nickel, Phys. Rev. Lett. 48 (1982) 630; M.E. Fisher, Rep. Prog. Phys. 30 (1967) 615. [24] A.M. Ferrenberg, D.P. Landau, Phys. Rev. B 44 (1991) 5081. [25] L.P. Kadanoff, Statistical Physics: Statics, Dynamics and Renormalization, World Scientific, Singapore, 2000.
Contents lists available at SciVerse ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Magnetic phase transition in Heisenberg antiferromagnetic films with easy-axis single-ion anisotropy Kok-Kwei Pan ∗ Physics Group, Center of General Education, Chang Gung University, No. 259, Wen-Hua 1st Road, Kwei-San, Tao-Yuan 33302, Taiwan, ROC
article
info
Article history: Received 14 August 2011 Received in revised form 5 November 2011 Available online 26 November 2011 Keywords: Heisenberg antiferromagnetic films Easy-axis single-ion anisotropy Critical behavior Shift exponent
abstract The staggered susceptibility of spin-1 and spin-3/2 Heisenberg antiferromagnet with easy-axis single-ion anisotropy on the cubic lattice films consisting of n = 2, 3, 4, 5 and 6 interacting square lattice layers is studied by high-temperature series expansions. Sixth order series in J /kB T have been obtained for free-surface boundary conditions. The dependence of the Néel temperature on film thickness n and easy-axis anisotropy D has been investigated. The shifts of the Néel temperature from the bulk value can be described by a power law n−λ with a shift exponent λ, where λ is the inverse of the bulk correlation length exponent. The effect of easy-axis single-ion anisotropy on shift exponent of antiferromagnetic films has been studied. A comparison is made with related works. The results obtained are qualitatively consistent with the predictions of finite-size scaling theory. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The study of the magnetic properties of thin films has attracted much attention in recent years [1]. Because of important progress in epitaxial techniques necessary for the precise preparation of thin films, our understanding of finite-size effects on the magnetic phase transition [2–4] in these quasi-two-dimensional systems has advanced considerably. The study of magnetic properties of magnets on a nanometer scale is in general not only of fundamental interest but also of technological importance. Of particular interest, from both the experimental and theoretical points of view, is the critical behavior of magnetic films, from which one can explore and test the universality hypothesis. In low-dimensional magnetic materials, single-ion anisotropy is important for establishing three-dimensional long-range order at a finite temperature [5] and plays a major role in determining the magnetic behavior of the system [6] with spin greater than 1/2. A number of theoretical works have been devoted over the years for investigating the magnetic properties and phase transition of the Ising and Heisenberg models in thin films [7–11] and superlattices [12]. However, systematic analytical studies of the influence of single-ion anisotropy on phase transition and critical behavior of magnetic thin films are still sparse, apart from theoretical studies on the magnetic ordering in Heisenberg thin films with single-ion anisotropy and dipolar interactions using the Green function theory [13] and the Monte Carlo simulation method [14]. In this work, we investigate the magnetic phase transition of the spin-1 and spin- 32 Heisenberg antiferromagnet with easy-axis single-ion anisotropy in cubic lattice films by the exact high-temperature series expansions [10,15]. We consider ∞ × ∞ × n simple cubic (sc) lattice which are infinite in two of its dimensions but of n finite layers in the third dimension. We impose free-surface boundary [7,8] condition, in which each surface spin lacks one nearest neighboring spin on the sc lattice. Restricting the size in one and more dimensions will affect the critical behavior of a system. If TN (n) denotes the Néel temperature for the n-layer film and TN (∞) is the three-dimensional (3D) bulk Néel temperature, the shifts
∗
Tel.: +886 3 2118800; fax: +886 3 2118033. E-mail address: [email protected].
