arXiv:cond-mat/9903118v2 [cond-mat.mtrl-sci] 13 Jul 1999 npr eas fawd ag fnvlphenom- novel of magnetoresistance, range colossal comes wide and interest giant experiments, a This spin-injection the of e.g., because decade. ena, part last in the over transport n experiments. ing et nF,C,N,adN losi h 90s They 1970’s. the in alloys results. Ni surprising and two Ni, found Co, experi- Fe, on ferromagnet-insulator-superconductor measure ments principle of states. in of series can density the one in film, polarization the field spin a the of by directly split plane Zeeman the is superconductor in the in density states the of where experiments the tunneling of superconductor product polarizations. the measures spin one experiments ferromagnet-insulator-ferromagne of tunneling polar- case this the of of In measure particu- density direct ization. a the In provide should measure in- they experiments states, Fermi or calculations. tunneling the directly since transport at either lar, most is states enters into ferromagnet of polarization directly a density This in the energy. in transport polarization polarized the spin of erties eaiesi oaiainfrC n i vnthough even Slater Ni, curve the explained Pauling and successfully Co have calculations structure for these and band polarization Fe level, for spin Fermi polarization negative spin the Assuming positive to- at predict the states calculations to positive. of proportional is density be Ni tal to and conductance tunneling Co, the Fe, for polarization ti o la h h lcrnsi oaiainmea- and surface, polarization Fermi spin the of electron property the a why tunneling, by clear metals, sured not transition Given the the is of near process. it structure electrons tunneling band the complicated only the in that participate expected the level is is On Fermi alloys it Ni moments. hand, the magnetic and other the Ni, to Co, proportional Fe, roughly for polarization spin the orlto ewe pnPlrzto n antcMoment Magnetic and Polarization Spin between Correlation hr a enrnwditrs nspin-polarized in interest renewed been has There erw eevy n olbrtr are u a out carried collaborators and Meservey, Tedrow, nre eeto xeiet,sm fwihotie diffe th obtained discuss which also of We some experiments, experiments. reflection tunneling Andreev the by observed as ASnmes 57.i 51.p 21.v 85.30.Mn 72.15.-v, 75.10.Lp, 75.70.-i, numbers: PACS calculations. our and ments aiaino h e est fsae sntcreae wit correlated of not states is of states density of the density of net the of larization otermgei oet,wihcnntb xlie ycons by (C explained approximation be potential not coherent can tight-binding which all a Ni moments, Using various magnetic and Ni, their Co, to Fe, for polarization spin measured tun ferromagnet-insulator-superconductor in larization h orlto ewe h antcmmn nferromagnet in moment magnetic the between correlation The 11 – 4 14 .INTRODUCTION I. – 9 5 o h antcmmns Second, moments. magnetic the for n ftems udmna prop- fundamental most the of One oee,i ferromagnet-insulator- in However, eateto hsc n ainlHg antcFedLabo Field Magnetic High National and Physics of Department 3 n pnplrzdtunnel- spin-polarized and 10 is,teeeto spin electron the First, a-agCo,Ja hn n emnHershfield Selman and Chen, Jian Choy, Tat-Sang s nvriyo lrd,Gievle L32611 FL Gainesville, Florida, of University lcrn screae ihtemgei oeti h same the in moment magnetic the with correlated is electrons 1 , 2 - t 1 r eae nsc ipeway. simple a sea, such whole Fermi in the related of are property a moment, magnetic the ue yMoodera by sured mea- al. as et For 43-46.5% is range. Ni Soulen by of same sured polarization the spin in the tunneling are example, the they in obtained although experi- those experiments, these from in different measured are polarization ments The spin the experiments. of contact ferromagnets values point of reflection polarization Andreev with spin the studied pendently h pnplrzto fNi of polarization spin the rcino h lcrn.I h oko Stearns of work the In electrons. the only by of dominated is fraction concluded states calculations of a density these tunneling tunnel- of the Meservey’s that Many and Tedrow experiments. of ing results the explain to Meservey. and made Tedrow measurements re- by tunneling Their the from moment. different are magnetic sults the of independent roughly h eeateetoi ttsaea are states electronic relevant the ieetdpnece ntedniyo ttsadthe and the states experiments. of account of sets two into density for take velocity the Fermi to on differ- have dependencies but may different related One measuring quantities. experiments ent the to due be auso h xeiet.Mr eety yuigAn- using particular by Nadgorny recently, of techniques, More instead reflection trend dreev experiments. the important the compare is of to it values theory that indicates a experiments for the in ancies yatgtbnigmdlwihhsonly has which model tight-binding a by oee yaprblcdseso erteFrisur- Fermi the the near Aoi and dispersion Hertz parabolic face. a by modeled h xeiet.Rcnl,Nguyen-Manh Recently, experiments. the concluded only They that insulator. the and ferromagnet the tween Tsym- waves. spin with Pettifor interaction and bal electron the to due eety Soulen Recently, hr aebe ubro hoeia calculations theoretical of number a been have There eigeprmnshsbe ytr.The mystery. a been has experiments neling s hl h unln pnplrzto fN mea- Ni of polarization spin tunneling the while , est fsae smdfidb aybd effects body many by modified as states of density h antcmmn,tepolarization the moment, magnetic the h A oe,w hwta hl h po- the while that show we model, PA) y spstv n ogl proportional roughly and positive is oys etrslsfo h unln experi- tunneling the from results rent pnplrzto esrmnsby measurements polarization spin e s caly n h unln pnpo- spin tunneling the and alloys ic lcrntneigwssffiin oexplain to sufficient was tunneling electron drn h e est fstates. of density net the idering 21 tal. et tde h unln rmF n Co and Fe from tunneling the studied 20 tal. et tal. et n 2 smaue yUpadhyay by measured as 32% and ocue httneigmeasures tunneling that concluded ratory, nFroantcAlloys Ferromagnetic in 15 x Fe s33%. is n Upadhyay and 1 − x lost eaot45%, about be to alloys 8 t h ieec may difference The 2 manner g lk adta is that band -like tal. et 17 ssσ tal. et h discrep- The tal. et 18 od be- bonds measured 16 22 inde- per- 19 , formed a self-consistent band structure calculation of the 0.5 Co/Al2O3 interface by a LMTO technique. Their calcu- NiCu (a) lation suggested that the interfacial cobalt d bands spin 0.4 NiCr polarize s-p bands in the barrier, resulting in a positive NiFe 17 NiMn tunneling spin polarization. On the other hand, Mazin 0.3 and Nadgorny et al.18 suggested that spin polarization NiTi measured by both the tunneling and Andreev reflection experiments should be found from the polarization of the 0.2 density of states times the Fermi velocity squared. The values of the polarization obtained from the all of the 0.1

