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On the theory of domain structure in ferromagnetic phase of diluted magnetic semiconductors
Physics Letters A 357 (2006) 407–412 www.elsevier.com/locate/pla
On the theory of domain structure in ferromagnetic phase of diluted magnetic semiconductors V.A. Stephanovich Opole University, Institute of Mathematics and Informatics, Opole 45-052, Poland Received 11 April 2006; accepted 24 April 2006 Available online 4 May 2006 Communicated by V.M. Agranovich
Abstract We present a comprehensive analysis of domain structure formation in ferromagnetic phase of diluted magnetic semiconductors (DMS) of ptype. Our analysis is carried out on the base of effective magnetic free energy of DMS calculated by us earlier [Yu.G. Semenov, V.A. Stephanovich, Phys. Rev. B 67 (2003) 195203]. This free energy, substituting DMS (a disordered magnet) by effective ordered substance, permits to apply the standard phenomenological approach to domain structure calculation. Using coupled system of Maxwell equations with those obtained by minimization of above free energy functional, we show the existence of critical ratio νcr of concentration of charge carriers and magnetic ions such that sample critical thickness Lcr (such that at L < Lcr a sample is monodomain) diverges as ν → νcr . At ν > νcr the sample is monodomain. This feature makes DMS different from conventional ordered magnets as it gives a possibility to control the sample critical thickness and emerging domain structure period by variation of ν. As concentration of magnetic impurities grows, νcr → ∞ restoring conventional behavior of ordered magnets. Above facts have been revealed by examination of the temperature of transition to inhomogeneous magnetic state (stripe domain structure) in a slab of finite thickness L of p-type DMS. Our theory can be easily generalized for arbitrary temperature and DMS shape. © 2006 Elsevier B.V. All rights reserved. PACS: 72.20.Ht; 85.60.Dw; 42.65.Pc; 78.66.-w
1. Introduction The structure of domains and domain walls in conventional ordered magnets has been well studied both experimentally and theoretically several decades ago (see, [1,2] and references therein). The first quantitative theory of domain structure in ordered ferromagnets had been suggested by Landau and Lifshits in their classical paper [3] (see also [1]). Second step had been done by Shirobokov [4], who found the periodic distribution of a magnetic moment in magnetically ordered crystal, neglecting, however, the inhomogeneous distribution of demagnetization field deep inside a sample. It was shown further (see [2,5] and references therein) that such approach is valid only at sufficiently low temperatures (T Tc , where Tc is a phase transition temperature), where the demagnetizing field is localized
E-mail address: [email protected] (V.A. Stephanovich). URL: http://cs.uni.opole.pl/~stef. 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.04.088
near crystal surface and the domain walls are sufficiently thin. In that case the domains are large and the magnetization inside them equals to that of a bulk “domainless” sample. As temperature grows approaching Tc , the demagnetizing field penetrates deep inside crystal. In other words, the magnetization inside domains begins to depend on that field and vice versa—the demagnetization field depends on above magnetization so that the problem becomes self-consistent. It was shown in the papers [2,5] that the properties of domain structure of any magnetically ordered substance is completely determined by joint solution of Maxwell equations for demagnetization field and those obtained by minimization of corresponding Landau free energy functional of a magnet. The shape of a sample is determined by corresponding boundary conditions. This approach has already become standard for the calculation of physical properties of domain structure of ordered magnets. When magnet is disordered, which is the case for diluted magnetic semiconductors (DMS), the application of above formalism is impossible as there is no corresponding free energy
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functional. In our opinion, this is the reason why in the papers, where the ferromagnetism in DMS have been reported and investigated, the important question about their domain structure have not been addressed, see [6,7]. The characteristics of the domain structure in the DMS films of III–V type have been investigated in Refs. [8,9]. For instance, in Ref. [9], the conventional stripe-shaped domain structure has been observed in Ga0.957 Mn0.043 As. As we have mentioned above, the main problem in theoretical description of DMS domain structure in the above papers was the lack of suitable “continuous” free energy functional. Recently in Ref. [10] we have derived such free energy functions microscopically from Ising and Heisenberg models of DMS, using effective averaging over the magnetic impurity ensemble of DMS. As a result, we obtain the effective free energy functional, where the coefficients before powers of magnetization contain not only temperature but magnetic ions concentration also. In other words, our free energy function gives a “mapping” of initial disordered substance onto some effective ordered magnet, where the effects of dilution (disorder) are considered as a variable magnetic ions concentration. In the language of phenomenological theory of magnetism this functional corresponds to so-called homogeneous exchange part of total phenomenological free energy of DMS. To describe the domain structure properties, these contributions should be completed by inhomogeneous exchange and magnetic anisotropy energies. This can be done in a standard phenomenological way. Namely, it was demonstrated experimentally (see Ref. [6] and references therein) that magnetic anisotropy exists in DMS of (Ga, Mn)As type. At the same time it was demonstrated in Ref. [6] that unstrained samples (which can be well described by Heisenberg model, Ref. [10]) have easy plane magnetic anisotropy, while uniaxially strained samples (Ising model, Ref. [10]) have anisotropy of the easy axis type. It is well known (see Refs. [1,12]) that at low temperatures the domain pattern formation is primarily due to the rotation of magnetization vector with constant modulus being saturation magnetization M0 . On the contrary, for the temperatures close to Tc , this structure is formed by the variation of modulus rather then its rotation. This means that above homogeof M neous exchange part of the magnetic energy of DMS will only contribute to its domain structure in the vicinity of Tc . At low temperatures the influence of disorder on domain structure of DMS is small so that it will resemble very much the domain structure of conventional magnetically ordered substances. Having complete free energy functional of a DMS, we can apply the above standard approach to calculate the properties of domain structure of DMS in the entire temperature range, where the ferromagnetism exists. However, far from phase transition temperature the solution of correspondent nonlinear partial differential equations can be done only numerically. That is why in the present Letter we suggest a theory of inhomogeneous magnetic state (stripe domain structure) in the DMS slab in the vicinity of ferromagnetic phase transition temperature. We analyze the sample of finite thickness L. We show that the impurity character of ferromagnetism in DMS results in substantial narrowing of the region of temperatures and sample thicknesses, where domain structure exists. For example, in disordered mag-
Fig. 1. Upper panel—schematic phase diagram of the DMS slab in the coordinates temperature–thickness. Tk is a temperature of phase transition into domain state. Region 1 corresponds to paramagnetic phase for both ordered (OM) and disordered (DMS) magnets, region 2—paramagnetic phase for DMS and domain state for OM, region 3—domain state for both cases. Horizontal asymptotes correspond to τc (ν) = Tc /TcMF , Tc is a temperature of phase transition into ferromagnetic homogeneous (i.e., domainless) state. Lower panel—geometry of the sample.
