JOURNAL OF APPLIED PHYSICS

VOLUME 87, NUMBER 9

1 MAY 2000

Surface anisotropy and the Kondo effect in restricted geometry „invited… ´ jsa´ghy,a) L. Borda, and A. Zawadowskib) O. U

Institute of Physics and Research Group of Hungarian Academy of Sciences, Technical University of Budapest, H-1521 Budapest, Hungary

In recent years, several groups have demonstrated experimentally that in thin films the Kondo effect has a smaller amplitude but roughly the same Kondo temperature compared to the bulk values. We have developed a theory of surface anisotropy proportional to K d S z2 , where S z is the component of the impurity spin perpendicular to the surface and K d is an amplitude which is proportional to the inverse of the distance measured from the surface. The anisotropy is due to the fact that conduction electrons interact not only with the magnetic impurity by the exchange interaction, but also with the host atoms by the spin-orbit interaction, which allows the impurity spin to get information about the shape of the sample. Near the surface of the sample and at low temperature, the spin degrees of freedom freeze in the singlet ground state for spin S⫽2 so that the impurities at the surface cannot contribute to the Kondo anomaly, in contrast to the case S⫽5/2 where the ground state is degenerate. The theory of the surface anisotropy is in good agreement with experiments examining Kondo resistivity and magnetoresistance in the ballistic regime, while in the dirty limit, a different theory is required, which has recently been developed by I. Martin, Y. Wan, and P. Phillips. © 2000 American Institute of Physics. 关S0021-8979共00兲93408-X兴

I. INTRODUCTION

arises for the impurity spin due to the spin-orbit interaction of the conduction electrons on the nonmagnetic host atoms. This anisotropy reflects the geometry of the sample and the position of the impurity, which can be described for planelike surfaces by the Hamiltonian H a ⫽K d (n"S) 2 , where n is the normal direction of the experienced surface element and S is the impurity spin. K d is the anisotropy constant, which is positive and inversely proportional to the distance d of the impurity from the surface. Recently, Fomin and his co-workers13 have worked out an elegant method which can be applied for general geometries. The surface anisotropy results in a splitting of the spin multiplet according to S z 共when the z axis is parallel to n兲 which results in freezing out of the spin states with higher energies as the temperature is lowered or the distance from the surface is reduced at a given temperature. Those splittings are different for integer and half-integer spin. For integer spin, the lowest level is a singlet with S z ⫽0, while for half-integer spin, it is S z ⫽⫾1/2. Thus, an impurity close enough to the surface, where the splitting is essentially larger than the Kondo energy k B T K , does not contribute to the Kondo resistivity for integer spin, but does give a contribution even for large anisotropy for half-integer spin. To explain the measured size dependence for both cases, the suppression in the Kondo resistivity due to the surface anisotropy was calculated using the multiplicative renormalization group method14,15 and excellent agreement with experiments was found. The magnetoresistance of thin films in the presence of the surface anisotropy was calculated by solving the Boltzmann equation in the relaxation time approximation for T ⰇT K , 16 giving also very good agreement with the experiment of Giordano.5 That experiment verifies the presence of the surface anisotropy even in the case where the Kondo effect is absent.

In the last decade, dilute magnetic alloys with restricted dimensions have been of considerable interest both experimentally and theoretically. The measurements1,2 were motivated first by searching for the Kondo compensation cloud3 and were focused on the Kondo resistivity4 in thin films and narrow wires of magnetic alloys with nearly integer spin impurities 关i.e., Au共Fe兲, Cu共Fe兲, and Cu共Cr兲 alloys兴. In most of the experiments,1 a suppression in the amplitude of the Kondo resistivity was found when the sample size was reduced or the disorder in the sample was increased. Later on, other appearances of the size effect 共thickness dependence兲 were found by examining the magnetoresistance,5 and the thermopower6 of thin films. More recently, the examination of the size effect in Kondo resistivity has been extended to alloys with nearly half-integer spin impurities 关i.e., Cu共Mn兲兴.7 On the theoretical side, the first possible explanation was related to the size of the Kondo screening cloud.8 This explanation, however, was ruled out both experimentally9 and theoretically10 as resistivity depends only on the density of states and level spacing of the conduction electrons. There are two theories which have turned out to be able to explain the size effects in different limits. The first theory based on weak localization by Phillips and co-workers11 might well explain the experiments in the limit of strong disorder. In the limit with long mean free path, the theory of the spin-orbit-induced surface anisotropy introduced by Gyorffy and the authors12 explains well the measurements made in dilute Kondo alloys with small disorder and restricted dimensions. According to this theory,12 a magnetic anisotropy a兲