above models are in the same range of the experimental tunneling spin polarization values; however, all these models are different and it is 0.0 not clear which model is closer to reality. ) Β To understand if there is indeed a subset of the elec- µ 3 (b) tronic states which can account for the tunneling density of states, in this article we present a microscopic cal- culation for both the magnetic moment and the various 2 density of states based on a self-consistent tight-binding coherent potential approximation (CPA)23 model. In a range of alloys we find that the s density of states fol- 1 lows the same trend as the measured tunneling density of states. This is the first microscopic calculation to see this correlation. Furthermore, we are able to understand 0 the correlation seen in our calculation and to show that it 6 7 8 9 10 11 is not universal, i.e., the tunneling density of states is not magnetic moment per atom ( Cr Mn Fe Co Ni Cu simply proportional to the magnetic moment. There may even be some alloys where they are inversely correlated. number of valence electrons per atom FIG. 1. Experimental data of (a) tunneling spin polariza- tion and (b) bulk magnetic moment of 3d alloys versus the av- II. REVIEW OF EXPERIMENTS erage number of valence electrons per atom. The similarities between (a) and (b) suggest the tunneling spin polarization In this section, we review the series of tunneling exper- and the magnetic moment are related. iments by Tedrow and Meservey which show a correlation between the spin polarization and the magnetic moment. We also discuss the spin polarization measurements by NiCr, NiCu, NiMn, and NiTi in Fig. 2. The spin po- Andreev reflection. larization data are taken from the experiment of Tedrow In the tunneling experiments of Tedrow and Meservey, and Meservey, and the magnetic moment data are taken spin polarized electrons tunnel from films made by al- from the bulk measurements of alloys with the same com- loying Ni and other 3d transition metals. In Fig. 1(a), positions. Figure 2 shows a clear correlation between the the tunneling spin polarization for the Ni alloys is plotted spin polarization and the magnetic moment in these al- against the average number of valence electrons per atom, loys. The data points for different alloys are roughly in which is changed by changing the composition of the al- a straight line passing through the origin. loys. Figure 1(a) looks very similar to the Slater-Pauling Finite spin polarization is obtained for a few samples curve shown in Fig.1(b), in which the bulk magnetic mo- with zero bulk magnetic moments. This is probably due ment of alloys are plotted against the average number to the difference between the estimated bulk magnetic number of valence electrons per atom. In NiFe and NiMn, moments, which are taken from experiments on alloys the spin polarization peak is close to the magnetic mo- with the same composition, and the surface magnetic mo- ment peak. In NiCu, NiCr, and NiTi, both the spin po- ments of the actual sample, which are not measured in larization and the magnetic moment decrease monotoni- the tunneling experiments. When the magnetic moment cally as impurity concentrations increase. The thresholds is zero, there should be no difference between the major- at which the spin polarization and the magnetic moment ity and minority spins, the spin polarization is expected drops to zero are close to each other. These similari- to be zero. ties suggest that the spin polarization and the magnetic In Fig. 3, we plotted the tunneling data shown in Fig. moment are correlated. 2 together with the tunneling measurements on other To see how the spin polarization relates to the mag- samples fabricated from different techniques by Tedrow netic moment, we follow Tedrow and Meservey10 and plot and Meservey and by Moodera. Also included are the the spin polarization against the magnetic moment for spin polarization measurements using Andreev reflection of Soulen et al., Nadgorny et al., and Upadhyay et al.