netic substances under consideration, the domain structure appears at some threshold value Ltr , depending on the ratio ν of charge carriers and magnetic ions concentrations (nc and ni respectively so that ν = nc /ni ). This can be approximated by a simple analytical dependence Ltr ∝ |ν − νcr |−b , where νcr and b depend on DMS model (Heisenberg or Ising, see Eq. (25)). This behavior is sketched on Fig. 1, where the region of parameters, where domain structure exists in disordered substance (DMS), is much smaller than that for ordered magnet. The area of that region is due to the value of ν-additional parameter, absent in ordered magnets. This effect makes the domain structure of DMS (disordered magnets) qualitatively different from that of conventional ordered magnets. The developed formalism can be easily extended for wide temperature range and for thin DMS films. 2. Theory Consider the slab-shaped sample of DMS with thickness L (Fig. 1, lower panel). Let z axis is magnetic anisotropy axis (and xy plane is the plane of anisotropy for Heisenberg model). The phenomenological free energy of DMS near Tc can be written in the form (see, e.g., Ref. [2]) 1 1 P 2 + fAN F = dv α(∇ M) (1) + f P (M) − M HD , 2 2 is a magnetization vector, α is inhomogeneous exwhere M P are the change constant, HD is a demagnetizing field, fAN P anisotropy energies and f (M) are the homogeneous exchange energies, where P equals to H or I , which stand for Heisenberg or Ising model, respectively. For disordered Heisenberg model
V.A. Stephanovich / Physics Letters A 357 (2006) 407–412
and Maxwell equations
of DMS (easy plane anisotropy, Ref. [10]) 1 H = βMx2 , β > 0, fAN 2 1 4 H 1 f H (m) = m2 1 − 2AH m A3 + · · · , 1 + 2 20 ∞ t H H −F0 (t/2T ) n An = B1/2 (πt)e F1 dt, 2T
(2) (3) (4)
where m = |m|, m = S /S, S is a spin of a magnetic ion. Bar means the averaging over spatial disorder in magnetic ion subsystem in DMS, while angular brackets mean the thermal averaging, see Ref. [11] for details. The relation between m and = 0) is M in this case is usual: m = M/M0 , where M0 = S (T saturation magnetization. For Ising model (easy axis anisotropy) (5) (6) (7)
0
Here H (x) = B1/2
1 + x coth x , 3 sinhx
I (x) = B1/2
1 , sinh x
n(r ) 1 − e−iJ (r )x d 3 r,
F0 (x) + iF1 (x) =
(8) (9)
V
F0,1 are the real and imaginary parts of Fourier image of distribution function of random magnetic fields, acting among magnetic impurities in DMS, J (r ) is an interaction between spins of magnetic impurities in the Heisenberg or Ising Hamiltonians (this is actually RKKY interaction, see Refs. [10,11] for details), n(r ) is (spatially inhomogeneous) concentration of magnetic ions. Index 1/2 in the functions (8) means that free energies (3) and (6) have been derived for spin 1/2 (see Ref. [10]). However, our analysis shows that all our results remain valid for arbitrary spin. We note here that in our random field method (see Refs. [10, 11]) mean field asymptotics formally corresponds to the limit nc /ni → 0 or ni → ∞. Latter relation shows that mean field approximation corresponds in fact to the case of ordered magnet, where all lattice cites are occupied by magnetic ions. In this approximation, we have well-known (see, e.g., Ref. [6]) expression for phase transition temperature into spatially uniform (i.e., domainless) state 1 n(r )J (r ) d 3 r. TcMF = (10) 4 Equilibrium distribution of magnetization in DMS can be obtained from the equation of state δF /δ m =0
rot h = 0,
div(h + 4π m) =0
with boundary conditions for slab geometry ∂m hx |z=± L = h(e) , = 0, x z=± L 2 2 ∂z z=± L 2 [hz + 4πmz ]z=± L = h(e) z z=± L , 2
0
1 I = β Mx2 + My2 , β > 0, fAN 2 1 1 f I (m) = m2 1 − AI1 + m4 AI3 + · · · , 2 24 ∞ t I I −F0 (t/2T ) n F1 dt. An = B1/2 (πt)e 2T
409
(11)
(12)
(13)
2
where h = HD /M0 , h(e) is a demagnetizing field in a vacuum. It was shown in Ref. [2] that for sufficiently thick slabs1 and β < 4π the following equation for distribution of magnetization in DMS can be derived from the equation of state (11) with respect to Eq. (1) 2 ∂2 ∂ mz ∂ 2 mz 3 μ⊥ 2 α (14) − b m − c m = 0, − 4π P z P z ∂x ∂x 2 ∂z2 H where μ⊥ = 1 + 4π/β, bH = 1 − 2AH 1 , cH = A3 /5, bI = I I 1 − A1 , cI = A3 /6, where in the ferromagnetic phase (T < Tc ) functions bP (T ) < 0 (P = H, I ). It should be noted here, that the different forms of anisotropy energies for Heisenberg (Eq. (2)) and Ising (Eq. (5)) models in the above suppositions do not influence the form of Eq. (14). It was shown in Ref. [2] (see also Ref. [5]), that transition from paramagnetic phase to the ferromagnetic phase with domain structure (domain state) occurs via phase transition of second kind. This means that for the determination of transition temperature Tk to the domain state it is sufficient to consider the linearized version of Eq. (14). We look for its solution in the form
mz = A cos qz cos kx.