0378-4371/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2011.11.048
K.-K. Pan / Physica A 391 (2012) 1984–1990
1985
in the Néel temperature TN (n) from the bulk value TN (∞) can be described by a finite-size scaling relation [7,8] that [TN (∞) − TN (n)]/TN (∞) varies with thickness n, as n−λ for large n with a shift exponent λ. The shift exponent λ is related to the bulk correlation length exponent ν by λ = 1/ν [2,16]. This paper is organized as follows. In Section 2, a description of the model and the formalism of the high-temperature series expansions (HTSE) method for the free energy and the staggered susceptibility of the Heisenberg antiferromagnet with easy-axis single-ion anisotropy on the sc lattice films are given. In Section 3, we present the analysis of the series and the results of the calculation. The conclusions are given in Section 4. 2. Model and method The Hamiltonian of the Heisenberg antiferromagnet model with easy-axis single-ion anisotropy is given by H =
−
Jij S⃗i · S⃗j − D
− − − (Sjz )2 − hs Siz + hs Sjz .
⟨i,j⟩
i∈A
j
(1)
j∈B
∑
The first term is the Heisenberg antiferromagnetic exchange interaction between spins and the sum ⟨i,j⟩ is over all nearest neighbor bonds. The second term represents the easy-axis single-ion anisotropy with a positive anisotropy parameter D. hs in the Zeeman energy terms are staggered magnetic fields applied along the axis of magnetic ordering on two sublattices A and B for calculating the staggered susceptibility. The easy-axis anisotropy in the Hamiltonian [Eq. (1)] favors the Néel order along the z axis and the system shows an Ising-like critical behavior [17,18]. The Hamiltonian is divided into two parts, the mean-field Hamiltonian
− − 1 {D(Siz )2 + hm Siz } − {D(Siz )2 − hm Sjz } + NJz (M + )2
H0 = −
i∈A
2
j∈B
(2)
and the perturbation Hamiltonian H1 = J
− J − + − [Siz − M + ][Sjz + M + ] + [Si Sj + Si− Sj+ ] 2 ⟨i,j⟩
⟨i,j⟩
(3)
where hm = { JzM + + hs } and z is the number of nearest neighbors and N is the total number of spins in a thin film. M + is the sublattice magnetization which minimizes the free energy of the system. Here we give a brief description of the essentials of the HTSE method. According to the fluctuation–dissipation theorem [19], the high-temperature series expansions for the staggered susceptibility series per site for an n-layer film is obtained from the expansion of static two-spin correlation function by using the linked-cluster expansion [10,20]
β − − Tr {(Skz ) (−Slz ) e−β H } N k∈A l∈B Tr {e−β H } ∫ ∫ β 1 −− β = χ0s + dτk dτl ⟨T {[Skz (τk )][−Slz (τl )] S (β)}⟩c , nN2 β k∈A l∈B 0 0
χns =
(4)
where S (β) =
β
∫ ∞ − (−1)m m=1
m!
dτ1
β
∫
0
0
dτ2 · · ·
β
∫
dτm [H1 (τ1 )H1 (τ2 ) · · · H1 (τm )],
(5)
0
β = (kB T )−1 , T is the Dyson τ -ordering operator and the angular brackets represent the canonical thermal average over the unperturbed Hamiltonian H0 . The subscript c denotes that the cumulant part of the τ -ordered product or only the connected diagrams have to be considered. N2 is the number of lattice sites in each spin layer and χ0s is the staggered susceptibility term corresponding to the mean-field Hamiltonian H0 . The staggered susceptibility diagrams are therefore two-rooted diagrams with two spin operators Skz and Slz placed together on one of the sites of the free energy diagrams. The staggered susceptibility series expansion in [Eq. (4)] can be written in power of {β J } as
χ =χ + s n
s 0
∞ − (−1)m m=1
m!
−
W (cm )L(cm )I (cm ) {β J }m .
(6)
cm
The calculation of the coefficients in the mth-order term of the staggered susceptibility series expansion includes three parts: (a) The finding and enumerating of all two-rooted unique diagrams or graphs cm and the weight of each unique graph or the number of times the graph appears in the mth-order expansion W (cm ). (b) Calculating the lattice constant of each graph or the number of times each graph can be embedded in the sc lattice film system L(cm ). (c) Evaluating the multiple integrals containing τ -ordered cumulant products of spin operators for each graph I (cm ).
1986
K.-K. Pan / Physica A 391 (2012) 1984–1990
Table 1 The coefficients anl (x, y) from (Eq. (10)) in the text for spin-1 system are written in the form as anl (x, y) = (1 + 2x)−(l+1) αln,p,q up to order l = 2 are listed for n = 2–6 layers.