2 0.15 0.6 NiCr NiCu NiCr NiMn NiCu NiTi NiMn NiFe 0.1 NiTi 0.4 Ni,Co,Fe Moodera Soulen Upadhyay Nadgorny

0.05 0.2 tunneling spin polarization tunneling spin polarization

0 0 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 magnetic moment per atom magnetic moment per atom 10 FIG. 2. Experimental tunneling spin polarization of FIG. 3. Spin polarization obtained by different experi- NiCr, NiCu, NiMn, and NiTi versus the bulk magnetic mo- ments versus the bulk magnetic moment of the correspond- ment of the corresponding alloy. The data points for different ing alloy. The open symbols are tunneling data and the rest alloys roughly line in the same straight line. The commonly are obtained from Andreev reflection experiments. The two accepted experimental bulk magnetic moments of alloys are dashed lines outline an area which roughly divides the graph used as the x-coordinates. Finite spin polarization is obtained into a fcc region (left) and a bcc region (right). for zero magnetic moments in some samples. This is proba- bly due to the difference between the bulk magnetic moments used on the graph and the surface magnetic moments of the III. MODEL actual samples, which are not measured. When the magnetic moment is zero, there should be no difference between the ma- To see if there is actually a relationship between the jority and minority spins, the spin polarization is expected to magnetic moment and the spin polarization of a subset be zero. of the electronic states, we calculate both the magnetic moment and the spin polarization using a tight-binding The two dashed lines outline an area which roughly di- model. The band structure of an alloy is calculated vides the graph into a fcc region (left) and a bcc region self-consistently within the coherent potential approxi- (right). When a data point is close to this area, the mation. The magnetic moments and the spin polariza- structure depends on the particular sample. One can vi- tion of the density of states for different bands at the sually identify three regimes in the figure: the lower left Fermi level can then be obtained from the band struc- regime (fcc NiCr, NiCu, NiMn and NiTi), the middle ture. It is found that when the p and d bands are in- regime (fcc NiFe with magnetic moment less than about cluded, the spin polarization of the density of states is 1.8 µB), and the right regime (bcc NiFe with magnetic obviously inconsistent with the experiments. For exam- moment greater than about 2 µB). There is a clear cor- ple, negative spin polarization is obtained for Ni when relation between the spin polarization and the magnetic the d bands are included. Therefore, we present only the moment in the lower left regime, which is the same as results of the spin polarization of the s electrons. Fig. 2. In the middle regime, the tunneling data and the The alloys we consider are substitution alloys, in which Andreev reflection data by Upadhyay et al. only show a some of the host atoms are replaced by impurity atoms roughly increasing relation; on the other hand, the An- without changing the crystal structure. The band struc- et al. et ture of these alloys can be found by the coherent potential dreev reflection data by Soulen and Nadgorny 23 al. suggest that the spin polarization is independent of approximation. the magnetic moment. In right regime of Fig. 3, no clear Before the band structure calculation of a magnetic 3d correlation is seen between the spin polarization and the alloy can be carried out, one has to know the splitting be- magnetic moment. However, this regime corresponds to tween the majority and minority spins, which is related the peak area near Fe in Fig. 1, which clearly indicates a to the magnetic moment. The splitting can be found correlation between the tunneling spin polarization and within our model from the number of majority and mi- the magnetic moment. nority electrons, which has to be found from the band structure. Therefore, the band structure, the number of majority and minority electrons, and the splitting have to be solved self-consistently. To calculate the alloy band structure, we consider a nine-band tight-binding Hamiltonian written as