(15)
Here cos qz determines the spatial inhomogeneity along z direction, while cos kx defines a “linear” domain structure (in x direction so that domain walls lie in yz plane) with a period d = 2π/k. Substitution of solution (15) into the linearized version of Eq. (14) gives the equation relating q and k μ⊥ 2 k ζP − αk 2 , ζP = −bP . q2 = (16) 4π To obtain the dependence of Tk on sample thickness, we need one more equation relating ζ (and by its virtue Tk , see Eqs. (4), (7)), k and q. Such equation can be gotten substituting Eq. (15) into boundary conditions (13) with respect to vacuum solu(e) (e) tions hz = C exp(−k|z|) cos kx, hx = C exp(−k|z|) sin kx. It 1 Eqs. (3) and (6) have been derived for 3D RKKY interaction between localized magnetic moments in DMS, Ref. [10]. For such consideration to be valid, the slab thickness L should be more then the range of RKKY interaction 1/kf (kf is Fermi wave vector). For DMS, the typical value of Fermi energy Ef ∼ 0.08 eV. Hence 1/kf = h/ ¯ 2m∗ Ef , where m∗ ∼ 0.1m0 is an effective mass of charge carrier, m0 is a free electron mass. So, in our consideration 1/kf ∼ 20 Å which is the lower limit for plate thickness. However, real plate thickness should be much more then this threshold value. Namely in the phenomenological theory of magnetic domains (see, e.g., Refs. [2,5]) the criterion of slab thickness reads L α 1/2 . For α 1/2 = 200 Å both above criteria coincide.
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This behavior is typical for stripe domain structure in ordered magnets (see Refs. [2,5]) and is seen on the figure. 3. Discussion Now we consider the explicit dependencies bP (ν, τ ), ν = nc /ni = xc /xi , xc,i = nc,i Ω, Ω is DMS unit cell volume, nc and ni are charge carriers (electrons or holes) and magnetic ions concentrations, respectively. Using dimensionless variables, we have ∞ τ ξτ P B1/2 Φ(ξ ) dξ − 1, ζP = 12ν 36π 2 ν 2 0 ϕ0 (ξ ) ϕ1 (ξ ), P = H, I, Φ(ξ ) = exp (20) 6πν Fig. 2. Left panel: dependence of ζH and ζI on normalized temperature at different ν. Points where ζP = 0 correspond to τc (ν). Dashed line labeled MF corresponds to the case of ordered magnet ζMF = 1/τ − 1. Right panel: equilibrium period of emerging domain structure and parameter ζ versus dimensionless sample thickness at μ⊥ = 10.
reads qL μ⊥ k tan (17) = . 2 q Eqs. (16) and (17) constitute a close set of equations for the instability temperature Tk and equilibrium domain structure period (width of domain stripes) as a function of sample thickness L and concentration ratio ν. Eqs. (16) and (17) can be reduced to√a single equation for the dependence of ζP = −bP on y = k α
μ⊥ 4πμ⊥ 2 , ζP − y 2 = πn + arctan y (18) 4π η ζP − y 2 where η2 = L2 /α. Eq. (18) is a single equation for ζ (y) at different η. This dependence has the form of a curve with a minimum. The real transition to domain state in DMS occurs when Tk reaches its maximal value as a function of y. This, similarly to Ref. [2], can be demonstrated by substitution of a solution of nonlinear equation (14) in the form of infinite series in small parameter proportional to |T − Tk | into free energy (1) with its subsequent minimization over y. Since ζP = −bP for both models is a decreasing function of temperature (e.g., for both models in a mean field approximation ζ = 1/τ − 1, τ = T /TcMF , see Refs. [10,11]), the coordinates of minimum ζPmin and y min of the curve ζ (y) determine the equilibrium temperature of a phase transition to the domain state Tk and equilibrium period of emerging domain structure (i.e., width of domain stripes) λ = 2π/y min as functions of dimensionless sample thickness η. We have taken the minimum of implicit function ζ (y) (18) numerically to get the dependencies λ(η) and ζPmin (η). They are reported on right panel of Fig. 2. It is seen that dependence ζPmin (η) decays rapidly as η → ∞, at η → 0 ζPmin → ∞. It can be shown that at large η √ √ λ ∝ η = L/ α. (19)
4/3
where ξ = J0 xc t/2T . In the expressions (20) we used RKKY potential in the simplest possible form corresponding to oneband carrier structure x cos x − sin x 4/3 J (r ) = −J0 xc F (2kf r), F (x) = , (21) x4 where J0 = (3/π)1/3 (3/2h¯ 2 )Jci2 Ω 2/3 md , Jci is a carrier-ion exchange constant, md is the density of states effective mass. Functions ϕ0 (ξ ) and ϕ1 (ξ ) have the form ∞ ϕ0 (ξ ) + iϕ1 (ξ ) = 1 − exp −iξ F (y) y 2 dy. 0
For potential (21) the expression for TcMF assumes the form, which will be used in subsequent numerical calculations 1 4/3 J0 xi ν 1/3 . (22) 24π The dependencies ζH (τ ) and ζI (τ ) at different ν are shown on Fig. 2 (left panel). In a mean field (MF) approximation ζH MF = ζI MF = ζMF = 1/τ − 1 is unbounded at T = 0, while beyond this approximation functions ζH and ζI have finite values at T = 0 ∞ 2 2 dξ Φ(ξ ) − 1. ζH 0 = ζI 0 = (23) 2 3 ξ 9π ν TcMF =
0
These finite values are indeed “responsible” for the emergence of concentrationally dependent threshold sample thickness Ltr for DMS. Having dependencies (20), we can solve them numerically for ζPmin (determined above from Eq. (18)) to obtain the dimensionless phase transition temperature √ τk = Tk /TcMF as a function of critical sample thickness Lcr / α. In other words, here we have the phase diagram of DMS slab in the coordinates (T , L). This is reported on Fig. 3. The presence of Lcr (ν) is clearly seen. The asymptotes for large Lcr are due to the dependence of equilibrium (i.e., to the ferromagnetic phase without domain structure) phase transition temperature τc = Tc /TcMF on the concentration ratio ν. Latter dependence is given by the conditions ζH = 0 and ζI = 0 (see Eqs. (3), (6)) for Heisenberg and Ising models, respectively. It
V.A. Stephanovich / Physics Letters A 357 (2006) 407–412
Fig. 3. Phase diagram of DMS slab in the coordinates T , L. Horizontal asymptotes—τc (ν) similar to Fig. 1, μ⊥ = 10.