∑
p,q
αln,p,q xp y−q . The coefficients
l
p
q
αl2,p,q
αl3,p,q
αl4,p,q
αl5,p,q
αl6,p,q
0 1 2 2 2 2 2 2 2 2 2 2 2 2
1 2 0 1 1 1 2 2 2 2 3 3 3 3
0 0 3 0 1 3 0 1 2 3 0 1 2 3
2.00 20.00 5.00 1.67 −10.00 10.00 −6.67 −10.00 −10.00 −5.00 160.00 −10.00 −20.00 −10.00
2.00 21.33 5.33 1.78 −10.67 10.67 −7.11 −10.67 −10.67 −5.33 186.67 −10.67 −21.33 −10.67
2.00 22.00 5.50 1.83 −11.00 11.00 −7.33 −11.00 −11.00 −5.50 200.00 −11.00 −22.00 −11.00
2.00 22.40 5.60 1.87 −11.20 11.20 −7.47 −11.20 −11.20 −5.60 208.00 −11.20 −22.40 −11.20
2.00 22.67 5.67 1.89 −11.33 11.33 −7.56 −11.33 −11.33 −5.67 213.33 −11.33 −22.67 −11.33
All two-rooted connected graphs, the corresponding weight factors and the lattice constants are calculated by computer programs [10,18]. The following standard basis operators [18,20] are used in order to facilitate the calculation of the multiple integrals containing τ -ordered products of spin operators in series expansion: Lmn ≡ | ϵm ⟩⟨ ϵn |
(7)
where | ϵm ⟩, | ϵn ⟩ are eigenstates of mean-field Hamiltonian H0 . The spin operators are then rewritten in terms of the standard basis operators. For spin-1 system, S+ =
−
⟨ ϵm | S + | ϵn ⟩ Lmn =
m,n
S− =
−
⟨ ϵm | S − | ϵn ⟩ Lmn =
√
2(L12 + L23 ),
√
2(L21 + L32 ),
(8)
m,n
Sz =
−
⟨ ϵm | S z | ϵn ⟩ Lmn = (L11 − L33 )
m,n
and for spin-3/2 system, S+ =
−
⟨ ϵm | S + | ϵn ⟩ Lmn =
m,n
S− =
−
⟨ ϵm | S − | ϵn ⟩ Lmn =
√
√ 3L12 + 2L23 +
√
√ 3L21 + 2L32 +
3L34 , 3L43 ,
m,n
Sz =
−
⟨ ϵm | S z | ϵn ⟩ Lmn =
m,n
3 2
(9)
1
(L11 − L44 ) + (L22 − L33 ). 2
The multiple integrals containing τ -ordered products of spin operators in each connected diagrams of the staggered susceptibility series are calculated by using the multiple-site Wick reduction theorem and the standard basis operators [18,20]. The zero-field (hs = 0) high-temperature series for the staggered susceptibility series per site of n-layer lattice with free surface is obtained as
β −1 χns =
∞ −
anl (x, y){β J }l
(10)
l =0
where the coefficients anl (x, y) are polynomials in the variables x = eβ D and y = β D. We obtained the staggered susceptibility series up to sixth order for the sc lattice films. The whole coefficients anl are too long, so here we present these coefficients only to the second order in Tables 1 and 2 for spin-1 and spin-3/2, respectively. The full expressions of the staggered susceptibility series expansion of spin-1 and spin-3/2 systems are available upon request. The series obtained have been checked in the D = 0 limit. In the D = 0 limit, the results of the present calculation for the spin-1 and spin- 32 antiferromagnetic Heisenberg model on sc lattice films with n = 2–6 layers agree completely with previous results [10]. In addition, our results for the bulk value(n = ∞) of spin-1 system for finite D agree with the hightemperature series of spin-1 antiferromagnetic Heisenberg model with easy-axis single-ion anisotropy on the sc lattice [18].
K.-K. Pan / Physica A 391 (2012) 1984–1990
1987
Table 2 The coefficients anl (x, y) from (Eq. (10)) in the text for spin- 32 system are written in the form as anl (x, y) = (1 + x2 )−(l+1) βln,p,q up to order l = 2 are listed for n = 2–6 layers.