3 σ † † H = uimcimσcimσ + tijmncimσcjnσ, (1) error is expected to increase. Although this simplified imσX hijXimnσ model requires only two more parameters in addition to the parameters of the host, it still produces correctly the † 25 where cimσ(cimσ) is the creation (annihilation) operator Slater-Pauling curve for the magnetic moment . of a spin-σ electron of orbital m on the lattice site i, The three quantities which we will compare are the σ uim is the on-site potential, tijmn is the hoping energy magnetic moment per atom, the polarization of the den- between neighbors. The band structure of the alloy de- sity of states as measured by tunneling experiments, and scribed by this Hamiltonian will be found using the coher- the polarization of the calculated s density of states. The ent potential approximation. Tight-binding parameters magnetic moment per atom, µ, is determined from the in the Slater-Koster two-center form are obtained from number of electrons per atom in the majority spin orien- fits to the local density approximation band structure.24 tation, N ↑, and the minority spin orientation, N ↓, via In principle all of the parameters in the alloy are changed ↑ ↓ as one varies the alloy composition. However, the 3d al- µ = µB (N − N ), (3) loys are similar and the differences in hoping and s and p on-site energies are not as significant as the differences where µB is the Bohr magneton. The s density of states in the d on-site energies. Therefore, we assume that the of spin σ at the Fermi level at lattice site i is defined as most important changes due to alloying are contained in 3 9 ↑ ↓ d k 2 the on-site energies, ui,d and ui,d, of the spin-up (ma- nσ = δ(E − E(k)) |hisσ|kmσi| , (4) i,s Z (2π)3 F jority) and spin-down (minority) d electrons. In other mX=1 words, the alloy is assumed to have site-independent hop- ing parameters and s and p on-site energies the same where |isσi is the s orbital with spin σ at site i and |kmσi as the host metal, and site-dependent d on-site energies. is the m-th eigenvector with a wavevector k and spin σ. This assumption will be justified later by the agreement The net s density of states is the weighted average of the with local density calculations of supercells in different s density of states at each lattice site: impurity concentrations. ↑ nσ = (1/N ) nσ , (5) The major contributions to the on-site energies, ui,d s at i,s ↓ Xi and ui,d come from the atomic core potential plus the Coulomb energy due to the opposite spin. In the itinerant where Nat is the number of atoms in the sample. The electron model, it is more convenient to work with the density of states for spin σ measured experimentally is ↑ σ number of spin-up and spin-down d electrons, Ni,d and denoted by n . Using these definitions, the polarization ↓ of the tunneling density of states, P , and the calculated Ni,d, at each site i. Therefore, we rewrite the parameters in the form of s density of states, Ps, are given by