411
Fig. 5. Temperature dependence of domain structure period. Critical divergence of the period at τ → τc (ν) is seen. Vertical asymptotes correspond to τc (0.05) ≈ 0.65 and τc (0.02) ≈ 0.85, respectively. P is defined by Eq. (24) for Heisenberg and Ising modwhere νcr els, respectively; bI ≈ 1.4, bH ≈ 1.15, a ≈ 1.2 is the same for both models. It is seen both from Fig. 4 and from Eq. (25) that as ν approaches νcr Ltr → ∞ so that DMS sample looses its ferromagnetism (both homogeneous and inhomogeneous). The period of domain structure also diverges as ν → νcr . The temperature dependence of equilibrium domain structure period (width of domain stripes) is reported on Fig. 5. It is seen the abrupt increase of domain stripes width as T → Tc . This fact well coincides with experimental results of Ref. [9]. Near τc = Tc /TcMF we can get following analytical result
Fig. 4. Threshold sample thickness (i.e., that at T = 0) and corresponding domain structure period (width of domain stripes) as functions of concentration H and Ising ν I are shown. ratio ν at μ⊥ = 10. Critical ratios for Heisenberg νcr cr For mean field approximation (ordered magnet) νcr → ∞.
was shown in Ref. [11] that impurity ferromagnetism in DMS P (P = H, I , Ref. [11]), is possible for 0 < ν < νcr H νcr = 0.0989,
I νcr = 0.2473,
(24)
so that τc (νcr ) = 0. This means that τc decays as ν grows and the region (T , L), where ferromagnetic domain state exists in DMS, diminishes substantially (compared to the case of ordered ferromagnets) and vanishes as ν → νcr . Note, that in MF approximation it is very easy to solve (20) analytically to get τcMF = 1/(ζP + 1). Resolving Eq. (23) for ζPmin we obtain the dependence of threshold thickness on concentration ratio ν. This dependence along with corresponding period of domain structure is shown on Fig. 4 for Heisenberg and Ising models. Our analysis shows, that dependence of sample critical thickness on parameter ν (Fig. 4) has critical character and can be well approximated by the function a Ltr , √ = P |bP α |ν − νcr
(25)
2π a(ν) (26) = , y τ − τc (ν) where for Heisenberg model and ν = 0.05 a(ν) ≈ 10.1, for ν = 0.02 a(ν) ≈ 8.6. Above divergences make it possible to control the critical thickness of DMS sample by changing ν. This, in turn, might give a possibility to engineer the domain structure in nanocrystals of DMS, which is useful for many technical applications (see Refs. [6,8] and references therein). Note that our formalism permits to calculate νcr and other characteristics of domain structure for a wide temperature range (away from Tc ) and arbitrary sample geometry.
λ=
4. Conclusions Now we present some numerical estimations. The major problem here is uncertain value of inhomogeneous exchange constant α. It can be estimated by the expression (see Ref. [1]) α ≈ kB Tc a 2 /(Ms μ0 ), where a = 4 Å is a typical value of lattice constant for DMS, Ms ≈ 50 mT (Ref. [6]) is a saturation magnetization (of localized spin moments) of DMS, Tc = TcMF ≈ 100 K (Ref. [6]) is a temperature of transition to homogeneous ferromagnetic state in a mean field approximation, kB and μ0 are Boltzmann constant and Bohr magneton, respectively. Evaluation gives α 1/2 ∼ 200 Å. From Fig. 4 for
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V.A. Stephanovich / Physics Letters A 357 (2006) 407–412
ν = 0.075 we have threshold sample thickness Ltr ∼ 50α 1/2 = 10 000 Å = 1 µm and corresponding period of domain structure is ∼ 25α 1/2 = 5000 Å = 0.5 µm for Heisenberg model. The same values for Ising model occur at ν ∼ 0.2. These values are in fair agreement with results of Ref. [8]. Moreover, for different ν we have quite different values of Ltr and ytr . This is the base for above discussed domain structure engineering. For our picture to give the quantitative description of experiment in real DMS, the precise experimental determination of inhomogeneous exchange constant α and anisotropy constant β is highly desirable. Here we presented a formalism for the calculation of properties of domain structure in DMS. Our present results about phase diagram of DMS is the simplest application of the formalism. Generally, it permits to calculate all desired properties of domain structure (like the temperature and concentration dependencies of domain structure period and domain walls thickness) in the entire temperature range as well as to account for more complex then slab sample geometries. Latter can be accomplished by applying different from Eq. (13) boundary conditions. The external magnetic field can also be easily taken into
account. However, far from Tk the solution of resulting nonlinear differential equations would require numerical methods. References [1] L.D. Landau, E.M. Lifshits, Electrodynamics of Continuous Media, Wiley, New York, 1984. [2] V.V. Tarasenko, E.V. Chenskii, I.E. Dikshtein, Sov. Phys. JETP 43 (1976) 1136. [3] L.D. Landau, E.M. Lifshits, Phys. Zs. UdSSR 8 (1935) 153. [4] M.Ya. Shirobokov, Zh. Eksp. Teor. Fiz. 15 (1945) 57. [5] V.G. Baryakhtar, B.A. Ivanov, Sov. Phys. JETP 45 (1977) 789. [6] T. Dietl, H. Ohno, F. Matsukura, Phys. Rev. B 63 (2001) 195205. [7] Recently the ferromagnetism had been discovered in DMS, see M. Tanaka, Semicond. Sci. Tech. 17 (2002) 327; T. Dietl, Semicond. Sci. Tech. 17 (2002) 377. [8] T. Dietl, J. König, A.H. MacDonald, Phys. Rev. B 64 (2001) 241201. [9] T. Shono, T. Hasegawa, T. Fukumura, F. Matsukura, H. Ohno, Appl. Phys. Lett. 77 (2000) 1363. [10] Yu. Semenov, V. Stephanovich, Phys. Rev. B 67 (2003) 195203. [11] Yu.G. Semenov, V.A. Stephanovich, Phys. Rev. B 66 (2002) 075202. [12] L.N. Bulaevskii, V.L. Ginzburg, Zh. Eksp. Teor. Fiz. 45 (1963) 772.
