PHYSICAL REVIEW B VOLUME 55, NUMBER 2 1 JANUARY 1997-II Ab initio calculation of the perpendicular giant magnetoresistance of finite Co/Cu 001... and Fe/Cr 001... superlattices with fluctuating layer thicknesses J. Mathon Department of Mathematics, City University, London EC1V 0HB, United Kingdom Received 10 July 1996 The results of rigorous quantum calculations of the current-perpendicular-to-plane giant magnetoresistance CPP GMR of finite Co/Cu 001 and Fe/Cr 001 superlattices with perfectly flat interfaces but with growth- induced fluctuations in layer thicknesses are reported. They are based on an exact numerical evaluation of the Kubo formula using tight-binding parametrization with s, p, d bands and hopping to first and second neigh- bors of an ab initio band structure. These calculations show that three distinct regimes of CPP transport occur. When there are no fluctuations, CPP transport is in the ballistic regime. The CPP GMR ratio RCPP of finite Co/Cu and Fe/Cr superlattices in the ballistic regime reach saturation values equal to RCPP of an infinite superlattice after only 3­5 repeats of a superlattice unit cell and the maximum values of RCPP are of the order of 100%. When small fluctuations in layer thickness corresponding to only one atomic plane at the interface being displaced are introduced, transport changes from ballistic to Ohmic. The calculated GMR ratio RCPP increases initially linearly with the number N of ferromagnet/spacer bilayers and then saturates for N 40­50. The theoretical maximum values of RCPP for Co/Cu and Fe/Cr superlattices in the Ohmic regime are in the region 800­1000 %. The zero-field and saturation-field resistances increase linearly with N good Ohm's law and the calculated zero-field resistance of the Co/Cu superlattice is within 10% of the resistance observed in a Co/Cu sample of the same composition and thickness. Small spontaneous growth-induced fluctuations in layer thickness can thus account well for the observed CPP GMR. When superlattices with large fluctuations in layer thickness are grown deliberately pseudorandom spin valves , the Ohmic regime changes into, experimentally as yet unexplored, Anderson localization regime. The results for Co/Cu and Fe/Cr super- lattices in which layer thicknesses are made to fluctuate typically between 2 and 10 atomic planes show that strong disorder of the sequence of ferromagnet/spacer interfaces has virtually no effect on the saturation-field resistance RFM , which remains as low as in the Ohmic regime. The zero-field resistance, on the other hand, increases approximately exponentially with the number of bilayers N due to Anderson localization with a localization length 30­40 nm. The CPP GMR ratio RCPP , therefore, also increases approximately exponen- tially with N and values as high as RCPP 3 104 are predicted for Fe/Cr valves with N 50 bilayers. Some- what smaller (RCPP 104) enhancement of the CPP GMR is obtained for Co/Cu pseudorandom spin valves. The conditions under which such enhancement should be observable are discussed. S0163-1829 97 04102-7 I. INTRODUCTION on an exact numerical evaluation of the Kubo formula for fcc Co/Cu 001 and bcc Fe/Cr 001 finite superlattices sand- I recently proposed1 that magnetic multilayers with delib- wiched between two semi-infinite leads. These calculations erately induced large fluctuations in layer thickness should show that, depending on the size of fluctuations in layer exhibit a very large enhancement of the current- thickness, three different regimes of CPP transport occur: perpendicular-to-plane giant magnetoresistance CPP GMR . ballistic, Ohmic, and Anderson localization. Such multilayers will be referred to as pseudorandom spin The first regime corresponds to no fluctuations in layer valves.2 A single-orbital tight-binding model used in Ref. 1 thickness, i.e., the case of transport in a perfectly periodic but predicts that the CPP GMR of a pseudorandom spin valve finite superlattice without any impurities. The superlattice is grows exponentially with its thickness and values of the sandwiched between two semi-infinite leads made of the GMR ratio RCPP as high as RCPP 105 % can be expected.1 same material as the nonmagnetic spacer Cu or Cr . When The large enhancement of the CPP GMR is due to quantum the number Nrpt of unit cells of such a finite superlattice is interference of electrons undergoing multiple reflections sufficiently large typically, Nrpt 3­5 is enough , the re- from a disordered sequence of ferromagnet/spacer interfaces. sults of Schep et al.3 for an infinite superlattice are recov- The predicted very large enhancement of the CPP GMR can ered. This provides an independent check on the validity and be observable only if the contribution to the resistance of a accuracy of the evaluation of the GMR from the Kubo for- pseudorandom spin valve due to quantum interference ef- mula since Schep et al.3 have used a completely different fects is so large that it dominates the total resistance of the method based on counting propagating states. Note that valve. To decide whether this is the case one needs to make their method can only be applied to a perfectly periodic in- a rigorous quantum calculation of the total resistances in the finite system. As already demonstrated by Schep et al.3, a antiferromagnetic AF and ferromagnetic FM configura- large RCPP 100% due to quantum effects is obtained. How- tions for a specific multilayer system using an ab initio band ever, the total resistance from this source for typical Co/Cu structure. I report here the results of such calculations based superlattices is only 3 10 15 m2, which is too small to 0163-1829/97/55 2 /960 10 /$10.00 55 960 © 1997 The American Physical Society 55 Ab initio CALCULATION OF THE PERPENDICULAR . . . 