Superconductivity, including high-temperature superconductivity
DOMAIN WALL CONTAINING A BLOCH POINT IN AN AFM
Download 2.75 Mb. Pdf ko'rish
|
- Bu sahifa navigatsiya:
- MAGNETIC CONFIGURATION IN THE PRESENCE OF A DISLOCATION
- CONCLUSION
|
DOMAIN WALL CONTAINING A BLOCH POINT IN AN AFM WITH IDEAL STRUCTURE In Ref. 12 it was shown for an isotropic easy-plane AFM that in the absence of dislocations the system of coupled equations for the fields of atomic displacements and spin deviations which is obtained in the proposed generalization of the Peierls model admits a magnetic vortex solution. The system of equations ͑11͒ and ͑12͒, which takes into account the anisotropy in the easy plane, should describe a domain wall containing a so-called Bloch point, in one turn around which the direction of the antiferromagnetism vector changes by 2
͑Fig. 2͒. The corresponding solution in the limit ␥ →0 must go over to the solution for a magnetic vortex. In an ideal AFM with no dislocation we have w ϭ0 and uϵ0. Then Eq. ͑12͒ reduces to ͑x,zѥ0͒ϭϯ 1
͵ Ϫϱ
K 0 ͩͩ ͑xϪx Ј ͒ 2 2 ϩ z 2 2 ͪ 1/2 ͪ sin
dx Ј . ͑13͒ At distances x ӷ the domain wall is almost uniform along the x direction, and the function describing the relative deviations at the boundary of the half spaces can be assumed approximately constant. In this approximation one can easily obtain from
Eq. ͑13͒ expressions for ϱ
(x →ϱ) ϭ
/2 ͱ ␥ /J 2 ϭ and for the distribution of the magnetiza- tion in the domain wall along the z direction at large values of x ͓see formula ͑9͔͒. Assuming in Eq. ͑13͒ that zϭϩb/2 and taking into ac- count that Ϫ
Ϫ ϩ in the given configuration ͑see Fig. 2
͒, we arrive at a one-dimensional equation for the rela- tive spin deviations at the boundary of the half spaces: ϭ
Ϫ 2 l˜ ͵
0 ͩͩ
Ј ͒ 2 2 ϩ b 2 4 2 ͪ 1/2 ͪ sin dx Ј . ͑14a͒ Differentiating ͑14a͒ with respect to x and setting bϭ0, we obtain the equation d
ϭ 2
l˜ ͵ K 1 ͩ ͉x Ϫx Ј ͉
ͪ sgn
͑xϪx Ј ͒sin dx Ј , ͑14b͒ which is close to the Peierls equation describing the structure of a dislocation in a two-dimensional model. 15 However, in our case the Cauchy kernel /(xϪx Ј ) of the Peierls integral equation is replaced by the kernel K 1 ( ͉x Ϫx Ј ͉/
Ϫx Ј ). Since K 1 ( p) Ϸ1/p at small values of the argument, these two kernels coincide for ͉x Ϫx Ј ͉
exponential decay of the function K 1 with distance, the ker- nel has a local character, and the field (x) is localized and exponentially approaches its asymptotic forms at large dis- tances.