On the theory of domain structure in ferromagnetic phase of diluted magnetic semiconductors V.A. Stephanovich Opole University, Institute of Mathematics and Informatics, Opole 45-052, Poland Received 11 April 2006; accepted 24 April 2006 Available online 4 May 2006 Communicated by V.M. Agranovich
Abstract We present a comprehensive analysis of domain structure formation in ferromagnetic phase of diluted magnetic semiconductors (DMS) of ptype. Our analysis is carried out on the base of effective magnetic free energy of DMS calculated by us earlier [Yu.G. Semenov, V.A. Stephanovich, Phys. Rev. B 67 (2003) 195203]. This free energy, substituting DMS (a disordered magnet) by effective ordered substance, permits to apply the standard phenomenological approach to domain structure calculation. Using coupled system of Maxwell equations with those obtained by minimization of above free energy functional, we show the existence of critical ratio νcr of concentration of charge carriers and magnetic ions such that sample critical thickness Lcr (such that at L < Lcr a sample is monodomain) diverges as ν → νcr . At ν > νcr the sample is monodomain. This feature makes DMS different from conventional ordered magnets as it gives a possibility to control the sample critical thickness and emerging domain structure period by variation of ν. As concentration of magnetic impurities grows, νcr → ∞ restoring conventional behavior of ordered magnets. Above facts have been revealed by examination of the temperature of transition to inhomogeneous magnetic state (stripe domain structure) in a slab of finite thickness L of p-type DMS. Our theory can be easily generalized for arbitrary temperature and DMS shape. © 2006 Elsevier B.V. All rights reserved. PACS: 72.20.Ht; 85.60.Dw; 42.65.Pc; 78.66.-w
1. Introduction The structure of domains and domain walls in conventional ordered magnets has been well studied both experimentally and theoretically several decades ago (see, [1,2] and references therein). The first quantitative theory of domain structure in ordered ferromagnets had been suggested by Landau and Lifshits in their classical paper [3] (see also [1]). Second step had been done by Shirobokov [4], who found the periodic distribution of a magnetic moment in magnetically ordered crystal, neglecting, however, the inhomogeneous distribution of demagnetization field deep inside a sample. It was shown further (see [2,5] and references therein) that such approach is valid only at sufficiently low temperatures (T Tc , where Tc is a phase transition temperature), where the demagnetizing field is localized
E-mail address: [email protected] (V.A. Stephanovich). URL: http://cs.uni.opole.pl/~stef. 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.04.088
near crystal surface and the domain walls are sufficiently thin. In that case the domains are large and the magnetization inside them equals to that of a bulk “domainless” sample. As temperature grows approaching Tc , the demagnetizing field penetrates deep inside crystal. In other words, the magnetization inside domains begins to depend on that field and vice versa—the demagnetization field depends on above magnetization so that the problem becomes self-consistent. It was shown in the papers [2,5] that the properties of domain structure of any magnetically ordered substance is completely determined by joint solution of Maxwell equations for demagnetization field and those obtained by minimization of corresponding Landau free energy functional of a magnet. The shape of a sample is determined by corresponding boundary conditions. This approach has already become standard for the calculation of physical properties of domain structure of ordered magnets. When magnet is disordered, which is the case for diluted magnetic semiconductors (DMS), the application of above formalism is impossible as there is no corresponding free energy
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V.A. Stephanovich / Physics Letters A 357 (2006) 407–412
functional. In our opinion, this is the reason why in the papers, where the ferromagnetism in DMS have been reported and investigated, the important question about their domain structure have not been addressed, see [6,7]. The characteristics of the domain structure in the DMS films of III–V type have been investigated in Refs. [8,9]. For instance, in Ref. [9], the conventional stripe-shaped domain structure has been observed in Ga0.957 Mn0.043 As. As we have mentioned above, the main problem in theoretical description of DMS domain structure in the above papers was the lack of suitable “continuous” free energy functional. Recently in Ref. [10] we have derived such free energy functions microscopically from Ising and Heisenberg models of DMS, using effective averaging over the magnetic impurity ensemble of DMS. As a result, we obtain the effective free energy functional, where the coefficients before powers of magnetization contain not only temperature but magnetic ions concentration also. In other words, our free energy function gives a “mapping” of initial disordered substance onto some effective ordered magnet, where the effects of dilution (disorder) are considered as a variable magnetic ions concentration. In the language of phenomenological theory of magnetism this functional corresponds to so-called homogeneous exchange part of total phenomenological free energy of DMS. To describe the domain structure properties, these contributions should be completed by inhomogeneous exchange and magnetic anisotropy energies. This can be done in a standard phenomenological way. Namely, it was demonstrated experimentally (see Ref. [6] and references therein) that magnetic anisotropy exists in DMS of (Ga, Mn)As type. At the same time it was demonstrated in Ref. [6] that unstrained samples (which can be well described by Heisenberg model, Ref. [10]) have easy plane magnetic anisotropy, while uniaxially strained samples (Ising model, Ref. [10]) have anisotropy of the easy axis type. It is well known (see Refs. [1,12]) that at low temperatures the domain pattern formation is primarily due to the rotation of magnetization vector with constant modulus being saturation magnetization M0 . On the contrary, for the temperatures close to Tc , this structure is formed by the variation of modulus rather then its rotation. This means that above homogeof M neous exchange part of the magnetic energy of DMS will only contribute to its domain structure in the vicinity of Tc . At low temperatures the influence of disorder on domain structure of DMS is small so that it will resemble very much the domain structure of conventional magnetically ordered substances. Having complete free energy functional of a DMS, we can apply the above standard approach to calculate the properties of domain structure of DMS in the entire temperature range, where the ferromagnetism exists. However, far from phase transition temperature the solution of correspondent nonlinear partial differential equations can be done only numerically. That is why in the present Letter we suggest a theory of inhomogeneous magnetic state (stripe domain structure) in the DMS slab in the vicinity of ferromagnetic phase transition temperature. We analyze the sample of finite thickness L. We show that the impurity character of ferromagnetism in DMS results in substantial narrowing of the region of temperatures and sample thicknesses, where domain structure exists. For example, in disordered mag-
Fig. 1. Upper panel—schematic phase diagram of the DMS slab in the coordinates temperature–thickness. Tk is a temperature of phase transition into domain state. Region 1 corresponds to paramagnetic phase for both ordered (OM) and disordered (DMS) magnets, region 2—paramagnetic phase for DMS and domain state for OM, region 3—domain state for both cases. Horizontal asymptotes correspond to τc (ν) = Tc /TcMF , Tc is a temperature of phase transition into ferromagnetic homogeneous (i.e., domainless) state. Lower panel—geometry of the sample.