961 explain the measured results. Moreover, the calculated resis- tances and GMR are independent of the sample thickness which disagrees with the experiment.4,5 One must, therefore, conclude that quantum effects in perfect periodic superlat- tices ballistic transport are not seen in the present experi- ments. However, they should be seen in future devices with smaller transverse dimensions ballistic contacts6 . The second regime occurs for finite Co/Cu and Fe/Cr su- perlattices without any impurities in which the thicknesses of both the magnetic and nonmagnetic layers are allowed to deviate at random from their nominal values. The CPP GMR was again evaluated exactly from the Kubo formula. Small fluctuations in layer thickness corresponding to only one FIG. 1. Schematic representation of a finite magnetic superlat- atomic plane at the interface being displaced were consid- tice in the CPP geometry. All the magnetic nonmagnetic layers ered. Such small fluctuations occur spontaneously due to ter- have the same thickness M (N) in Sec. II but the thicknesses of race formation even in most carefully grown superlattices. both magnetic and nonmagnetic layers are allowed to fluctuate at The effect of such relatively small growth imperfections on random in Secs. III and IV. the CPP GMR is profound. The transport changes from bal- listic to Ohmic and the CPP GMR increases initially linearly tacts lead wires . This is one step closer to reality than the with the superlattice thickness and then saturates. This is pioneering work of Schep et al.3 who calculated the GMR precisely the behavior of the CPP GMR observed by for a perfect infinite superlattice using an ab initio band Schroeder et al.4 The calculated CPP GMR ratio RCPP ranges structure. With a finite superlattice, one has a translationally from about 50 to 1000 % depending on the sample thickness inhomogeneous system and the method of counting all the and unit cell composition. Moreover, the total resistance in propagating states in an infinite superlattice, employed by the antiferromagnetic configuration of a Co/Cu sample of the Schep et al.,3 is no longer applicable. The Kubo formula has same composition and thickness as in Ref. 4, calculated to be used instead. It was evaluated using a tight-binding without any adjustable parameters, is 165 10 15 m2. parametrization with s,p,d bands and hopping to first and This is almost exactly the same value as observed by second nearest neighbors of an ab initio band structure. The Schroeder et al.4. The calculated results, therefore, indicate tight-binding parameters for ferromagnetic fcc Co were that the whole observed CPP GMR can be explained by taken from Ref. 7 and those for Cu, Fe, and paramagnetic Cr quantum scattering from small fluctuations in layer thick- were taken from Ref. 8. A small lattice mismatch between ness. Co/Cu and Fe/Cr was neglected. Finally, when large fluctuations in layer thickness are in- The system for which the Kubo formula was evaluated is duced deliberately, one is in the pseudorandom spin valve shown schematically in Fig. 1. It consists of the left and right regime.1 Pseudorandom spin valves can be fabricated by semi-infinite leads contacts made of the spacer material Cu growing layers whose thicknesses follow a predetermined or Cr which are attached to N pseudorandom sequence.1 Once a specific pseudorandom se- rpt repeats of a superlattice unit cell. Each magnetic unit cell consists of a ferromagnetic quence is chosen to grow an experimental sample, the CPP layer containing M atomic planes followed by N atomic GMR of that particular sample can be evaluated exactly from planes of a nonmagnetic spacer, a second ferromagnetic the Kubo formula in which the same pseudorandom growth layer containing again M atomic planes and, finally, a second sequence is used, and vice versa. I report here the results for nonmagnetic layer of N atomic planes. For the purpose of Co/Cu and Fe/Cr pseudorandom spin valves in which the comparing the calculated results with the experiment, the thicknesses of the magnetic and nonmagnetic layers were more commonly used number of bilayers N (N 2N typically made to fluctuate between 2 and 10 atomic planes. rpt) will be used. The calculations based on an ab initio band structure confirm The Kubo formula has to be evaluated separately for up- the results obtained earlier for a single-orbital tight-binding and down-spin carriers in the FM configuration and for car- model.1 The calculated CPP GMR of Co/Cu and Fe/Cr pseu- riers of either spin orientation in the AF configuration. Since dorandom spin valves increases approximately exponentially the in-plane translational invariance is preserved, the wave with the valve thickness and the maximum calculated RCPP is of the order of 5 104 % both for Co/Cu and Fe/Cr valves. vector k parallel to the layers remains a good quantum num- ber. It follows that the total conductance in a spin channel can be written for any magnetic configuration of the su- II. FINITE PERFECTLY PERIODIC SUPERLATTICE perlattice as a sum of partial conductances The only reliable quantum-mechanical method for calcu- lating the GMR without any adjustable parameters is an ex- act numerical evaluation of the Kubo formula using a fully e2/h k 1 realistic band structure. In general, this is, of course, an im- k possible task. However, there are well defined cases for mag- netic multilayers accessible to experiment for which this can be done. The simplest case is that of a perfect finite super- where (k ) is the partial conductance in a channel lattice without any impurities sandwiched between two con- (k , ) measured in units of the quantum conductance e2/h 962 J. MATHON 55 and the sum in Eq. 1 is over all k and right overlayers. I use notation g for the left and g for from the two- 00 11 dimensional Brillouin zone BZ . The partial conductance the right overlayer surface Green's functions. Finally, the (k exact Green's functions G , G , and G for the connected ) is given by the Kubo formula.6,9­11. For a general 00 11 01 multiorbital band structure, the Kubo formula expressed in multilayer are obtained from g 00 and g11 using the Dyson terms of the one-electron propagators takes the following equation.1 form The Green's function g 00 is simply the surface Green's function of the left semi-infinite lead Cu or Cr . The surface k 4Tr G 00 k t01 k G 11 k t10 k Green's function g11 of the right overlayer is generated recursively10,15 from the surface Green's function of the right Re G 01 k t10 k G 01 k t10 ...]. 