∑
p,q
βln,p,q (x2 )p y−q . The coefficients
l
p
q
βl2,p,q
βl3,p,q
βl4,p,q
βl5,p,q
βl6,p,q
0 0 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
0 1 0 1 2 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3
0 0 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3
0.25 2.25 0.31 5.62 25.31 −0.10 1.52 0.70 −7.85 0.05 32.46 −14.30 −0.35 86.72 4.10 −17.11 7.85 227.81 3.16 −2.11 0.35
0.25 2.25 0.33 6.00 27.00 −0.08 1.62 0.75 −8.38 0.90 34.62 −15.25 −0.38 100.09 4.37 −18.25 8.38 265.78 3.38 −2.25 0.38
0.25 2.25 0.34 6.19 27.84 −0.07 1.68 0.77 −8.64 1.32 35.71 −15.73 −0.39 106.78 4.51 −18.82 8.64 284.77 3.48 −2.32 0.39
0.25 2.25 0.35 6.30 28.35 −0.06 1.71 0.79 −8.79 1.58 36.36 −16.01 −0.39 110.79 4.59 −19.16 8.79 296.16 3.54 −2.36 0.39
0.25 2.25 0.35 6.38 28.69 −0.05 1.73 0.80 −8.90 1.75 36.79 −16.20 −0.40 113.47 4.65 −19.39 8.90 303.75 3.59 −2.39 0.40
3. Results and discussions 3.1. Effect of anisotropy on the thickness dependence of the Néel temperature The series are extrapolated to study the crossover from three-dimensional (3D) critical behavior at high temperatures to two-dimensional (2D) critical behavior in critical temperature region. The Néel temperatures TN (n) for the n-layer films are estimated from the divergence of the paramagnetic staggered susceptibility series (Eq. (10)) by using the ratio method and Padé approximants [21]. We have applied [22] the ratio plot for (anl /anl−2 )1/2 versus {l(l − 1)}−1/2 to estimate the Néel temperatures TN (n) for the n-layer films. For each value of D/J, the Néel temperature TN (n) for the n-layer film is estimated self-consistently from the extrapolation given by kTN (n) J
= µn (l, m) =
√ √ [( l(l − 1) ) νln − ( m(m − 1)) νmn ] √ √ l(l − 1) − m(m − 1)
(11)
where νln = [anl (x, y)/anl−2 (x, y)]1/2 . In Figs. 1 and 2, we show the Néel temperature kTN /J as a function of single-ion anisotropy D/J of sc lattice films with n = 2, 3, 4, 5, 6, and ∞ layers for spin-1 and spin-3/2, respectively. They are plotted as a solid line which is estimated from the average of the sixth-order µn (6, 5) and µn (6, 4) extrapolations. The extrapolation shows good convergence in the sense that each extrapolation value differs from the average value by less than 1%. We have also used the Padé approximant analysis of the series to estimate the Néel temperature. The Néel temperature is estimated from the poles of direct Padé approximants to the series {χns }1/γ . The critical exponent for the 2D Ising [23], γ = 1.75, is assumed for all finite thickness films in order to obtain the best possible estimates of the Néel temperature. For the bulk (n = ∞), the critical temperature is obtained from the series with the 3D Ising [24] critical exponent γ = 1.25. The estimates of the Néel temperature obtained from the average of [2/3], [2/4] and [3/3] the Padé approximant for spin-1 and spin- 32 are also plotted in Figs. 1 and 2. The over-all results of the Padé approximants analysis of the series are in quantitative agreement with the ratio results with the uncertainties reaching 5% for the films of n = 5 and 6 layers. 3.2. The Néel temperature shifts and the shift exponent The shift of the reduced Néel temperature kTN (n)/J from the 3D bulk value kTN (∞)/J can be described by a power law characterized by a shift exponent λ according to the scaling theory [7,8] TN (∞) − TN (n) TN (∞)
≈ n−λ .