We shall solve equation ͑14a͒ by successive approxima- tions. As a first approximation, in view of the local nature of the distribution of the field (x) and its nonzero asymptotic expression at infinity, we approximate the function sin on the right-hand side of Eq. ͑14a͒ as follows: sin
2 ͩ ␥ J 2 ͪ 1/2 sgn
͑x͒Ϫ2 d dx ␦ ͩ x ͪ . ͑15͒
I 1 ϭϪ 1 ͵ Ϫϱ ϩϱ
0 ͩͩ
2 2 ϩ b 2 4 2 ͪ 1/2 ͪ sgn ͑xϪu͒du ϷϪ exp ͩ Ϫ b 2 ͪ ϩ2
ͭͩ x 2 2 ϩ
2 4
2 ͪ 1/4 ϫexp ͫ Ϫ ͩ x 2 2 ϩ
2 4
2 ͪ 1/2 ͬͮ x Ϫ1 , ͑16͒ where the function K 0 (k) is replaced by its asymptotic ex- pression at large values of the argument: K 0 (k) Ϸ ͱ /2k e Ϫk , k ӷ1. In an analogous way we obtain the value of I 1 for x Ͻ0. The contribution of the second term in ͑15͒ is calculated exactly: I 2 ϭϪ 4 l˜ xK 0 ͩͩ x 2 2 ϩ
2 4
2 ͪ 1/2 ͪͩ x 2 2 ϩ
2 4
2 ͪ Ϫ1/2 . ͑17͒
Finally, for the second approximation for we obtain ͑xѥ0͒Х ϮI 1 ϩI 2 ,
where I 1 and I 2 are given by expressions ͑16͒ and ͑17͒. From formulas
͑14͒–͑18͒ we obtain the asymptotic expressions for (x) at x →Ϯϱ: ͉ x →Ϫϱ
→2 Ϫ /2 ͱ ␥ /J 2 and
͉
→ϩϱ →
/2 ͱ ␥ /J 2 , which agree with the result ͑9͒ for a uniform domain wall. The behavior of the function (x) at the center of the vortex is found by differentiating Eq. ͑14a͒ with respect to x and approximating the kernel K 1 (k) in the resulting expres- sion by the function 1/sinh(k), which leads to the equation d
ϭ 2
l˜ ͵ Ϫϱ ϩϱ ͫ sinh ͩ x Ϫx Ј
Ϫ1 sin
dx Ј . ͑19͒ Equation
͑19͒ has an exact soliton solution, 16 and its asymptotic behavior of interest to us is linear in x at small x and has a value of at the center of the vortex: Х Ϫ2 x , ͑20͒ 606
Low Temp. Phys. 26 (8), August 2000 O. K. Dudko and A. S. Kovalev and the gradient of the relative spin deviations is maximum in the core region of the magnetic vortex. Applying the approximation ͑15͒ to expressions ͑13͒, we obtain the distribution of the magnetization over the entire volume of the AFM: ͑x,zѥ0͒ХϮ 2 exp ͩ Ϫ
ͪ
ͩ 2 ͪ 1/2
ͱ re Ϫr x/ Ϫ
2 l˜ xK 0 ͑r͒ r , ͑21͒ where r ϭ(x 2 /
2 ϩz 2 /
2 ) 1/2 , the signs ϯ corresponding to the upper and lower half spaces, respectively. It follows from ͑21͒ that the nonuniformity in the ordering of the spins de- cays exponentially in the z direction, perpendicular to the orientation of the domain wall, and the ordering becomes ideal as z →Ϯϱ:
(x,z) ͉
→ϩϱ
→0, (x,z) ͉ z →Ϫϱ
→Ϫ . Thus the distribution of the magnetization ͑21͒ can be written in the form ͑x,z͒ϭ f 1 ͑z͒ϩ f 2 ͑r,cos ͒ϩ f 3 ͑r,cos ͒, ͑22͒ where the function f 1 describes the distribution of in the
domain wall at large values of x, where the wall is practically uniform with respect to x; f 3 ͓the contribution of the second term in ͑15͔͒ is due to the contribution of the vortex to the magnetization field at small values of x; f 2 is a correction to f 3 and describes the influence of the vortex far from its lo- calization region (x ӷ). The characteristic dimension of the vortex along the z direction is ⌬
ϭ
ͱ J 2 / ␥ and its di- mension along the x axis is ⌬
ϭϭa ͱ
1 /
. Thus in our anisotropic model ⌬