2 semi-infinite lead. All the atomic planes of the right over- layer are deposited one by one on the right lead and the Equation 2 is a straightforward generalization of the result overlayer surface Green's function is updated after each obtained earlier by Lee and Fisher10 see also Ref. 11 for a deposition from the Dyson equation: single-orbital tight-binding model. The indices 0, 1 in Eq. 2 label any two neighboring principal planes12 parallel to the g k k k layer structure, G new 1 gisol 1 t01gold t10 , 4 i, j(k ) (1/2i) Gi, j(k ) Gi, j(k ) , and G where g (k i, j(k ), Gi, j(k ) are the matrix elements between principal isol ) is the Green's function of an isolated princi- planes i, j of the advanced and retarded one-electron Green's pal layer of the material that is being deposited. Provided the functions evaluated at the Fermi energy E surface Green's function of the lead is known, the recursive F . Similarly, t method based on repeated application of Eq. 4 method of 01(k ) is the tight-binding hopping matrix between the prin- cipal planes 0,1. Because of the current conservation, the adlayers15 involves no approximations and, therefore, gives choice of the planes 0, 1 is arbitrary. The trace is taken over the Green's functions of the connected multilayer with a ma- all the orbital indices that are contained implicitly in the chine accuracy. Moreover, the technique is not restricted to a principal layer indices 0,1. Since hopping to nearest and sec- periodic system and this flexibility of the method of adlayers ond nearest neighbors is considered, each principal plane will be exploited to the full in Secs. III and IV. contains two atomic 001 planes and, therefore, all the The only remaining problem is, therefore, the calculation Green's functions and hopping matrices in Eq. 2 are of the surface Green's functions of the left and right semi- 18 18 matrices. Finally, the usual GMR ratio, defined in infinite leads. In our previous calculations14 of the exchange terms of the conductances for the ferromagnetic FM and coupling in Co/Cu 001 , we used an iterative decimation antiferromagnetic AF configurations of the magnetic lay- technique.16 In this method, the surface Green's function is ers, is given by approximated by its value at the surface of a thick stack of atomic planes. However, to obtain a truly surface Green's R , , function, it is necessary to add in the decimation method a CPP FM FM 2 AF /2 AF . 3 small imaginary part to the energy to disrupt quantum in- The input in Eq. 2 are the matrix elements of the one- terference between the two surfaces of the slab. When is electron Green's function in and between the principal small, the convergence of the decimation method becomes planes 0, 1. This is exactly the same information that is poor. This is not a problem in total energy calculations since needed in the calculation of the oscillatory exchange cou- one integrates over a contour in the complex energy plane. pling from the spin-current formula.13,14 The formal similar- However, there is no energy integral in the Kubo formula ity between the spin current formula for the coupling13 and transport takes place at the Fermi surface and has to be the Kubo formula for CPP transport merely reflects the fact very small in order not to introduce a spurious resistance due that the two effects are closely related. The oscillatory cou- to finite lifetime effects. I have, therefore, used an entirely pling is determined by the spin current between the magnetic new noniterative technique for generating the surface layers and the CPP GMR by the electric current but both Green's function17 in which the convergence problem does currents are, of course, carried by the same electrons. This not arise. A value 10 8 Ry, which was used in all the observation alone is a compelling reason for making a fully calculations, is so small that it has no effect on the conduc- quantum calculation of the CPP GMR. Without quantum in- tance. terference effects, there would be no coupling and it is, there- Finally, the BZ sum in Eq. 1 over k fore, most likely that the quantum effects are very important needs to be carried out. The convergence in k also in the CPP transport. is not such a serious problem as in the calculation of the oscillatory exchange coupling. This From the technical point of view, the formalism for cal- is because, unlike the coupling, the GMR effect does not culating local one-electron Green's functions has already decrease with increasing thickness of the multilayer. I am been developed for oscillatory exchange coupling14 and it not interested here in details of quantum oscillations of the can be easily adapted to the present problem. One uses the GMR about its average value18,19 which would require a far trick of cutting formally the multilayer between the planes 0 and 1 into two disconnected parts by setting the hopping higher accuracy. Nevertheless, a large number of k points matrix h in the two-dimensional 2D BZ is needed to determine even 01 equal to zero. The two disconnected parts are referred to as the left and right overlayers on semi-infinite the nonoscillatory part of the GMR. In all the calculations leads.1 It is convenient to make the cut between the finite reported here, I first used 104k points in the 2D BZ and then superlattice and the left semi-infinite lead. The next step is checked the result for convergence with 4 104 points. This the calculation of the surface Green's functions for the left number of k points is sufficient to achieve convergence for 55 Ab initio CALCULATION OF THE PERPENDICULAR . . . 963 hand, the observed4 total resistance in the AF figuration can be as large as 175 10 15 m2. Moreover, the observed resistance increases linearly with increasing thickness of the multilayer whereas the ballistic resistance is independent of the thickness. This indicates that purely ballistic effects are not seen in the present experiments. However, it would be quite wrong to conclude that quantum reflections from per- fectly flat interfaces play no significant role in CPP transport without first investigating the effect of small fluctuations in layer thickness which inevitably occur even in most carefully grown superlattices. III. FINITE SUPERLATTICE WITH SMALL FLUCTUATIONS IN LAYER THICKNESS In Sec. II perfect finite superlattices were investigated. Real samples contain imperfections. Even if the impurity concentration is negligible e.g., for samples smaller than the FIG. 2. Dependence of the CPP GMR ratio of a finite Co5Cu5 mean free path , there are always growth imperfections in superlattice on the number of bilayers N. any layer structure. They arise because the control over layer multilayers of up to 150 nm thick. This means that, for su- thicknesses in deposition cannot be perfect and also because perlattices with about fifteen atomic planes per unit cell, the of spontaneous terrace formation. The method of adlayers total number of bilayers that can be handled is