↑ 0 x ↓ ↑ ↓ ui,d = Ui + Ui Ni,d n − n P = ↑ ↓ , (6) u↓ = U 0 + U xN ↑ , (2) n + n i,d i i i,d ↑ ↓ ns − ns 0 x Ps = ↑ ↓ . (7) where Ui and Ui are the on-site parameter and effec- ns + ns ↑ tive Coulomb energy per pair at site i. Both Ni,d and ↓ Ni,d in Eq. (2) are obtained from the band structure, ↑ ↓ IV. RESULTS which in turn depends on ui,d and ui,d. Thus, Eq. (2) has to be solved self-consistently with the band structure calculated from the Hamiltonian of Eq. (1). As mentioned, when the density of states of the p and All parameters for the host atoms can be found from d 24 bands are included, the spin polarization is inconsistent fits to the local density calculations of the pure metal. with experiments. Therefore, to compare with the exper- 0 x The only two parameters left, Uimp and Uimp of impurity iments, we plot the spin polarization of the s density of atoms, are obtained from fits to the local density calcula- states, Ps, against the calculated magnetic moment in the tions of supercells composed the two types of atoms. For same graph as the experimental data. In Fig. 4, we plot example, parameters for Fe as impurities in fcc Ni host the calculated spin polarization of NiCr and NiCu (filled are obtained from fits to fcc Ni3Fe band structures. As a symbols), together with the experimental tunneling spin check, parameters obtained from the fit is used to calcu- polarization, P , shown in Fig. 2 (unfilled symbols). We late the tight-binding band structure of fcc Ni31Fe, which choose to calculate NiCr and NiCu because they repre- agrees with the local density approximation band struc- sent different dependencies of the magnetic moment on ture. This indicates that the model works well at least the average number of electrons. As seen in Fig. 1(b), the when impurity concentration is less than 25%. When the magnetic moment of NiCr increases as the average num- impurity concentration is large, or when the difference in ber of electrons increases, while the the magnetic moment hoping between the host and the impurity is large, the

4 of NiCu decreases as the average number of electrons in- al. and Nadgorny et al. In the right regime (bcc NiFe), creases. For both alloys, the calculated spin polarization the calculated results are significantly higher than the varies from zero (non-magnetic) to about 27% (pure Ni), measured values. and has the same dependency on the magnetic moment. 1 When the magnetic moment is zero, the majority and mi- NiCr nority spin bands are the same, and the spin polarization NiCu are also zero. NiMn 0.8 NiTi 0.3 NiFe Ni,Co,Fe Moodera NiCr 0.6 Soulen NiCu Upadhyay NiMn Nadgorny 0.2 NiTi 0.4

tunneling spin polarization 0.2

0.1 0 0 0.5 1 1.5 2 2.5 tunneling spin polarization magnetic moment per atom FIG. 5. Calculated spin polarization of the s density of 0 states of NiCr (filled square), NiCu (filled diamond), and NiFe 0 0.2 0.4 0.6 0.8 1 (filled downward triangle) alloys versus the magnetic moment magnetic moment per atom plotted with the experimental data (open symbols) shown in Fig 3. FIG. 4. Calculated spin polarization of the s density of states of NiCr (filled square) and NiCu (filled diamond) al- loys versus the magnetic moment plotted with the experimen- Thus far we have presented calculations of the bulk tal data (open symbols) shown in fig 2. The spin polarization density of states. However, experiments have suggested measured for these series of films are lower than the calculated that the spin-polarized tunneling electrons originate from 4 10 values. One reason is that a growth technique that seems to the first two to three layers at the surface. , To estimate reduce the spin polarization was used in order to mix the how the surface affects the spin polarization, we study the elements. For example, the tunneling spin polarization mea- variation of the s density of states as a function of depth sured for the Ni films in the series are only about 8%, which from the surface. The band structure of bcc (100) Fe, hcp is much lower than the values of 27 − 33% measured for the (001) Co, and fcc (111) Ni slabs of 18 atomic layers thick Ni films grown by better techniques. are calculated by using fixed tight-binding parameters. This calculation only serves as a crude estimate of the Both calculated and experimental results show that the surface effects. Self-consistent iterations are not used in spin polarization is positive and roughly proportional to the band structure calculation. Effects such as the change the magnetic moment. However, the experimental values in surface magnetic moment and surface roughness are are much lower than the calculated ones. One reason may also neglected. We study the spin polarization of the be that in order to mix the elements, the samples in this cumulative s density of states as a function of the number series of experiments were grown with technique which of layers. The cumulative s density of states of spin-σ 10 σ seems to reduce the spin polarization. For example, the in the first l layers from the surface is defined as nl,s = tunneling spin polarization measured for the Ni films in l nσ , where nσ is the s density of states in the j-th this series is only about 8%, which is much lower than j=1 j,s j,s layer.P The spin polarization of the cumulative s density the measured values of 27 − 33% for the Ni films grown of states for the first l layers from the surface is given by by better techniques.10,8 To compare with more experiments, we plot in Fig. n↑ − n↓ 5 the calculated spin polarization of the s electrons for l,s l,s P l,s = ↑ ↓ . (8) NiCr, NiCu, and fcc and bcc NiFe (filled symbols) to- nl,s + nl,s gether with the experimental spin polarization shown in Fig. 3 (unfilled symbols). As explained above, the calcu- This quantity is related to the tunneling spin polarization lation for the lower left regime agrees with the tunneling because it takes into account the electrons contributed experiment. The calculation for the middle regime (fcc from the first few layers. However, this is only an ap- NiFe) agrees well with the tunneling experiments and the proximation to the tunneling spin polarization because Andreev reflection experiments by Upadhyay et al., but the contribution from different layers may not be uni- not with the Andreev reflection experiments of Soulen et formly weighted.