Electronic mail: [email protected] Also at Research Institute for Solid State Physics, POB 49, H-1525 Budapest, Hungary.

b兲

0021-8979/2000/87(9)/6083/5/$17.00

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© 2000 American Institute of Physics

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J. Appl. Phys., Vol. 87, No. 9, 1 May 2000

Ujsaghy, Borda, and Zawadowski

New experiments examining the thermopower6 in samples with restricted dimensions are also in accordance with the surface anisotropy mechanism which was proposed to explain them. In this article, we summarize our understanding about the surface anisotropy, why and when it arises and what consequences it has for the Kondo effect and magnetoresistance of samples with restricted dimensions. II. SPIN-ORBIT-INDUCED SURFACE ANISOTROPY

The existence of the surface anisotropy was shown in a simplified model12 where an infinite half space is considered. Host atoms with spin-orbit interaction are homogeneously dispersed 共crystal structure effects are ignored兲 and the magnetic impurity is placed at a distance d from the surface. The conduction electrons move in the unlimited whole space and only the distribution of the spin-orbit scatterers reflects the ‘‘shape’’ of the sample. In real sample actually, the conduction electrons are confined in the sample and they are scattered by the surface, but it is expected that the qualitative results of the anisotropy are not sensitive to this aspect of the model, as in a mesoscopic sample the surface scattering is rather incoherent 共see Ref. 13兲. The electrons are treated free-electron-like, thus with elastic mean free path l el⫽⬁, while in reality l el is finite. This results in an exponential decay in the Green’s function connecting the impurity and the spin-orbit scatters which ensures that in reality the anisotropy is influenced only by those spin-orbit scatterers which are inside the region of the elastic mean free path 共ballistic region of the surface兲. For the interaction between the conduction electrons and the magnetic impurity, we have chosen the simplest model with orbital quantum numbers where the azimuthal quantum number 共m兲 is preserved in the scattering on the impurity. This model is valid only for a perfectly developed spin (S ⫽(2l⫹1)/2, i.e., S⫽5/2 for l⫽2兲.17 It is believed, however, that the concept of the surface anisotropy would remain valid in a more realistic model and that only the quantitative results would change. For the spin-orbit interaction on the host atoms, a simple model was introduced where the spin-orbit interaction takes place on the d-levels of the host which hybridize with the conduction electrons.12 Thus, the model Hamiltonian is H⫽

兺

k,m, ␴

⫹J

† ⑀ k a km ␴ a km ␴

兺 k,k ,m,m ⬘

␴␴ ⬘

⫹⑀0

兺

nm ␴

⬘

† S共 a km ␴ ␴␴␴ ⬘ a k ⬘ m ⬘ ␴ ⬘ 兲 ␦ mm ⬘

b 共mn␴兲 † b 共mn␴兲 ⫹␭

兺

nmm ⬘ ␴␴ ⬘

具 m 兩 L兩 m ⬘ 典

共n兲

⫻具 ␴ 兩 ␴兩 ␴ ⬘ 典 b 共mn␴兲 † b m ⬘ ␴ ⬘ ⫹

共 V kmm ⬘ 共 Rn 兲 b 共mn␴兲 † a km ⬘ ␴ ⫹h.c. 兲 , 兺 nkmm ␴ ⬘

共1兲

FIG. 1. 共a兲 Electron propagator leaving and arriving at the impurity. The heavy lines represent the localized d-electron propagators, and V and ␭ indicate the hybridization with the localized orbital and the spin-orbit interaction, respectively. The indices are according to the local system where the z axis is parallel to Rn . 共b兲–共d兲 Self-energy diagrams for the impurity spin. The double line represents the spin, the single one the conduction electrons. The solid circles stand for the exchange interaction and the x labeled by n for the effective spin-orbit interaction on the orbital of the host atom at Rn .

