Large morphological sensitivity of the magnetothermopower in Co/Cu multilayered systems
LLarge morphological sensitivity of themagnetothermopower in Co/Cu multilayeredsystems
Voicu Popescu and Peter Kratzer
Faculty of Physics and Center for Nanointegration (CENIDE), University ofDuisburg-Essen, Lotharstraße 1, 47057 Duisburg, GermanyE-mail: [email protected]
PACS numbers: 72.10.-d, 72.15.Jf, 73.50.Jt
Abstract.
We present results of first-principles calculations on the transportproperties, both under an electric field or a temperature gradient, in the Co/Cumultilayered systems. The various effects brought about by the changes in themorphological parameters, such as the number of repeats and the layer thickness,are discussed in a systematic way. Our calculations show that the Seebeckcoefficient and the magnetothermopower (MTP) converge rather rapidly with thenumber of Co repeats. In the range of thin Co layers, we find strong variationsin amplitude and sign of both the Seebeck coefficient and the MTP. These largevariations, which have no correspondent in the (magneto)conductance, are shownto be the result of quantum well states present in the minority spin channel ofthin Co layers.
1. Introduction
Metallic heterostructures of alternating magnetic and non-magnetic materials havebeen in the focus of research for more than two decades. These intense experimentaland theoretical investigations have been triggered by the giant magnetoresistance(GMR) effect [1]. Currently the key-stone of standard magnetic field sensors, theGMR denotes the large change in the resistance caused by the switching froman anti-parallel to a parallel magnetic alignment of the adjacent magnetic layersunder an external magnetic field. An analogous phenomenon could be observed inmultilayered structures subject to a temperature gradient , in which case the centralquantity measuring the magnetic response was the magneto-thermopower (MTP).These experiments, performed both in the current-in-plane (CIP) [2, 3, 4, 5] as well asin the current-perpendicular-to-the-plane (CPP) geometry [6], mark the first successfulattempts of linking the heat flow with the spin degree of freedom, paving the waytowards the emerging field of spin calorics [7].In recent years, the ability to fabricate multilayer samples in the formof nanopillars has opened up the possibility to detect their internal state ofmagnetisation. The small diameter of the pillars, resulting in a small thermalconductance, in conjuction with a strong heating by pulsed laser illumination of thepillar top allows one to build up sizeable temperature gradients [8, 9, 10, 11, 12], a r X i v : . [ c ond - m a t . m t r l - s c i ] D ec arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers g asdefining quantity to express the magneto-conductance (MC) ratio as:MC(%) = g P − g AP g P × . (1)The MTP ratio can be introduced quite analogously:MTP( T )(%) = S P ( T ) − S AP ( T ) S AP ( T ) × , (2)using the temperature dependent Seebeck coefficients for the two magneticconfigurations S P ( T ) and S AP ( T ). Since S AP ( T ) and S P ( T ) may differ not onlyin magnitude, but also in sign, one can imagine that the magnetic contrast in athermoelectric measurement, as expressed by the MTP ratio, may become largerthan the MC ratio for a specific sample. Such an expectation could be confirmedexperimentally, for example by Gravier et al [9]. In multilayered Co/Cu nanowiresthese authors found an MTP ratio of −
30 %, larger than the 20 % measured GMRratio. This behaviour is usually traced back to the fact that the conductance(equivalently, the conductivity σ ) is essentially a Fermi surface related property.The thermoelectric voltage (or the Seebeck coefficient), on the other hand, is ameasure of the energy dependence of the relaxation rate near the Fermi energy E F [4]. As expressed by Mott’s formula [13], the Seebeck coefficient is proportional to thelogarithmic derivative of σ ( E ): S = − π e k T d ln σ ( E )d E (cid:12)(cid:12)(cid:12)(cid:12) E = E F , (3)where e and k B are, respectively, the elementary charge and the Boltzmann constant.On its basis, one could derive a rather simple relation between the two quantities, theMTP and the MC ratios [8].Phenomenological models, while being useful in identifying general trends, domiss the important link between the described quantity and the underlying electronicstructure. Precisely the opposite philosophy is adopted in first-principles basedinvestigations, as the ones presented here: perform appropriate modifications of theelectronic structure and track the evolution of a given property with the ultimatepurpose of achieving specific design rules for a desired target value.For this purpose, we have considered one of the GMR prototypes, the Co/Cumultilayered system. Many of its ground-state properties as well as the CIP- or CPP-GMR effects have been already addressed on an ab initio level [14, 15, 16, 17, 18, 19, 20,21, 22, 23, 24], In contrast, first-principles calculations of the magneto-thermoelectricproperties of several Co/Cu heterostructures, that require a significantly largercomputational effort, gained only recently an increased attention [25, 26].The multilayered structure subject to our investigations can be seen as a stackingof Co m /Cu q bilayers of thickness m and q embedded in Cu(001). Accompanying thevariations in the metallic layers thickness, the number of repeats N was also treatedas a variable, leading to the actual configuration Cu[( N − m /Cu q )/Co m ]Cu,as schematically shown in Figure 1(b). Note that, by construction, N was taken arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers CoCoCoCo CoCoCoCoCuCuCuCuCu CuCuCuCuCu
MTP setup • Parallel Anti-parallel • ~ ∇ T (a) • N = 3, m = 4, p = 4 • N = 2, m = 6, p = 4 • N = 2, m = 6, p = 8 xy z ~ ∇ T (b) Cu[( N − m /Cu p )/Co m ]Cu Figure 1.