N 100. For combined with the Kubo formula allows us to determine larger unit cells, reliable results can be obtained only for exactly the CPP GMR and the individual conductances in the N 50. FM and AF configurations for multilayers in which layer The calculated CPP GMR ratio for Co5Cu5 superlattice is thicknesses deviate at random from their nominal values. I plotted in Fig. 2 against the number of bilayers N. The re- will make the most optimistic assumption that individual sults for Co/Cu superlattices with other compositions of the layer thicknesses in experimental samples are controlled so unit cell and those for Fe/Cr superlattices are qualitatively well that they fluctuate at random by no more than one the same. A typical feature is that after only about three atomic plane. This restriction will be relaxed in Sec. IV. repeats of the unit cell the CPP GMR ratio RCPP for all the The method for calculating the conductances and GMR superlattices investigated reaches a saturation value equal to ratio from Eqs. 1 ­ 3 is exactly the same as for a periodic RCPP of an infinite superlattice. Oscillations about the satu- superlattice but the layer thicknesses used in the adlayering ration value seen in Fig. 2 are a genuine effect and occur due procedure 4 are now selected according to the following to size quantization in a superlattice of a finite thickness. prescription. Pseudorandom sequences Mi and Ni of in- Given that the average nonoscillatory component of the tegers distributed uniformly over intervals Mmin , Mmax and CPP GMR reaches its saturation value so rapidly, there is no Nmin , Nmax are generated and the thicknesses Mi and Ni of need to discuss here finite superlattices in any great detail the magnetic and nonmagnetic layers measured in numbers since the results of Schep et al.3 for infinite superlattices are of atomic planes are chosen to follow these sequences. already a very good guide to their behavior. In the case of Since fluctuations of only one atomic plane are allowed, the Co5Cu5 superlattice, the saturation value of GMR is conditions Mmax Mmin 1 and Nmax Nmin 1 are imposed. RCPP 150 %, which is very close to the result obtained by To study systematically the conductances and CPP GMR Schep et al. 120%) for an infinite Co/Cu superlattice of the of Co/Cu samples, the nominal thickness of Cu spacer was same composition. The small discrepancy between the fixed between Nmin 5 and Nmax 6 and nominal Co thick- present results and those of Schep et al.3 is due to differences nesses were in the range 2 Mmin 8 (3 Mmax 9). The in the band structures used. Schep et al.3 considered a tetra- choice of the Cu thickness is dictated by the fact that the hedral distortion of the Co lattice to allow for a small lattice coupling should be antiferromagnetic. The thickness of Co mismatch between Co and Cu which is neglected in the layers was restricted to relatively small values to keep the present work. When the conductance calculated from the computer time within reasonable limits. The calculated CPP Kubo formula for pure Cu is compared with the conductance GMR for a Co8 9Cu5 6 superlattice is plotted as a function obtained by counting the propagating states, the results are of the number of bilayers N in Fig. 3 for 2 N 50. The identical19 and equal to the value of the ballistic conductance superlattice with this particular composition of the unit cell of pure Cu quoted by Schep et al.3,6 was selected for Fig. 3 because Co/Cu 111 samples with the The fact that the calculated CPP GMR is almost as large same Co and Cu layer thicknesses were investigated by as the observed effect4 is not sufficient. It is also necessary to Schroeder et al.4 check whether the total resistance of the sample, particularly It can be seen from Fig. 3 that small fluctuations in layer in its AF configuration, is large enough to be measurable. thickness have a profound effect on the GMR. The magni- The results of the present calculation, in agreement with tude of the calculated GMR ratio RCPP is now so large that it Schep et al.,3 give 2 AF 3 1014 1 m 2. It follows that can easily account for the whole observed effect this is not the ballistic resistance RAF , which is independent of the su- the case for the purely ballistic contribution discussed in Sec. perlattice length, is only about 3 10 15 m2. On the other II . Moreover, the transport is clearly no longer ballistic and 964 J. MATHON 55 FIG. 3. CPP GMR ratio of a Co FIG. 5. Resistances in the ferromagnetic FM and antiferro- 8 9Cu 5 6 superlattice with small fluctuations in layer thickness plotted against the number of magnetic AF configurations of a Co8 9Cu5 6 superlattice with bilayers N. small fluctuations in layer thickness plotted against the number of bilayers N. Circles denote the resistance R , AF squares the resistance the calculated CPP GMR appears to increase linearly with R FM ; and triangles the resistance RFM . N. However, the linear dependence of the CPP GMR on the The crucial test of the theory is, therefore, whether the number of bilayers N holds only for relatively small values absolute values of the resistances R of N. For larger N, the CPP GMR reaches a saturation value. FM and RAF calculated without any adjustable parameters possess these two proper- This cannot be seen in Fig. 3 because, for computational ties. Moreover, they must also be of the same order of mag- reasons, the maximum N is limited to N 50. However, for a nitude as the measured resistances. The theory has one ad- Co 2 3Cu5 6 superlattice, fully converged results can be ob- tained for N 100. They are shown in Fig. 4 open circles vantage over the experiment in that the resistances RFM and together with the corresponding results for Fe RFM in the up- and down-spin channels in the FM configu- 2 3Cr 5 6 su- perlattice full circles . It can be seen that the GMR ratio ration can be calculated separately. They are plotted in Fig. 5 , R together with the resistance RAF 1/2 R in the AF con- CPP reaches a saturation value of the order of 800­1000 % AF for both the Co/Cu and Fe/Cr superlattices. This is precisely figuration against the number of bilayers N for the the behavior reported by Schroeder et al.4 for their Co/Cu Co8 9Cu5 6 superlattice. samples. The observed initial increase of the CPP GMR fol- It can be seen from Fig. 5 that the calculated resistances in lowed by a saturation is due to two factors: i the measured all three channels