† where a klm ␴ (a klm ␴ ) creates 共annihilates兲 an electron with momentum k, angular momentum l, m and spin ␴, J is the effective Kondo coupling, ␴ stands for the Pauli matrices and the origin is placed at the impurity site. In Eq. 共1兲, the host atom orbitals are labeled by n referring to the position Rn and also by the quantum numbers l, m, ␴. Keeping only the l⫽2 channels both for the magnetic impurity and for the host 共e.g., l⫽2 for Cu and Au host兲, the indices l are (n)† (n) dropped. b m ␴ (b m ␴ ) creates 共annihilates兲 the host atom orbital at site n, V kmm ⬘ (Rn ) is the Anderson hybridization matrix element,18 which depends on Rn since a spherical wave representation with origin at the magnetic impurity is used. ␭ is the strength of the spin-orbit coupling, and L is the orbital momentum at site n. Both the exchange interaction and the spin-orbit interaction are assumed to be weak, so that perturbation theory can be applied. In the framework of the presented model, first the electron propagator leaving and arriving at the impurity 关see Fig. 1共a兲兴 was calculated in first order of spin-orbit coupling.12 In the next step, the self-energy corrections for the impurity spin according to Figs. 1共b兲–1共d兲 was calculated12 by using Abrikosov’s pseudofermion representation19 for the impurity spin and Matsubara’s diagram technique applied for the exchange interaction given by Eq. 共1兲.12 It turns out that the leading contribution in spin-orbit coupling to the anisotropy comes from the second order diagram in Fig. 1共c兲, so that we must carry out a summation over the two host atoms in the diagram. This summation corresponds in our model to an integration with respect to the coordinates of the two host atoms, which were estimated in the leading order in 1/(k F d) 12. The result for the anisotropy constant for k F dⰇ1 is12

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J. Appl. Phys., Vol. 87, No. 9, 1 May 2000

K d ⫽16D 共 J ␳ 0 兲 2

⌬ 2␭ 2

⑀ 40

f

冉冊

␻ 1 P共 k Fr 0 兲 ⬎0, D 共 k Fa 兲6 k Fd

Ujsaghy, Borda, and Zawadowski

共2兲

where ␳ 0 is the density of the states of the conduction electrons for one spin direction with bandwidth D, ⌬ is the width of the d-level due to the hybridization, 1/⑀ 0 is a constant by which the Green’s function of the d-level can be replaced for ␻ Ⰶmax兵⑀d ,⌬其. ␻ is the energy variable for the spin degree of freedom, the function f ( ␻ )⬃2 共Ref. 12兲 gives the analytical part of the diagram, and a 3 is the size of the volume per host atom. P(k F r 0 ) is a numerical factor depending strongly on r 0 共a short distance cutoff in range of the atomic radius兲 and it is positive at least for k F r 0 ⫽⬎0.1. Oscillations occur only in the next order 关 ⬃1/(k F d) 2 兴 . It is important to note that the calculation is beyond the Hartree–Fock approximation, and thus it cannot be obtained from a simple band structure calculation in agreement with.20 The result is quite general and in all of the cases K d ⬎0.12 A numerical estimation for K d was given in Ref. 12 as 0.01 1 eV⬍K eV. 共 d/Å 兲 共 d/Å 兲

共3兲

The strongest ambiguity in the calculation comes from P(k F r 0 ) with the short-range cutoff r 0 . III. THE KONDO EFFECT IN SAMPLES WITH RESTRICTED GEOMETRY

The experiments that have concentrated on the study of the Kondo effect in restricted geometry 共thin films, narrow wires兲1,2 have examined the effect of reduced dimensions on the Kondo temperature and the amplitude of the Kondo resistivity. A change in the Kondo temperature turned out to be almost negligible, but in most of the experiments, a size effect in the amplitude of the Kondo resistivity was observed.1 A set of experiments21,22 was performed where the film of dilute alloys was covered by a second layer of pure metal. In these experiments, the so-called proximity effect was observed, that is, the covering by a second layer results in a partial recovery of the suppression in the Kondo resistivity depending on the thickness of the second layer. In this section, we present the calculation of the Kondo resistivity both for integer and half-integer impurity spin14,15 and compare it to the experiments made in dilute magnetic alloys with small disorder. In the presence of the spin-orbit-induced surface anisotropy, the Kondo Hamiltonian is14 H K⫽

兺 k, ␴ ⫹

⑀ k a k†␴ a k ␴ ⫹KS z2

兺 k,k , ␴ , ␴ ⬘

M ,M ⬘

⬘

J M M ⬘ SM M ⬘ 共 a k†␴ ␴␴␴ ⬘ a k ⬘ ␴ ⬘ 兲 ,

共4兲

(a k ␴ ) creates 共annihilates兲 a conduction electron where with momentum k, spin ␴ and energy ⑀ k measured from the Fermi level. The conduction electron band is taken with constant density of states ␳ 0 for one spin direction, with a sharp and symmetric bandwidth cutoff D. ␴ stands for the Pauli a k†␴