netic substances under consideration, the domain structure appears at some threshold value Ltr , depending on the ratio ν of charge carriers and magnetic ions concentrations (nc and ni respectively so that ν = nc /ni ). This can be approximated by a simple analytical dependence Ltr ∝ |ν − νcr |−b , where νcr and b depend on DMS model (Heisenberg or Ising, see Eq. (25)). This behavior is sketched on Fig. 1, where the region of parameters, where domain structure exists in disordered substance (DMS), is much smaller than that for ordered magnet. The area of that region is due to the value of ν-additional parameter, absent in ordered magnets. This effect makes the domain structure of DMS (disordered magnets) qualitatively different from that of conventional ordered magnets. The developed formalism can be easily extended for wide temperature range and for thin DMS films. 2. Theory Consider the slab-shaped sample of DMS with thickness L (Fig. 1, lower panel). Let z axis is magnetic anisotropy axis (and xy plane is the plane of anisotropy for Heisenberg model). The phenomenological free energy of DMS near Tc can be written in the form (see, e.g., Ref. [2]) 1 1 P 2 + fAN F = dv α(∇ M) (1) + f P (M) − M HD , 2 2 is a magnetization vector, α is inhomogeneous exwhere M P are the change constant, HD is a demagnetizing field, fAN P anisotropy energies and f (M) are the homogeneous exchange energies, where P equals to H or I , which stand for Heisenberg or Ising model, respectively. For disordered Heisenberg model
V.A. Stephanovich / Physics Letters A 357 (2006) 407–412
and Maxwell equations
of DMS (easy plane anisotropy, Ref. [10]) 1 H = βMx2 , β > 0, fAN 2 1 4 H 1 f H (m) = m2 1 − 2AH m A3 + · · · , 1 + 2 20 ∞ t H H −F0 (t/2T ) n An = B1/2 (πt)e F1 dt, 2T
(2) (3) (4)
where m = |m|, m = S /S, S is a spin of a magnetic ion. Bar means the averaging over spatial disorder in magnetic ion subsystem in DMS, while angular brackets mean the thermal averaging, see Ref. [11] for details. The relation between m and = 0) is M in this case is usual: m = M/M0 , where M0 = S (T saturation magnetization. For Ising model (easy axis anisotropy) (5) (6) (7)
0
Here H (x) = B1/2
1 + x coth x , 3 sinhx
I (x) = B1/2
1 , sinh x
n(r ) 1 − e−iJ (r )x d 3 r,
F0 (x) + iF1 (x) =
(8) (9)
V
F0,1 are the real and imaginary parts of Fourier image of distribution function of random magnetic fields, acting among magnetic impurities in DMS, J (r ) is an interaction between spins of magnetic impurities in the Heisenberg or Ising Hamiltonians (this is actually RKKY interaction, see Refs. [10,11] for details), n(r ) is (spatially inhomogeneous) concentration of magnetic ions. Index 1/2 in the functions (8) means that free energies (3) and (6) have been derived for spin 1/2 (see Ref. [10]). However, our analysis shows that all our results remain valid for arbitrary spin. We note here that in our random field method (see Refs. [10, 11]) mean field asymptotics formally corresponds to the limit nc /ni → 0 or ni → ∞. Latter relation shows that mean field approximation corresponds in fact to the case of ordered magnet, where all lattice cites are occupied by magnetic ions. In this approximation, we have well-known (see, e.g., Ref. [6]) expression for phase transition temperature into spatially uniform (i.e., domainless) state 1 n(r )J (r ) d 3 r. TcMF = (10) 4 Equilibrium distribution of magnetization in DMS can be obtained from the equation of state δF /δ m =0
rot h = 0,
div(h + 4π m) =0
with boundary conditions for slab geometry ∂m hx |z=± L = h(e) , = 0, x z=± L 2 2 ∂z z=± L 2 [hz + 4πmz ]z=± L = h(e) z z=± L , 2
0
1 I = β Mx2 + My2 , β > 0, fAN 2 1 1 f I (m) = m2 1 − AI1 + m4 AI3 + · · · , 2 24 ∞ t I I −F0 (t/2T ) n F1 dt. An = B1/2 (πt)e 2T
409
(11)
(12)
(13)
2
where h = HD /M0 , h(e) is a demagnetizing field in a vacuum. It was shown in Ref. [2] that for sufficiently thick slabs1 and β < 4π the following equation for distribution of magnetization in DMS can be derived from the equation of state (11) with respect to Eq. (1) 2 ∂2 ∂ mz ∂ 2 mz 3 μ⊥ 2 α (14) − b m − c m = 0, − 4π P z P z ∂x ∂x 2 ∂z2 H where μ⊥ = 1 + 4π/β, bH = 1 − 2AH 1 , cH = A3 /5, bI = I I 1 − A1 , cI = A3 /6, where in the ferromagnetic phase (T < Tc ) functions bP (T ) < 0 (P = H, I ). It should be noted here, that the different forms of anisotropy energies for Heisenberg (Eq. (2)) and Ising (Eq. (5)) models in the above suppositions do not influence the form of Eq. (14). It was shown in Ref. [2] (see also Ref. [5]), that transition from paramagnetic phase to the ferromagnetic phase with domain structure (domain state) occurs via phase transition of second kind. This means that for the determination of transition temperature Tk to the domain state it is sufficient to consider the linearized version of Eq. (14). We look for its solution in the form
mz = A cos qz cos kx.