obey an almost perfect Ohm's law and resistances R start from approximately the same value for small N. A lin- FM and RAF in the FM and AF configurations obey Ohm's law; ii the values of R ear dependence on N clearly indicates that the calculated FM and RAF extrapolated to N 0 are very nearly equal to one another. CPP resistances are due mainly to scattering from interfaces. In fact, an alternative way of viewing a superlattice with small fluctuations in layer thickness is to regard such a sys- tem as a perfectly periodic superlattice in which single atomic planes of a wrong type Cu instead of Co, and vice versa are inserted at random at the interfaces. The calculated CPP resistances can be then explained as being due to scat- tering from ``impurity'' planes located at the interfaces. The linearity of the effect indicates that the scattering from dif- ferent interfaces is uncorrelated in this regime. The calculated zero-field resistance RAF of the Co8 9Cu5 6 superlattice circles in Fig. 5 can be compared with the experimental results of Schroeder et al.4 It is only necessary to extrapolate linearly the calculated RAF from N 50 to N 150, which is the thickness of the sample in- vestigated in Ref. 4. The theoretical resistance for N 150 is RAF 165 10 15 m2. This is almost exactly the same value as the resistance RAF 175 10 15 m2 measured by Schroeder et al.4 for a Co/Cu 111 superlattice of the same composition and thickness (N 150). The results for the FIG. 4. CPP GMR ratios of Co2 3Cu5 6 open circles and Co2 3Cu5 6 and Fe2 3Cr5 6 superlattices, shown in Fig. 6, Fe2 3Cr5 6 full circles superlattices with small fluctuations in are very similar and demonstrate that a good Ohm's law layer thickness plotted against the number of bilayers N. holds also for larger N, which provides a justification for the 55 Ab initio CALCULATION OF THE PERPENDICULAR . . . 965 only modification that the roles of the up- and down-spin channels are interchanged. This is due to the fact that it is now the down-spin minority band in Fe that matches al- most perfectly the bands of Cr. Before I leave this section, a comment on the nature of randomness in samples with small fluctuations in layer thick- ness is called for. I used a pseudorandom number sequence to model such fluctuations. One could argue that a configu- ration average is required. However, when different pseudo- random sequences are tried, one finds that the calculated con- ductances are insensitive to the choice of the sequence. This might seem surprising to those familiar with a large body of theoretical work on one-dimensional disordered wires. How- ever, the explanation is simple. One must remember that the multilayer is not a strictly one-dimensional system. For any fixed configuration of interfaces, electrons in different k channels sample different pseudorandom potential reliefs. FIG. 6. Total resistances R Since the total conductance given by Eq. 1 is the sum over AF and RFM in the antiferromagnetic AF and ferromagnetic FM configurations of Co2 3Cu5 6 open all k , some averaging over disorder is built in naturally. symbols and Fe2 3Cr5 6 full symbols superlattices with small fluctuations in layer thickness plotted against the number of bilayers IV. CO/CU 001... AND FE/CR 001... N. Circles denote the resistance RAF and squares the resistance PSEUDORANDOM SPIN VALVES RFM . Having established in Sec. III that quantum reflections linear extrapolation used above. Only the total resistances from perfectly flat interfaces in multilayers with small fluc- RFM squares and RAF circles in the FM and AF configu- tuations in layer thickness lead to CPP resistances that not rations are shown in Fig. 6. Qualitatively the same behavior only display the observed Ohmic behavior but also have the is obtained for all the Co/Cu and Fe/Cr superlattices investi- correct magnitude, one can address with some confidence the gated with nominal Co Fe thickness ranging from two to interesting questions concerning transport in pseudorandom nine atomic planes and nominal Cu Cr thicknesses fixed spin valves with deliberately induced large fluctuations in between 5 and 6 atomic planes. layer thickness.1 The behavior of the resistance R FM in the up-spin channel The results of Sec. II show that the CPP GMR ratio in the FM configuration of the Co/Cu superlattice squares in R Figs. 5 and 6 is also very interesting. Because the matching CPP for a superlattice in the ballistic regime is independent of its thickness for N of the up-spin bands in Co to the Cu bands is almost perfect, rpt 3 5) and the maximum attain- able R there is virtually no scattering at the interfaces and, therefore, CPP is only of the order of 100 %. The GMR ratio R R CPP of a superlattice in the Ohmic regime also saturates as a FM increases only very slowly with N. Up-spin majority function of the number of bilayers N and the maximum theo- electrons thus provide a low-resistance channel which shunts retical attainable value of RCPP for Co/Cu and Fe/Cr super- the high-resistance channel R FM triangles in Fig. 5 . It fol- lattices is of the order of 1000 %. Saturation of RCPP is a lows that, for all practical purposes, R FM denoted by squares consequence of the Ohmic behavior of the resistances in all in Fig. 5 can be regarded as the total saturation field resis- three conductance channels (R FM , RFM , RAF) and is, there- tance RFM . The slow increase of RFM with N is again very fore, inevitable in this regime. The saturation value of RCPP similar to the observed behavior.4 However, the calculated is determined by the magnetic contrast of an individual in- values of RFM are a factor of 3 smaller than the observed terface. This in turn depends on the difference between the results. This is the main reason why the theoretical CPP strengths of the scattering potentials for the majority- and GMR ratio is also higher by approximately the same factor minority-spin electrons at a ferromagnet/spacer interface. than the observed RCPP . One can think of two most likely Since nature provides us with a limited number of explanations for this discrepancy. The first one is that there is ferromagnet/nonmagnet combinations, and Co/Cu or Fe/Cr some additional weak spin-independent scattering in the ex- are probably the best combinations, this places an upper perimental samples that is not included in the present calcu- bound on what can be achieved with conventional periodic lation. The