(12)
A log–log plot of [TN (∞) − TN (n)]/TN (∞) versus n for spin-1 with various values of easy-axis anisotropy D/J is shown in Fig. 3. The inset in Fig. 3 shows a log–log plot of reduced Néel temperature shifts versus n for spin- 32 . The error bars represent the range of uncertainty in the estimate of the Néel temperature for each n-layer film. The uncertainties for the
1988
K.-K. Pan / Physica A 391 (2012) 1984–1990
Fig. 1. The Néel temperature as a function of easy-axis anisotropy of sc lattice films for spin-1 and n = 2, 3, 4, 5, 6, and ∞ layers. The solid line shows the results of the series expansion by using the ratio method. The triangles are the results of Padé approximants with estimated errors.
Fig. 2. The Néel temperature as a function of easy-axis anisotropy of sc lattice films for spin- 32 and n = 2, 3, 4, 5, 6, and ∞ layers. The solid line shows the results of the series expansion by using the ratio method. The triangles are the results of Padé approximants with estimated errors.
films of n = 5 and 6 are much larger. A good straight-line fit to various values of D/J for both spin-1 and spin- 23 is obtained. The estimated values of shift exponent λ for spin-1 and spin- 23 are λ ≃ 1.28 ± 0.3 and λ ≃ 1.25 ± 0.3, respectively. The uncertainties of the fits are determined by the lack of precision in the values of the Néel temperatures TN (n). In Fig. 4, we show [TN (∞) − TN (n)]/TN (∞) versus (1/n)λ for spin-1 with various values of easy-axis anisotropy D/J with λ ≃ 1.28 which corresponds to λ = 1/ν with ν ≃ 0.78. A good straight-line fit to various values of D/J is obtained. The corresponding plot for spin- 32 with λ ≃ 1.25(ν ≃ 0.80) is shown in the inset. These values are close to the critical exponent ν = 0.79 for Ising thin films by Monte Carlo simulation [11]. The results obtained are also consistent with the general universality principles [25] that the shift exponents as well as critical exponents depend only on spin and lattice dimensionality. 4. Conclusions In Section 4, we have studied the staggered susceptibilities of the spin-1 and spin-3/2 Heisenberg antiferromagnet with easy-axis single-ion anisotropy on a n-layer sc lattice films with free-surface boundary condition theoretically on the basis
K.-K. Pan / Physica A 391 (2012) 1984–1990
1989
Fig. 3. The Log–log plot of the shift of the reduced Néel temperature [TN (∞) − TN (n)]/TN (∞) versus layers n for spin-1 sc lattice films with various values of easy-axis anisotropy D/J. The error bars represent the range of uncertainty for each value of n. The straight line shows a linear fitting and the slope of the straight line yields the shift exponent λ. In the inset, we show a log–log plot of reduced Néel temperature shifts versus n for spin- 32 .
Fig. 4. The Néel temperature shift [TN (∞) − TN (n)]/TN (∞) versus (1/n)λ for spin-1 sc lattice films with various values of easy-axis anisotropy D/J with λ ≃ 1.28. The error bars represent the range of uncertainty for each value of n. In the inset, we show [TN (∞) − TN (n)]/TN (∞) versus (1/n)λ plot for spin- 32 with λ ≃ 1.25.