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FIG. 2. Coefficient B/B bulk as a function of t/ ␣ , solid line: for S⫽2 and T K ⫽0.3 K 关i.e., Au共Fe兲兴, dashed line: for S⫽5/2 and T K ⫽10⫺3 K, 关i.e., Cu共Mn兲兴.

matrices, J M M ⬘ ’s are the effective Kondo couplings and the quantization axis is chosen to be parallel to n. For the initial values, J M M ⬘ ⫽J holds. Corresponding to the freezing out of different intermediate states due to the surface anisotropy, a multiple step scaling was performed using the Callan–Symantzik multiplicative renormalization group method.14 The running couplings j M M ⬘ ⫽ ␳ 0 J M M ⬘ at D⫽T obtained in that manner were used to calculate the Kondo resistivity by solving the Boltzmann equation. It turned out14 that the Kondo temperature is only slightly affected by the anisotropy, in agreement with the experiments. The amplitude of the Kondo resistivity shows similar behavior as in the experiments with size effects.1,7 That behavior, however, is quite different for integer and half-integer impurity spin. In the half-integer spin case, the impurities close enough to the surface still contribute to the Kondo resistivity, and the size dependence for half-integer spin is drastically suppressed, in contrast to the integer spin case14,15 and in agreement with the experiments.1,7 To compare the results quantitatively to the experiments, the Kondo resistivity of a thin film was calculated as14 ¯␳ Kondo共 t,T 兲 ⬇

1 t

冕␳ t

0

Kondo共 K 共 x,t 兲 ,T 兲 dx,

共5兲

where the anisotropy constant

␣ ⬎␣ K 共 d,t 兲 ⫽K d ⫹K t⫺d ⫽ ⫹ d t⫺d

共6兲

is used. Two surfaces of the film contribute in an additive way to the anisotropy. In Eq. 共6兲, ␣ is the proportionality factor of the surface anisotropy and t is the thickness of the film. Fitting the logarithmic function ⫺B ln T to the Kondo resistivity, the thickness dependence of the coefficient B was examined both for integer 共i.e., S⫽2 as for Fe兲 and halfinteger 共i.e., S⫽5/2 as for Mn兲 impurity spins.14,15 The plots of B/B bulk can be seen in Fig. 2 where we can see well the difference between the integer and half-integer impurity spin cases. First of all, the Kondo amplitude compared to the bulk value is reduced in both cases, but the thickness dependence is much weaker for S⫽5/2 than for S⫽2. Second, for small

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J. Appl. Phys., Vol. 87, No. 9, 1 May 2000

Ujsaghy, Borda, and Zawadowski

t/ ␣ ’s, the coefficient goes to zero for S⫽2, but for S⫽5/2, it has a minimum below which the slope changes sign and it tends to increase. This difference in behavior reflects the difference that for S⫽2 the lowest energy state in the presence of the surface anisotropy is S z ⫽0, which does not contribute to the Kondo resistivity, while for S⫽5/2, the lowest energy state is the S z ⫽⫾1/2 doublet, which gives a contribution to the Kondo resistivity even for large anisotropy 共i.e., small distance from the surface兲. As the minimum in the case of S⫽5/2 comes from the region, where j 1/2 1/2 is already in the strong coupling limit where the calculation is not reliable, it may only be a sign of the breakdown of the weak coupling calculation.15 The proximity effect can be well understood with the help of the surface anisotropy because the covering layer provides new spin-orbit scatterers for the electrons.14 An experiment was proposed12 where in the covering layer the spin-orbit interaction is negligible 共e.g., Al or Mg兲, so that the proximity effect does not work. As the anisotropy looses its meaning for S⫽1/2, we do not expect any size dependence in that case, in agreement with the experiment.23 With the help of the surface anisotropy, the thickness dependence of the magnetoresistance can also be explained, as was proposed by Giordano.5 In the presence of a magnetic field, the Hamiltonian of the magnetic moment is H⫽K d 共 n"S兲 2 ⫹g B B"S.