/ ⌬ z ϭa ͱ
1 /J 2 /b ӷ1, and the vortex is strongly flattened out along the x axis. MAGNETIC CONFIGURATION IN THE PRESENCE OF A DISLOCATION Let us return to the general case of an AFM containing a dislocation. Assuming in Eq. ͑11͒ that zϭϮb/2, we arrive at a one-dimensional equation for the relative atomic displace- ments w at the boundary of the half spaces. 12 In the limit of a small ratio of the magnetic interaction to the elastic, which is usually the case in real physical situations, this equation goes over to the Peierls equation, with the solution
ϭϪarctan(
duced the ‘‘elastic length’’ l ϭa ͱ ␣ /  . When the Peierls so- lution is substituted into Eq. ͑12͒, the latter becomes ͑for z Ͼ0):
͑x,zϾ0͒ϭ 1
͵ Ϫϱ
0 ͩͩ
Ј ͒ 2 2 ϩ z 2 2 ͪ 1/2 ͪ ϫ
Ј ͱ
Ј 2 ϩl 2 sin
dx Ј . ͑23͒ Since, as a rule, the ‘‘magnetic length’’ is much greater than an atomic dimension and the size of the dislocation, we shall investigate Eq. ͑23͒ in the limit of a point dislocation:
→0. We shall show below that the finite size of the core of the dislocation has little influence on the magnetization in comparison with other factors. Noting that in this configura- tion
ϭϪ Ϫ ͑see Fig. 1͒, we obtain a one-dimensional equation, analogous to ͑14a͒, for the relative spin deviations at the boundary of the half spaces in the presence of a point dislocation: ϭ 2
͵
0 ͩͩ ͑xϪx Ј ͒ 2 2 ϩ b 2 4 2 ͪ 1/2 ͪ sgn ͑x Ј ͒sin ͑x Ј ͒dx Ј . ͑24͒ A good approximation for the function sin is the substitu- tion sin
→P͑x͒Ϫ 2
␥ J 2 ͪ 1/2 S ͑x͒, ͑25͒ where P(x) ϭϪ1 in the interval ͉x͉р and Pϭ0 outside this interval, and the Heaviside step function S(x) is equal to 0 for x Ͻ0 and to 1 for xϾ0. In this approximation the solution of equation ͑24͒ for xӷ has the form ͑xϾ0͒ХϪ exp
ͩ Ϫ
2
ϩ ͩ 2 ͪ 1/2 ͩ x 2 2 ϩ
2 4
2 ͪ Ϫ1/4 /x ϫexp
ͫ Ϫ ͩ x 2 2 ϩ
2 4
2 ͪ 1/2 ͬ Ϫ l˜ K 1 ͩ x 2 2 ϩ
2 4
2 ͪ . ͑26͒ For x Ͻ0 the first term in ͑26͒ is equal to zero. An estimate of the gradient of the relative deviations of the spins near the core of a point dislocation gives
͑xϭ0͒ dx Ϸln͑
͒ϩln͑l˜͒. ͑27͒ When the finite size of the core of the dislocation is taken into account, an additional term ln(l) appears in the sum
͑27͒ ͑see Ref. 12͒, where l is the ‘‘elastic length’’ intro- duced above. We have used the assumption that the elastic and magnetic properties are spatially anisotropic, i.e., the condition l, l˜ ӷa,b. Actually, however, the really large pa- rameter is the magnetic length ӷb, and the inequality ӷl,l˜ holds. Then the main contribution to d (x ϭ0)/dx is given by the parameter , and the approximation of a point dislocation is physically reasonable. Using the approximation ͑25͒, we find the solution of equation
͑23͒ in the limit of a point dislocation: ͑x,zϾ0͒ХϪ 2 exp ͫ Ϫ
b ͩ ␥ J 2 ͪ 1/2 ͬ ϩ 1 2 ͩ 2 ͪ 1/2 ͱ
Ϫr
Ϫ
2 l˜ K 1 ͑r͒, ͑28͒ ͑x,zϽ0͒Х 1 2 ͩ 2 ͪ 1/2 ͱ
Ϫr
Ϫ
2 l˜ K 1 ͑r͒. For x →ϩϱ Eq. ͑28͒ yields the domain wall solution ͑9͒: (x,z) →Ϫ( /2)exp( Ϫz/ ). For x →Ϫϱ and z→Ϯϱ the magnetization tends to zero, and the ordering of the spins becomes ideal.