of exact high-temperature series expansions. The effect of easy-axis single-ion anisotropy on the thickness dependence of the Néel temperature TN and shift exponent λ of antiferromagnetic films has been studied. The shift of the Néel temperature TN for various values of easy-axis anisotropy D/J is described by a scaling relation with a shift exponent of λ ≃ 1.28 ± 0.3 for spin-1 and λ ≃ 1.25 ± 0.3 for spin- 32 . Our estimates of the shift exponent for spin-1 and spin- 32 are in agreement with
1990
K.-K. Pan / Physica A 391 (2012) 1984–1990
Monte Carlo simulations results. They are also consistent with the finite-size scaling prediction for the 3D Ising universality class and the general universality principles of the spin independent shift exponents. The lack of precision in the values of Néel temperatures and the limited number of layers due to the shortness of the series lead to uncertainties in the estimates of the shift exponent λ. Longer series are essential to obtain a more precise estimate of the shift exponent. Acknowledgments This research was supported by the Chang Gung University of Republic of China under Grant No. UMRPD590081. We are grateful to the National Center for High-performance Computing (NCHC) for computer time and facilities. References [1] C. Lu, S. Bader, in: L.M. Falicov, F. Mejia-Lira, J.L. Moran-Lopez (Eds.), Magnetic Properties of Low Dimensional System II, Springer Verlag, Berlin, 1990. [2] M.N. Barber, in: C. Domb, J.L. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, vol. 8, Academic, New York, 1983. [3] S. Andrieu, C. Chatelain, M. Lemine, B. Berche, Ph. Bauer, Phys. Rev. Lett. 86 (2001) 3883; Malte Henkel, Stéphane Andrieu, Philippe Bauer, Michel Piecuch, Phys. Rev. Lett. 80 (1998) 4783. [4] T. Ambrose, C.L. Chen, Phys. Rev. Lett. 76 (1996) 1743; Yi Li, K. Baberschke, Phys. Rev. Lett. 68 (1992) 1208. [5] W. Knafo, C. Meingast, K. Grube, S. Drobnik, P. Popovich, P. Schweiss, P. Adelmann, Th. Wolf, H.v. Löhneysen, Phys. Rev. Lett. 99 (2007) 137206. [6] J.E. Crow, R.P. Gruertin, and T.W. Mihalisin (Eds.), Crystalline Electric Field and Structural Effect in f -Electron Systems, Plenum, New York, 1980. [7] G.A.T. Allan, Phys. Rev. B 1 (1970) 352; T.W. Capehart, M.E. Fisher, Phys. Rev. B 13 (1976) 5021. [8] D.S. Ritchie, M.E. Fisher, Phys. Rev. B 7 (480) (1973) 480. [9] H.T. Diep, Phys. Rev. B 43 (1991) 8509. [10] K.K. Pan, Phys. Rev. B 71 (2005) 134524; K.K. Pan, Phys. Rev. B 64 (2001) 224401. [11] P. Cossio, J. Mazo-Zuluaga, J. Restrepo, Physica B 384 (2006) 227; J. Mazo-Zuluaga, J. Restrepo, Physica B 384 (2006) 224. [12] H. Bakrim, M. Hamedoun, A. Hourmatallah, J. Magn. Magn. Mater. 261 (2003) 415; S. Zhang, G. Zhang, J. Appl. Phys. 75 (1994) 6685; A.S. Carriço, R.E. Camley, Phys. Rev. B 45 (1992) 13117. [13] P. Fröbrich, P.J. Kuntz, Phys. Rep. 432 (2006) 223. [14] J.J. Alonso, J.F. Fernández, Phys. Rev. B 74 (2006) 184416. [15] M. Wortis, in: C. Domb, M.S. Green (Eds.), Phase Transitions and Critical Phenomena, 3, Academic, New York, 1974. [16] M.E. Fisher, in: C. Domb, M.S. Green (Eds.), Critical Phenomena, Academic, New York, 1971. [17] J. Oitmaa, C.J. Hamer, Phys. Rev. B 77 (2008) 224435. [18] K.K. Pan, Phys. Rev. B 79 (2009) 134414, and references therein. [19] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Clarendon Press, Oxford, 1971. [20] C. Wentworth, Y.L. Wang, Phys. Rev. B 36 (1987) 8687; Y.L. Wang, C. Wentworth, B. Westwanski, Phys. Rev. B 32 (1985) 1805. [21] D.S. Gaunt, A.J. Guttman, in: C. Domb, M.S. Green (Eds.), Phase Transitions and Critical Phenomena, 3, Academic, New York, 1974. [22] K. Binder, P.C. Hohenberg, Phys. Rev. B 9 (1974) 2194; K. Binder, P.C. Hohenberg, Phys. Rev. B 6 (1972) 3461. [23] J.H. Chen, M.E. Fisher, B.G. Nickel, Phys. Rev. Lett. 48 (1982) 630; M.E. Fisher, Rep. Prog. Phys. 30 (1967) 615. [24] A.M. Ferrenberg, D.P. Landau, Phys. Rev. B 44 (1991) 5081. [25] L.P. Kadanoff, Statistical Physics: Statics, Dynamics and Renormalization, World Scientific, Singapore, 2000.