5 1 σ σ for both spins. Since ns increases as N increases, it fol- ↑ ↓ lows that the spin polarization, Ps ∝ (ns − ns), increases ↑ ↓ −DOS Fe with increasing magnetic moment, µ = µB(N − N ). s 0.8 Co Ni 0.04 0.6 pure Ni Ni in NiFe Fe in NiFe 0.03 0.4 Ni in NiCu Cu in NiCu Ni in NiCr 0.2 0.02 Cr in NiCr

polarization of cummulative 0 1 3 5 7 9 0.01 number of layers counted from surface FIG. 6. Calculated spin polarization of the cumulative s density of states, Pl,s, as a function of the number layers −DOS at Fermi energy (states/eV) 0.00 counted from the surface. We studied the spin polarization of s 3.0 4.0 5.0 the cumulative s density of states because tunneling may be σ sensitive to the first few layers. As shown in the graph, the N i spin polarization of the s density of states at the bcc Fe surface σ FIG. 7. The s density of states ns,i at the Fermi energy is significantly lower than that in the bulk. This reduction for spin-up (filled symbols) and spin-down (unfilled symbols) is less important in the hcp and fcc metals. Therefore, our channels versus the number of spin-up and spin-down valence estimation of the tunneling spin polarization using the bulk ↑ ↓ electrons, Ni and Ni , at site i (i = Ni, Fe, Cu, . . . ) in fcc spin polarization is good for the fcc metals, while it is too alloys. The solid (dashed) curve is obtained by varying the large in the case of the bcc metals, as shown in fig 5. Fermi energy in the spin-up (spin-down) density of states of σ σ pure Ni. In the range of interest, (Ni > 3) ns,i is an increas- σ ↑ ↓ ing function of Ni . It follows that Ps ∝ (ns − ns ) increases As shown in Fig. 6, there is a reduction of the spin ↑ ↓ as µ = µB (N − N ) increases, explaining the correlation in polarization near the surface. The reduction is more im- Fig. 1-3. portant in bcc metals than in fcc metals. For example, the spin polarization of bcc Fe surface is less than one It is important to note that Ps is only roughly propor- third of the bulk value, while the spin polarization of tional to µ because the curve in Fig. 7 is quite different the fcc Ni surface is about two thirds of the bulk value. σ from a straight line even in the range of 3.0

6 0.04 2.0 0.04 (a) pure Fe 1.5 0.03 Fe in FeCr 0.03 Cr in FeCr 1.0 d−band 0.02 Fe in FeNi Ni in FeNi 0.5 s−band 0.01 −DOS (states/eV) −DOS (states/eV) s 0.02 d 2.0 0.04 (b) 1.5 d−band 0.03 0.01 1.0 0.02