background resistance due to such scattering superlattices in the ballistic and Ohmic regimes. masks the intrinsic scattering from Co/Cu interfaces in the Since the CPP GMR ratio 3 expressed in terms of the up-spin channel and determines the observed R FM . Alterna- resistances in the FM and AF configurations has the form tively, the matching of the Co and Cu bands in the up-spin channel may not be so perfect when one allows for relaxation RAF RFM effects due to a small Co/Cu lattice mismatch. On the other RCPP R 5 hand, the resistances in the down-spin channel and in the AF FM configuration are clearly totally dominated by the intrinsic it is clear that the only way to enhance the CPP GMR ratio spin-dependent scattering from Co/Cu interfaces and any beyond its saturation value is to fabricate a magnetic background scattering if present is unimportant. Exactly multilayer that operates in a regime in which the dependence the same arguments apply to the Fe/Cr superlattice with the of the resistance on the number of bilayers N is nonlinear. To 966 J. MATHON 55 achieve this goal, I proposed1 that one should grow superlat- tices with deliberately induced large fluctuations in layer thickness. Since CPP transport in a multilayer takes place in independent k channels, the whole multilayer can be re- garded as a system of one-dimensional wires connected in parallel. Large fluctuations in layer thickness mean1 that electrons in every channel move in a one-dimensional strongly disordered potential. It follows that Anderson localization20 must set in in every k channel provided the number of bilayers N is large enough. Since the resistance in each channel increases in the localization regime exponen- tially with the number of bilayers N, the condition that the resistances RFM and RAF in the FM and AF configuration should be nonlinear functions of N can be satisfied. It is well known20 that the localization length decreases with increasing degree of disorder. The crucial point for magnetic multilayers is1 that the degrees of disorder seen by FIG. 7. CPP GMR ratios of Co electrons in the FM and AF configuration and, hence, the 2 6Cu 5 8 squares and Fe2 10Cr4 10 circles pseudorandom spin valves plotted on a loga- corresponding localization lengths are very different. More- rithmic scale against the number of bilayers N. over, since a strong enough applied magnetic field can effect transition from the AF to the FM configuration, the degree of its were imposed so that the interlayer exchange coupling disorder can be controlled by the applied field. In fact, we remains antiferromagnetic. For the same reason, the thick- have shown in Sec. III that matching of the Co up-spin band nesses of Cr layers were made to fluctuate between 4 and 10 to the Cu bands is almost perfect. It follows that, regardless atomic planes. There is no real restriction on the range of of the size of fluctuations in layer thickness, up-spin elec- fluctuations in thickness of the ferromagnetic layers. How- trons in the FM configuration are only weakly scattered. One ever, for computational reasons convergence of the BZ can, therefore, expect that localization either does not occur sum , the total thickness of the valve cannot exceed 150 in this channel at all or is extremely weak all the localiza- nm. I have, therefore, restricted rather arbitrarily the mean tion lengths for up-spin electrons are long . On the other thickness of Co layers to 4 atomic planes (Mmin 2, hand, electrons in the down-spin channel in the FM configu- Mmax 6 and the mean thickness of Fe layers to 6 atomic ration and electrons of either spin orientation in the AF con- planes (Mmin 2, Mmax 10 . figuration experience highly disordered potentials and should As expected, the CPP GMR increases approximately ex- undergo strong localization localization lengths in all these ponentially with the number of bilayers N both for the Co/Cu channels should be short . and Fe/Cr pseudorandom valves. The maximum RCPP Exactly the same arguments apply to Fe/Cr pseudoran- achieved for the Fe/Cr valve with N 50 is approximately dom spin valves with the only modification that the roles of 3 104 %, which is about two hundred times greater than the the up- and down-spin channels in the FM configuration are maximum RCPP observed in the Ohmic regime.4 This is, of interchanged. In either case, the channel with a weak local- course, not the upper theoretical limit but merely a limit ization in the FM configuration shorts the channel with a imposed by the computer time available. The theoretical CPP strong localization and, therefore, RFM should increase only GMR increases with increasing number of bilayers without slowly with the number of bilayers N. On the other hand, any saturation as long as the valve remains in the localization RAF should increase exponentially with a large exponent and, regime. therefore, the CPP GMR ratio defined by Eq. 5 should also To understand the precise reason for such a large en- grow exponentially with N. hancement of the CPP GMR, one needs to examine the in- These general arguments were already presented in Ref. 1 dividual resistances in the FM and AF configurations. The and illustrated by model calculations for a single-orbital resistances R , FM , RFM , and RAF for the Co/Cu valve are tight-binding band. However, the pertinent question is plotted in Fig. 8 on a logarithmic scale against N. The cor- whether the localization lengths in the AF channel are short responding results for the Fe/Cr valve are shown in Fig. 9. enough in real systems so that localization can influence the Consider first the Co/Cu valve. The resistance in the AF CPP GMR. This question can only be answered by an exact configuration circles in Fig. 8 and the resistance in the evaluation of the Kubo formula for specific Co/Cu and Fe/Cr down-spin channel in the FM configuration triangles in- pseudorandom spin valves. The formalism developed in crease approximately exponentially with N due to Anderson Secs. II and III remains valid in the Anderson localization localization. Their values for N 50 are, therefore, a factor regime and can be readily applied to Co/Cu and Fe/Cr pseu- of twenty larger than for a superlattice in the Ohmic regime dorandom spin valves. I have made such calculations for Figs. 5 and 6 . On the other hand, the resistance in the Co2 6Cu5 8 and Fe2 10Cr4 10 valves and their CPP GMR up-spin channel in the FM configuration squares remains as ratios are plotted on a logarithmic scale against the number low as in the Ohmic regime. The reason for this behavior is of bilayers N in Fig. 7 squares for Co/Cu and circles for that localization does not set in for such small N in this Fe/Cr . The thicknesses of Cu layers in the Co/Cu valve were channel because disorder is almost negligibly weak due to made to fluctuate between 5 and 8 atomic planes. These lim- excellent matching of the up-spin bands in Co to the Cu 55 Ab initio CALCULATION OF THE PERPENDICULAR . . . 