(15)
Here cos qz determines the spatial inhomogeneity along z direction, while cos kx defines a “linear” domain structure (in x direction so that domain walls lie in yz plane) with a period d = 2π/k. Substitution of solution (15) into the linearized version of Eq. (14) gives the equation relating q and k μ⊥ 2 k ζP − αk 2 , ζP = −bP . q2 = (16) 4π To obtain the dependence of Tk on sample thickness, we need one more equation relating ζ (and by its virtue Tk , see Eqs. (4), (7)), k and q. Such equation can be gotten substituting Eq. (15) into boundary conditions (13) with respect to vacuum solu(e) (e) tions hz = C exp(−k|z|) cos kx, hx = C exp(−k|z|) sin kx. It 1 Eqs. (3) and (6) have been derived for 3D RKKY interaction between localized magnetic moments in DMS, Ref. [10]. For such consideration to be valid, the slab thickness L should be more then the range of RKKY interaction 1/kf (kf is Fermi wave vector). For DMS, the typical value of Fermi energy Ef ∼ 0.08 eV. Hence 1/kf = h/ ¯ 2m∗ Ef , where m∗ ∼ 0.1m0 is an effective mass of charge carrier, m0 is a free electron mass. So, in our consideration 1/kf ∼ 20 Å which is the lower limit for plate thickness. However, real plate thickness should be much more then this threshold value. Namely in the phenomenological theory of magnetic domains (see, e.g., Refs. [2,5]) the criterion of slab thickness reads L α 1/2 . For α 1/2 = 200 Å both above criteria coincide.
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This behavior is typical for stripe domain structure in ordered magnets (see Refs. [2,5]) and is seen on the figure. 3. Discussion Now we consider the explicit dependencies bP (ν, τ ), ν = nc /ni = xc /xi , xc,i = nc,i Ω, Ω is DMS unit cell volume, nc and ni are charge carriers (electrons or holes) and magnetic ions concentrations, respectively. Using dimensionless variables, we have ∞ τ ξτ P B1/2 Φ(ξ ) dξ − 1, ζP = 12ν 36π 2 ν 2 0 ϕ0 (ξ ) ϕ1 (ξ ), P = H, I, Φ(ξ ) = exp (20) 6πν Fig. 2. Left panel: dependence of ζH and ζI on normalized temperature at different ν. Points where ζP = 0 correspond to τc (ν). Dashed line labeled MF corresponds to the case of ordered magnet ζMF = 1/τ − 1. Right panel: equilibrium period of emerging domain structure and parameter ζ versus dimensionless sample thickness at μ⊥ = 10.
reads qL μ⊥ k tan (17) = . 2 q Eqs. (16) and (17) constitute a close set of equations for the instability temperature Tk and equilibrium domain structure period (width of domain stripes) as a function of sample thickness L and concentration ratio ν. Eqs. (16) and (17) can be reduced to√a single equation for the dependence of ζP = −bP on y = k α
μ⊥ 4πμ⊥ 2 , ζP − y 2 = πn + arctan y (18) 4π η ζP − y 2 where η2 = L2 /α. Eq. (18) is a single equation for ζ (y) at different η. This dependence has the form of a curve with a minimum. The real transition to domain state in DMS occurs when Tk reaches its maximal value as a function of y. This, similarly to Ref. [2], can be demonstrated by substitution of a solution of nonlinear equation (14) in the form of infinite series in small parameter proportional to |T − Tk | into free energy (1) with its subsequent minimization over y. Since ζP = −bP for both models is a decreasing function of temperature (e.g., for both models in a mean field approximation ζ = 1/τ − 1, τ = T /TcMF , see Refs. [10,11]), the coordinates of minimum ζPmin and y min of the curve ζ (y) determine the equilibrium temperature of a phase transition to the domain state Tk and equilibrium period of emerging domain structure (i.e., width of domain stripes) λ = 2π/y min as functions of dimensionless sample thickness η. We have taken the minimum of implicit function ζ (y) (18) numerically to get the dependencies λ(η) and ζPmin (η). They are reported on right panel of Fig. 2. It is seen that dependence ζPmin (η) decays rapidly as η → ∞, at η → 0 ζPmin → ∞. It can be shown that at large η √ √ λ ∝ η = L/ α. (19)
4/3
where ξ = J0 xc t/2T . In the expressions (20) we used RKKY potential in the simplest possible form corresponding to oneband carrier structure x cos x − sin x 4/3 J (r ) = −J0 xc F (2kf r), F (x) = , (21) x4 where J0 = (3/π)1/3 (3/2h¯ 2 )Jci2 Ω 2/3 md , Jci is a carrier-ion exchange constant, md is the density of states effective mass. Functions ϕ0 (ξ ) and ϕ1 (ξ ) have the form ∞ ϕ0 (ξ ) + iϕ1 (ξ ) = 1 − exp −iξ F (y) y 2 dy. 0
For potential (21) the expression for TcMF assumes the form, which will be used in subsequent numerical calculations 1 4/3 J0 xi ν 1/3 . (22) 24π The dependencies ζH (τ ) and ζI (τ ) at different ν are shown on Fig. 2 (left panel). In a mean field (MF) approximation ζH MF = ζI MF = ζMF = 1/τ − 1 is unbounded at T = 0, while beyond this approximation functions ζH and ζI have finite values at T = 0 ∞ 2 2 dξ Φ(ξ ) − 1. ζH 0 = ζI 0 = (23) 2 3 ξ 9π ν TcMF =
0
These finite values are indeed “responsible” for the emergence of concentrationally dependent threshold sample thickness Ltr for DMS. Having dependencies (20), we can solve them numerically for ζPmin (determined above from Eq. (18)) to obtain the dimensionless phase transition temperature √ τk = Tk /TcMF as a function of critical sample thickness Lcr / α. In other words, here we have the phase diagram of DMS slab in the coordinates (T , L). This is reported on Fig. 3. The presence of Lcr (ν) is clearly seen. The asymptotes for large Lcr are due to the dependence of equilibrium (i.e., to the ferromagnetic phase without domain structure) phase transition temperature τc = Tc /TcMF on the concentration ratio ν. Latter dependence is given by the conditions ζH = 0 and ζI = 0 (see Eqs. (3), (6)) for Heisenberg and Ising models, respectively. It
V.A. Stephanovich / Physics Letters A 357 (2006) 407–412
Fig. 3. Phase diagram of DMS slab in the coordinates T , L. Horizontal asymptotes—τc (ν) similar to Fig. 1, μ⊥ = 10.