0.5 s−band 0.01 −DOS at Fermi energy (states/eV) −DOS (states/eV) 0.00 −DOS (states/eV) s s 3.0 4.0 5.0 d σ 0.0 0.00 N i −1.6 −1.2 −0.8 −0.4 0 0.4 0.8 σ FIG. 8. The s density of states ns,i at the Fermi energy Shifted Energy (eV) for spin-up (filled symbols) and spin-down (unfilled symbols) FIG. 9. The (a) spin-up and (b) spin-down d and s density channels versus the number of spin-up and spin-down valence of states at the Ni sites in pure Ni (solid lines), Ni0.8Fe0.2 N ↑ N ↓ i i electrons, i and i , at site ( = Ni, Fe, or Cr) in bcc (dashed lines) and Ni0.6Fe0.4 (dot-dashed lines) versus the alloys. The solid (dashed) curve is obtained by varying the shifted energies. The energy of each spin channel for each Fermi energy in the spin-up (spin-down) density of states of alloy is shifted such that the number of states at the Ni sites σ σ pure Fe. In this case, ns,i is still an increasing function of Ni below 0eV is the same as pure Ni in the corresponding spin in most region; however, the function is more complicated channel. The d peaks roughly coincide and the s density of than that of the fcc alloys. states collapse onto a single curve. Therefore the primary effect of alloying is to shift the bands together relative to the Fermi level. This explains why data for different alloys and of states is responsible for the tunneling. The s and d spins in Fig. 7 are on the same curve. The positive slope in density of states are related by s-d hybridization. Fig. 7 can be explained by noting that the Fermi levels lie in As an example, we have plotted in Fig. 9 the s and the range where the s density of states increases as a function d density of states near the Fermi level for (a) the ma- σ of energy and hence the number of electrons Ni . jority band and (b) the minority band at the Ni sites of pure Ni, Ni0.8Fe0.2, and Ni0.6Fe0.4. We have shifted the energy such that within the same spin channel, the in- V. CONCLUSION tegrated density of states from the bottom of the bands to 0eV (not the Fermi level) are the same for every alloy. In this paper, a microscopic calculation for both the The Fermi energies in these plots are indicated by the magnetic moment and the spin polarization of subsets vertical lines. The s density of states fall on the same of the density of states of the 3d alloys are studied. By curve and the d peaks have the same energy. The only interpreting the tunneling spin polarization as the spin difference among the alloys are the Fermi levels. Thus, polarization of the s density of states, the trends in the the primary effect of alloying at the Ni sites is to shift tunneling experiments of Tedrow and Meservey are ob- the s and d bands together relative to the Fermi level. tained. The correlation between the magnetic moment σ We see from Fig. 9 that the s density of states ns of the and the density of states are understood by showing that fcc alloys is an increasing function of the Fermi energy in the s density of states for both spin orientations is an the range where the Fermi level lies. On the other hand, increasing function of the number of electrons and by σ N , the integrated density of states up to the Fermi level, showing that the primary effect of alloying is to shift the σ also increases with the Fermi energy. Therefore ns is an bands together relative to the Fermi level. The correla- σ increasing function of N as shown in Fig. 7. tion between the magnetic moment and the polarization It should be noted that the shifting of the d bands is of the s density of states is not universal. All of this due to energy considerations as in the itinerant electron work supports the picture that the tunneling current is theory. It results in the change of the magnetic moments, dominated by s electrons. µ. On the other hand, the shifting of the s band is due While we have explained the correlation between the to its hybridization with the shifted d bands. It results spin polarization and the magnetic moment, there are in the change of the spin polarization of the s density of still a number of open questions. Within this model, the states, Ps. tunneling current is assumed to be dominated by s elec- trons, but the mechanism behind it is not clear. Tsym- bal and Pettifor21 suggested that the hoping from the d