Let us start with the matter of the chosen orientation of the domain wall. We assume that the plastic deformation 607
Low Temp. Phys. 26 (8), August 2000 O. K. Dudko and A. S. Kovalev creating the dislocation involved a shift along the x axis, and this shift gave rise to the domain wall. Since the latter is associated with an excess surface energy density, the energy of the domain wall is proportional to its length. Conse- quently, it will be favorable for the system to terminate the domain wall on another dislocation in the same glide plane at a distance R 1 from the first ͑analogous to the situation de- scribed in Ref. 11 for a 1D system ͒. It can happen, however, that a dislocation in another glide plane is located nearer. Then these two dislocations will be ‘‘connected’’ by a do- main wall that is inclined to the x axis if the cost in anisot- ropy energy is compensated by a decrease in the distance R 2 between these two dislocations ͑R 2 ͑ ͱ J 2 cos 2 ϩ ͱ J 1 sin 2 ͒ϽR 1 ͱ
2 ͒.
other axis is also worthy of study. In the present paper we have used the proposed model to investigate the distribution of the magnetization: a ͒ in an easy-plane anisotropic AFM with ideal crystal structure and containing a domain wall with a Bloch line; we have found the characteristic sizes and
cal feature along the x and z axes; b ͒ in an easy-plane anisotropic AFM containing an edge dislocation. We have shown that the dislocation necessarily involves termination of the domain wall lying along the slip line of the dislocation for ␣ ӷ  . We have estimated the con- tribution of the elastic and magnetic parameters to the gradi- ent of the relative rotations of the spins. The authors thank A. M. Kosevich for interest in this study and for valuable comments. * E-mail: odudko@ilt.kharkov.ua 1 ͒ Domain wall with other orientations will be discussed in the Conclusion. 1 K. Hirakawa, H. Yoshizawa, J. D. Axe, and G. Shirane, J. Phys. Soc. Jpn. 52, 19 ͑1983͒.
2 D. G. Wiesler, H. Zabel, and S. M. Shapiro, Physica B 156 Õ157, 292 ͑1989͒.
3 D. G. Wiesler, H. Zabel, and S. M. Shapiro, Physica B 93, 277 ͑1994͒. 4
͑1984͒. 5 D. I. Head, B. N. Blott, and D. Melville, J. High Temp. Chem. Processes 8, 1649 ͑1988͒.
6 A. I. Zvyagin, V. N. Krivoruchko, V. A. Pashchenko, A. A. Stepanov, and D. A. Yablonski , Zh. E´ksp. Teor Fiz. 92, 311 ͑1987͒ ͓Sov. Phys. JETP 65, 177 ͑1987͔͒.
7 A. A. Stepanov, M. I. Kobets, and V. A. Pashchenko, Fiz. Nizk. Temp. 20, 267 ͑1994͒ ͓Low Temp. Phys. 20, 211 ͑1994͔͒. 8 H. Yamazaki and V. Mino, Suppl. Prog. Theor. Phys. 94, 400 ͑1989͒. 9
͑1977͒ ͓Sov. J. Low Temp. Phys. 3, 125 ͑1977͔͒. 10 I. E. Dzyaloshinski , JETP Lett. 25, 98 ͑1977͒. 11 A. S. Kovalev, Fiz. Nizk. Temp. 20, 1034 ͑1994͒ ͓Low Temp. Phys. 20, 815
͑1994͔͒. 12 O. K. Dudko and A. S. Kovalev, Fiz. Nizk. Temp. 24, 559 ͑1998͒ ͓Low Temp. Phys. 24, 422 ͑1998͔͒. 13 A. B. Borisov and V. V. Kiselev, Physica D 111, 96 ͑1998͒. 14 M. E. Gouvea, G. M. Wysin, A. R. Bishop, and F. G. Mertens, Phys. Rev. B 39, 11840 ͑1989͒.
15 A. M. Kosevich, Theory of the Crystal Lattice ͓in Russian͔, Vishcha Shkola, Kharkov ͑1988͒. 16
͑1991͒. Translated by Steve Torstveit 608 Low Temp. Phys. 26 (8), August 2000 O. K. Dudko and A. S. Kovalev |