共7兲

Thus, in the presence of the anisotropy, the magnetic field splits the levels further in addition to the splitting caused by the anisotropy, and because of the level crossing, a larger magnetic field is required to separate the lowest energy state from the other levels in order to saturate the magnetoresistance. The calculation of the magnetoresistance in the presence of the surface anisotropy16 gave a very good agreement with the experiment5 共see Fig. 3兲. The recent measurement of the thermopower of Au共Fe兲 samples with an appropriate shape6 shows that the thermopower decrease when the wire becomes narrower, also can be understood with help of the surface anisotropy. IV. CONCLUSIONS

In this article, we summarized our knowledge about the origin of the surface anisotropy and its influence on the Kondo effect in samples with restricted dimensions. The spin-orbit-induced magnetic anisotropy for magnetic impurities in magnetic alloys with restricted dimensions originates from the spin-orbit interaction of the conduction electrons on the nonmagnetic host atoms, independently of whether Kondo effect occurs or not. It is very important that angular momentum channels different from zero (l⫽0) are considered in the electron scattering on the magnetic impurity, because otherwise no anisotropy axis can be exhibited.12 Despite the approximations made in the calculation, such as the consideration of an infinite sea of electrons, the use of the simplest, Hamiltonian with orbital quantum numbers, the treatment of the conduction electrons as free 共no randomness was considered兲 and the most simple geometry, we think that the concept of the anisotropy and the qualitative results re-

FIG. 3. Magnetoresistance in a thin Au共Fe兲 film. The resistance unit ␳ 0 can 2 2 be expressed in terms of the parameters of the theory, ␳ ⫺1 0 ⫽ 关 ␳ ( ⑀ F )/J 兴 ⫻关 (8 ␲ 3 m e5 )/12e 2 兴 . The inset shows the splitting of the impurity spin energy levels for S⫽2 in the presence of the surface anisotropy and external magnetic field perpendicular to the surface of the film 共dashed lines for S z ⫽⫾2, solid lines for S z ⫽⫾1 and dotted lines for S z ⫽0兲.

main valid in more realistic cases, too. The elegant extension of the calculation for arbitrary geometries by Fomin and co-workers18 supports this confidence. The calculation of the Kondo resistivity in the presence of the surface anisotropy has shown that there is a size dependence in the resistivity, but the way of the reduction in the amplitude of the Kondo resistivity is quite different for integer and half-integer impurity spin. The size dependence is drastically suppressed for the half-integer case compared to the integer case. The Kondo temperature is only slightly affected. The magnetoresistance16 and the proximity effects can be explained, given the existence of the surface anisotropy. These results are in very good agreement with the experiments.1,5,7 The values of the ␣ parameter in Eq. 共6兲 obtained from fits on the experimental data of the Kondo resistivity in Au共Fe兲21,14 and Cu共Fe兲,22,24 and of the magnetoresistance for Au共Fe兲5,16 are shown in Table I. The differences between them likely come from the different mean free paths of the samples. It is interesting, however, that for Au共Fe兲 and Cu共Fe兲, almost the same value of ␣ was obtained. The uncertainties in the experimental data7 for Cu共Mn兲 were too large to resolve quantitatively the slight thickness dependence. The size dependence of the thermopower of samples of dilute magnetic alloys with restricted dimensions6 can also be well understood with the help of the surface anisotropy. TABLE I. Fitted values of ␣ defined by K(d,t)⫽( ␣ /d)⫹( ␣ /t⫺d) on the experimental data of the experiments.

␣ 共Å K兲

Experiment a

Kondo res. in Au共Fe兲 Kondo res. in Cu共Fe兲b magnetores. in Au共Fe兲c

248 247 42

a

Ref. 21. Ref. 22. c Ref. 5. b

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J. Appl. Phys., Vol. 87, No. 9, 1 May 2000

Summarizing, the theory of the surface anisotropy gives a coherent description of the size effects in samples of dilute magnetic alloys with restricted dimensions in the ballistic region. The effect of surface induced anisotropy in heavy fermion materials has recently been discussed by Schiller and co-workers, who showed that similarly to the mechanism presented in this article, the surface induced local density fluctuations may generate a large splitting of the orbitally degenerate states and kill the two-channel Kondo effect in the vicinity of the surface.25 ACKNOWLEDGMENTS

This work was supported by Grant OTKA Nos. T024005 ´ .兲 was partially supand T029813. One of the authors 共O. U ported by OTKA Postdoctoral Fellowship D32819. A. Blachly and N. Giordano, Phys. Rev. B 51, 12537 共1995兲 共for a review兲; J. F. DiTusa, K. Lin, M. Park, M. S. Isaacson, and J. M. Parpia, Phys. Rev. Lett. 68, 678 共1992兲. G. Neuttiens, J. Eom, C. Strunk, V. Chandrasekhar, C. van Haesendonck, and Y. Bruynserade, Europhys. Lett. 34, 623 共1996兲. 2 V. Chandrasekhar, P. Santhanam, N. A. Penebre, R. A. Webb, H. Vloeberghs, C. van Haesendonck, and Y. Bruynserade, Phys. Rev. Lett. 72, 2053 共1994兲. 1

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