967 an ab initio band structure of the constituent metals thus confirms the very large enhancement of the CPP GMR pre- dicted in Ref. 1. The enhancement is due to multiple scatter- ing of electrons in the AF configuration from a highly disor- dered sequence of ferromagnet/spacer interfaces. Such scattering gives rise to electron localization with short local- ization lengths of the order of twenty bilayers 30­40 nm . As in Sec. III, a comment on the nature of randomness in pseudorandom spin valves is required. The situation for a pseudorandom spin valve is qualitatively different from that for a superlattice with small spontaneous fluctuations in layer thickness. Once a pseudorandom spin valve is prepared with layer thicknesses following a predetermined pseudorandom number sequence, the position of each interface in it is known precisely. Any particular multilayer for which a cal- culation of the GMR is made can, therefore, be reproduced experimentally by growing the layers with the same known pseudorandom sequence, and vice versa. In other words, the FIG. 8. Resistances in the ferromagnetic FM and antiferro- relevant quantity to be calculated is the sample specific GMR magnetic AF configurations of a Co2 6Cu5 8 pseudorandom spin and, therefore, any configuration averaging over disorder of valve plotted on a logarithmic scale against the number of bilayers , the interfaces would be completely inappropriate. N. Circles denote the resistance RAF ; squares the resistance RFM and triangles the resistance R FM . bands. It follows that the very large enhancement of the CPP V. CONCLUSIONS GMR for the Co/Cu pseudorandom spin valve is due, en- tirely, to the Anderson localization of electrons in the AF The conventional explanation of the GMR effect is based configuration. on spin-dependent scattering of electrons from magnetic im- The magnetic contrast of the Fe/Cr pseudorandom valve purities located at ferromagnet/spacer interfaces interfacial is even more enhanced by the Anderson localization. The spin-dependent scattering . The resulting transport problem resistances for N 50 in the AF configuration and in the is solved either within the classical Boltzmann formalism21 up-spin channel in the FM configuration are two orders of or within a linear response theory with a simplified band magnitude greater than the corresponding resistances in the structure parabolic bands .22 Realistic modelling of interfa- Ohmic regime Fig. 6 , whereas the resistance in the down- cial roughness combined with a rigorous quantum evaluation spin channel in the FM configuration is virtually unaffected of the CPP GMR from the Kubo formula was made by by disorder. The reason is, of course, a very good matching Asano et al.11 but only for a single-orbital tight-binding band of Fe and Cr bands in the down-spin channel. structure. More recently, Butler et al.,23 Zahn et al.24, and An exact numerical evaluation of the Kubo formula for Nesbet25 solved the Boltzmann equation with a fully realistic Co/Cu 001 and Fe/Cr 001 pseudorandom spin valves using band structure. However, a common feature of all these theo- ries, with the exception of Ref. 11, is that the GMR effect disappears when the spin-dependent impurity scattering is switched off. Since interfacial impurity scattering is linked directly to interfacial roughness, the implication of all the above theories is that the GMR effect vanishes or is negli- gibly small26 for perfectly flat interfaces. This conventional point of view was challenged by Schep et al.3 They considered the simplest case of GMR without impurity scattering, i.e., an infinite perfectly periodic super- lattice. Using an ab initio band structure, they obtained CPP GMR ratios in excess of 100 %. These very high values of CPP GMR are due entirely to quantum scattering from per- fectly flat interfaces. In this paper, I have included an additional important in- gredient, i.e., fluctuations in layer thickness, and investigated comprehensively the CPP GMR due to scattering from oth- erwise perfectly flat interfaces GMR without impurity scat- tering for Co/Cu 001 and Fe/Cr 001 finite superlattices sandwiched between two semi-infinite contacts. Using an ab FIG. 9. Resistances in the ferromagnetic FM and antiferro- initio band structure for all the constituent metals and solving magnetic AF configurations of a Fe the quantum transport problem exactly numerical evaluation 2 10Cr 2 10 pseudorandom spin valve plotted on a logarithmic scale against the number of of the Kubo formula , I find that CPP GMR without impurity bilayers N. Circles denote the resistance R , AF ; squares the resis- scattering is far from negligible and can easily explain the tance R FM ; and triangles the resistance RFM . whole observed effect. Moreover, depending on the size of 968 J. MATHON 55 fluctuations in layer thickness, CPP GMR without impurity the fact that it is required to account quantitatively for the scattering occurs in three distinct regimes: ballistic, Ohmic, observed CPP GMR effect must be significant. and Anderson localization pseudorandom spin valve . The only discrepancy between the calculated and ob- When there are no fluctuations in layer thickness, CPP served results in the Ohmic regime is that the calculated transport is in the ballistic regime. The ballistic CPP GMR saturation-field FM resistance is a factor of three smaller ratio R than the observed value. One possible explanation is that CPP of a finite superlattice saturates rapidly as a func- tion of the number of bilayers only 5 bilayers are needed there is some additional scattering mechanism, other than and reaches a value equal to R spin-dependent scattering from interfaces, which is not in- CPP for an infinite superlattice. The saturation values of R cluded in the present calculation. However, a more likely CPP obtained from the Kubo for- mula are of the order of 100 % both for Co/Cu and