411
Fig. 5. Temperature dependence of domain structure period. Critical divergence of the period at τ → τc (ν) is seen. Vertical asymptotes correspond to τc (0.05) ≈ 0.65 and τc (0.02) ≈ 0.85, respectively. P is defined by Eq. (24) for Heisenberg and Ising modwhere νcr els, respectively; bI ≈ 1.4, bH ≈ 1.15, a ≈ 1.2 is the same for both models. It is seen both from Fig. 4 and from Eq. (25) that as ν approaches νcr Ltr → ∞ so that DMS sample looses its ferromagnetism (both homogeneous and inhomogeneous). The period of domain structure also diverges as ν → νcr . The temperature dependence of equilibrium domain structure period (width of domain stripes) is reported on Fig. 5. It is seen the abrupt increase of domain stripes width as T → Tc . This fact well coincides with experimental results of Ref. [9]. Near τc = Tc /TcMF we can get following analytical result
Fig. 4. Threshold sample thickness (i.e., that at T = 0) and corresponding domain structure period (width of domain stripes) as functions of concentration H and Ising ν I are shown. ratio ν at μ⊥ = 10. Critical ratios for Heisenberg νcr cr For mean field approximation (ordered magnet) νcr → ∞.
was shown in Ref. [11] that impurity ferromagnetism in DMS P (P = H, I , Ref. [11]), is possible for 0 < ν < νcr H νcr = 0.0989,
I νcr = 0.2473,
(24)
so that τc (νcr ) = 0. This means that τc decays as ν grows and the region (T , L), where ferromagnetic domain state exists in DMS, diminishes substantially (compared to the case of ordered ferromagnets) and vanishes as ν → νcr . Note, that in MF approximation it is very easy to solve (20) analytically to get τcMF = 1/(ζP + 1). Resolving Eq. (23) for ζPmin we obtain the dependence of threshold thickness on concentration ratio ν. This dependence along with corresponding period of domain structure is shown on Fig. 4 for Heisenberg and Ising models. Our analysis shows, that dependence of sample critical thickness on parameter ν (Fig. 4) has critical character and can be well approximated by the function a Ltr , √ = P |bP α |ν − νcr
(25)
2π a(ν) (26) = , y τ − τc (ν) where for Heisenberg model and ν = 0.05 a(ν) ≈ 10.1, for ν = 0.02 a(ν) ≈ 8.6. Above divergences make it possible to control the critical thickness of DMS sample by changing ν. This, in turn, might give a possibility to engineer the domain structure in nanocrystals of DMS, which is useful for many technical applications (see Refs. [6,8] and references therein). Note that our formalism permits to calculate νcr and other characteristics of domain structure for a wide temperature range (away from Tc ) and arbitrary sample geometry.
λ=
4. Conclusions Now we present some numerical estimations. The major problem here is uncertain value of inhomogeneous exchange constant α. It can be estimated by the expression (see Ref. [1]) α ≈ kB Tc a 2 /(Ms μ0 ), where a = 4 Å is a typical value of lattice constant for DMS, Ms ≈ 50 mT (Ref. [6]) is a saturation magnetization (of localized spin moments) of DMS, Tc = TcMF ≈ 100 K (Ref. [6]) is a temperature of transition to homogeneous ferromagnetic state in a mean field approximation, kB and μ0 are Boltzmann constant and Bohr magneton, respectively. Evaluation gives α 1/2 ∼ 200 Å. From Fig. 4 for
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V.A. Stephanovich / Physics Letters A 357 (2006) 407–412
ν = 0.075 we have threshold sample thickness Ltr ∼ 50α 1/2 = 10 000 Å = 1 µm and corresponding period of domain structure is ∼ 25α 1/2 = 5000 Å = 0.5 µm for Heisenberg model. The same values for Ising model occur at ν ∼ 0.2. These values are in fair agreement with results of Ref. [8]. Moreover, for different ν we have quite different values of Ltr and ytr . This is the base for above discussed domain structure engineering. For our picture to give the quantitative description of experiment in real DMS, the precise experimental determination of inhomogeneous exchange constant α and anisotropy constant β is highly desirable. Here we presented a formalism for the calculation of properties of domain structure in DMS. Our present results about phase diagram of DMS is the simplest application of the formalism. Generally, it permits to calculate all desired properties of domain structure (like the temperature and concentration dependencies of domain structure period and domain walls thickness) in the entire temperature range as well as to account for more complex then slab sample geometries. Latter can be accomplished by applying different from Eq. (13) boundary conditions. The external magnetic field can also be easily taken into
account. However, far from Tk the solution of resulting nonlinear differential equations would require numerical methods. References [1] L.D. Landau, E.M. Lifshits, Electrodynamics of Continuous Media, Wiley, New York, 1984. [2] V.V. Tarasenko, E.V. Chenskii, I.E. Dikshtein, Sov. Phys. JETP 43 (1976) 1136. [3] L.D. Landau, E.M. Lifshits, Phys. Zs. UdSSR 8 (1935) 153. [4] M.Ya. Shirobokov, Zh. Eksp. Teor. Fiz. 15 (1945) 57. [5] V.G. Baryakhtar, B.A. Ivanov, Sov. Phys. JETP 45 (1977) 789. [6] T. Dietl, H. Ohno, F. Matsukura, Phys. Rev. B 63 (2001) 195205. [7] Recently the ferromagnetism had been discovered in DMS, see M. Tanaka, Semicond. Sci. Tech. 17 (2002) 327; T. Dietl, Semicond. Sci. Tech. 17 (2002) 377. [8] T. Dietl, J. König, A.H. MacDonald, Phys. Rev. B 64 (2001) 241201. [9] T. Shono, T. Hasegawa, T. Fukumura, F. Matsukura, H. Ohno, Appl. Phys. Lett. 77 (2000) 1363. [10] Yu. Semenov, V. Stephanovich, Phys. Rev. B 67 (2003) 195203. [11] Yu.G. Semenov, V.A. Stephanovich, Phys. Rev. B 66 (2002) 075202. [12] L.N. Bulaevskii, V.L. Ginzburg, Zh. Eksp. Teor. Fiz. 45 (1963) 772.