7 bands of the ferromagnet to the s band of the barrier is 1 M. N. Baibich et al., Phys. Rev. Lett. 61, 2472 (1988). essentially zero. In 3d metals, the s-d hoping is normally 2 M. Johnson, Phys. Rev. Lett. 70, 2142 (1993); Science 260, a few times smaller than the s-s hoping, so it is rea- 320 (1993). 3 sonable to assume a similar ratio for the hoping at the S. Jin et al., Science 264, 413 (1994). 4 metal-insulator interface. However, we note that since R. Meservey and P.M. Tedrow, Phys. Rep. 238, n.4, 173 (1994). the d density of states is about two orders of magnitude 5 higher than that of the s density of states at the Fermi M. Julliere, Phys. Lett. 50A, n.3, 225 (1975). 6 MAG- level, the s-d hoping has to be much smaller than a tenth S. Maekawa and U. Gafvert, IEEE Trans. Magn. 18 of the s-s hoping to explain the dominance of the tunnel- , 707 (1982). 7 139 ing current by the s electrons. Therefore, it is unclear if T. Miyazaki and N. Tezuka, J. Magn. Magn. Mater. , L231 (1995). such a requirement is physical. 8 22 J. S. Moodera et al., Phys. Rev. Lett. 74, 3273 (1994); In another model, suggested by Nguyen-Manh et al., Phys. Rev. Lett. 80, 2941 (1998). the s band in the insulator is spin-polarized by the d 9 S. Zhang et al., Phys. Rev. Lett. 79, 3744 (1997). band of the ferromagnet due to hybridization, causing a 10 P. M. Tedrow and R. Meservey, Phys. Rev. Lett. 26, n.4, positive spin polarization in the s current. The model 192 (1971); R. Meservey, D. Paraskevopoulos, and P.M. predicts that in the insulator there is very small but spin Tedrow, Physica 91B, 91 (1977); D. Paraskevopoulos, R. polarized density of states at the Fermi level. Meservey, and P.M. Tedrow, Phys. Rev. B 16, 4907 (1977). On the other hand, a different model was suggested 11 8 17 18 J. C. Slater, J. Appl. Phys. , 385 (1937). by Mazin and Nadgorny et al.. They argued that the 12 J. Friedel, Nuovo Cimento 10 Suppl. no. 2, VII, 287 (1958) current, both in tunneling and Andreev reflection exper- 13 J. Crangle and G. C. Hallam, Proc. R. Soc. London Ser. A iments, is proportional to the density of states times the 272, 119 (1963). Fermi velocity squared. The low(high) density of states 14 P. H. Derichs et al., Journ. Magn. Magn. Mat. 100, 241 of the s(d) electrons are therefore compensated by their (1991). high(low) Fermi velocities. Thus, both the s and the d 15 R. J. Soulen et al, Science 282, n.5386, 85 (1998). 16 electrons are important to the tunneling current. The S. K. Upadhyay et al., Phys. Rev. Lett. 81, 3247 (1998). 17 spin polarization they calculated is roughly independent I. I. Mazin, cond-mat/9812327. 18 B. Nadgorny et al., cond-mat/9905097. of the magnetic moment, much like the Andreev reflec- 19 tion data the group obtained. At this point, it is unclear M. B. Stearns, Phys. Rev. B 8, 4383 (1973). 20 J. A. Hertz and K. Aoi, Phys. Rev. B 8, 3252 (1973). which of the above models gives a better physical picture. 21 Another open question is whether the spin polariza- E. Yu. Tsymbal and D. G. Pettifor, J. Phys.: Condens. 9 tion measured by the tunneling experiments and the An- Matter , L411 (1997). 22 492 dreev reflection point contact experiments are the same. D. Nguyen-Manh et al., Mat. Res. Soc. Symp. Proc. , 17 18 319 (1998). Mazin and Nadgorny et el. argued that the two are 23 R. J. Elliott, J. A. Krumhansl, and P. L. Leath, Rev. Mod. the same. However, the situation here is more compli- Phys. 46, 465 (1974). cated. There are even disagreements among the results 24 D. A. Papaconstantopoulos, Handbook of the band structure obtained from the Andreev reflection experiments. The 15 18 of elemental solids, (Plenum Press, New York, 1986). results obtained by Soulen et al. and Nadgorny et al. 25 86 16 T.-S. Choy, J. Chen, S. Hershfield, J. Appl. Phys. , 562 are not the same as those obtained by Upadhyay et al. (1999). It is unclear whether this is due to the differences in the sample preparation or the method of data analysis. There are also qualitative differences between the pre- diction of different models. While our calculations show that the spin polarization increases with magnetic mo- ments, the calculations by Mazin17 and Nadgorny et al.18 show that the spin polarization is independent of the magnetic moments. However, the spin polarization can remain constant at most in a certain regime. In regimes such as Ni1−xCrx with x> 0.15, Ni1−xCux with x> 0.55, or the invar regime near Ni0.36Fe0.64, the mag- netic moment drops to zero. When the magnetic moment is zero the spin polarization is expected be zero because there is no difference between the majority and minority spins. It would be helpful to compare experiments and theories in these regimes.

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