Fe/Cr explanation is that the matching between the Co majority and superlattices, which is in a very good agreement with the Cu bands Fe minority and Cr bands , which determines the results obtained earlier by Schep et al.3,6 for infinite Co/Cu saturation-field resistance, may not be so perfect when lattice and Fe/Cr superlattices. However, the fact that the absolute relaxation effects are included. In fact, the saturation-field value of the resistance of a superlattice in the ballistic regime resistance calculated with bulk Co and Cu parameters is al- most certainly underestimated. The matching in the up-spin is far too low compared with the experiment4 and also that channel for bulk Co and Cu bands is so perfect that any RCPP saturates so rapidly clearly indicates that ballistic ef- modification of the band structure due to lattice relaxation is fects are not seen in present experiments. bound to make the matching poorer, and hence, increase the When small fluctuations in layer thickness corresponding saturation-field resistance. This would, at the same time, to only one atomic plane at the interface being displaced are bring down somewhat the calculated CPP GMR and, hence, introduced, transport changes from ballistic to Ohmic. The make the agreement between the theory and experiment bet- calculated GMR ratio RCPP increases initially linearly with ter. the number of bilayers N and then saturates for N 40­50. When large fluctuations in layer thickness are introduced Such a behavior is a signature of the Ohmic regime and was deliberately, the Ohmic regime changes into, experimentally observed for Co/Ag and Co/Cu by Schroeder et al.4 The as yet unexplored, Anderson localization regime in which the maximum calculated saturation values of RCPP are in the re- proposed pseudorandom spin valves1 operate. If high-quality gion 800­1000 %. However, much more significantly, the Co/Cu and Fe/Cr pseudorandom spin valves could be fabri- absolute values of the zero-field AF and saturation-field cated, the results of Sec. IV show that they would have CPP FM resistances calculated without any adjustable param- GMR ratios at least a factor hundred higher than the highest eters increase linearly with N good Ohm's law , which is as currently attainable values. observed,4 and the zero-field AF resistance of the Co/Cu Successful operation of Co/Cu and Fe/Cr pseudorandom superlattice of the same thickness and composition as the spin valves depends on two conditions: i the saturation- Co/Cu 111 sample investigated in Ref. 4 has a value of field FM resistance of a pseudorandom spin valve must not 165 10 15 m2, which is within 10 % of the observed be much higher than the calculated resistance; ii the zero- resistance. field AF resistance must be sufficiently enhanced by These results indicate very strongly that it is scattering Anderson localization. from fluctuations in layer thickness rather than the conven- The first condition is easy to satisfy. The FM resistance tional interfacial scattering that determines the CPP GMR RFM of present Co/Cu superlattices is only a factor 3 higher observed in present experiments. Further support for this than the calculated RFM . The results of Secs. III and IV mechanism comes from an analysis of x-ray scattering data27 show that disordering the sequence of interfaces has virtually which shows that fluctuations in layer thickness of the order no effect on RFM . There is, therefore, no reason to expect of one atomic plane are always present in experimental that RFM of a carefully prepared Co/Cu pseudorandom spin samples. One can go even further and argue that, without valve should be any higher than RFM in the present Ohmic fluctuations in layer thickness, the observed CPP GMR can- regime. not be explained at all. This is because fluctuations in layer For a pseudorandom spin valve with all dimensions thickness increase the CPP GMR ratio whereas interfacial smaller than the mean free path, the calculations of Sec. IV roughness decreases it. The latter result was proved quite are exact for real Co/Cu and Fe/Cr systems and the second rigorously by Asano et al.11 for a single-orbital tight-binding condition is, therefore, also satisfied. It follows from Fig. 7 band. They showed that CPP GMR ratio has its maximum that CPP GMR of about 5000 % should be achieved for a value for perfectly flat interfaces and is always reduced from valve of about 50 nm thick. The thickness of the valve is, the maximum value when ferromagnet and spacer atoms are therefore, not a serious problem since 50 nm is comparable intermixed at the interface. Given that interfacial roughness to the mean free path. One should clearly try to keep all the is detrimental for CPP GMR and ab initio calculations for layers as thin as possible while maintaining large fluctuations periodic Co/Cu superlattices with perfectly flat interfaces in their thickness in order to squeeze as many bilayers as give an upper limit on CPP GMR of about 120­150 %, possible into a total thickness comparable to the mean free which is smaller than the maximum observed4 effect of about path. The only question that remains unresolved is whether 170 %, it is clear that, without fluctuations in layer thickness, the transverse dimensions of the valve must also be small, the magnitude of the observed CPP GMR cannot be ex- i.e., comparable to the mean free path. One can argue that plained. One may, therefore, conclude that fluctuations in impurity scattering in a macroscopic valve would lead to layer thickness is an important source of CPP GMR that has mixing of k channels which might eventually destroy one- not been considered in previous theoretical treatments and dimensional localization. However, what the upper bound is 55 Ab initio CALCULATION OF THE PERPENDICULAR . . . 969 on the valve diameter, if any, is difficult to estimate theoreti- valves should certainly be observable in Co/Cu and Fe/Cr cally. valves with all dimensions smaller than the mean free path at Finally, all the calculations presented here are at zero tem- low temperatures. It is, however, quite likely that a substan- perature. Inelastic scattering at finite temperatures might also tial enhancement persists in valves with more macroscopic spoil the localization in the AF configuration and thus reduce transverse dimensions and also at finite temperatures. the calculated GMR ratios. 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