Schematic representations of (a) the generic setup of a multilayeredCo/Cu system providing an MTP signal; and (b) selected structural models ofthe Cu[( N − m /Cu q )/Co m ]Cu systems investigated in this work, illustratingthe meaning of the various geometrical parameters N , m , and q . N is the numberof Co repeats embedded in Cu, while m and q represent the thickness in atomicmonolayers (MLs) of the Co (yellow) and Cu (dark green) layers. The figure onlyshows the scattering region, with the half-infinite Cu leads extending left andright along the z direction. The whole system is periodic in the ( x, y ) plane. Thethermopower is calculated along the temperature gradient which is taken to beperpendicular to the interface. finite , that is, no periodic boundary conditions along the (001) growth directionwere imposed. The transport properties of these systems are investigated byperforming first-principles calculations of the underlying electronic structure by meansof a spin-polarised relativistic Green’s function method [27, 28, 29]. The resultsobtained for the conductance and the Seebeck coefficient in a CPP geometry —thetemperature gradient taken perpendicular to the interface— are analysed in view ofthe modifications in the electronic structure induced by varying the morphology ofthe heterostructure, either through the number of repeats N or of the thickness m ( q )of the constituent Co (Cu) layers.The close lattice match of Co and Cu, as well as the advanced fabricationtechnique of the nanopillars by electrodeposition, allows the experimentalists to buildstacks with a large number of Co repeats. If the dominating scattering mechanismof the electrons is scattering by the Co/Cu interfaces, it is to be expected that theresistivity of a stack increases with the number of repeats N , while the MC ratio isalmost independent of N . For the Seebeck coefficient, which has the physical meaningof a voltage, its dependence on N is not obvious. Our calculations show that both S ( T ) and the MTP ratio converge rather rapidly with the number of Co repeats inthe Co/Cu stacks, reasonably converged values being attained already at N = 4.A modulation of the electronic density of states due to quantum confinementeffects in ultra-thin layers may affect the resistivity, but to an even higher degreethe Seebeck coefficient of multilayered structures. The minority spin quantum wellstates formed in thin Co layers lie at the origin of an oscillatory behaviour observedfor many physical properties of these systems, ranging from the interlayer exchangecoupling [14, 15] or the magnetic anisotropy energy [20] to the recently investigatedSeebeck magnetic anisotropy [25]. Our calculations show that these quantum wellstates hybridise with a high-mobility band crossing the Fermi energy. As a result,we find strong variations in amplitude and sign of both the Seebeck coefficient and arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers N − m /Cu q )/Co m ]Cu multilayers, withdetailed discussions on their N , m , and q dependence.
2. Geometry of the system and theoretical background
The calculations for the Cu[( N − m /Cu q )/Co m ]Cu multilayer systems wereperformed using a spin-polarised relativistic (SPR) [29, 30] version of the screenedKorringa-Kohn-Rostoker Green’s function (KKR-GF) method [31, 32, 33]. We applythe same procedure as described in our previous investigations on the Cu/Co m /Cutrilayers [25], that essentially consists of three steps: (i) setting up the geometry of thesystem; (ii) the self-consistent determination of the ground state potentials; and (iii)using these as input for the transport calculations which are based on the Landauer-B¨uttiker formula as implemented in the KKR-GF method [34, 35] within a relativisticrepresentation [27, 28]. Our approach, discussed to some extent in this section, hasthe one-electron retarded Green’s function G + ( (cid:126)r, (cid:126)r (cid:48) ; ε ) at energy ε = E + i δ as centralquantity. We model the systems under investigation by taking two half-infinite Cu leads with aninteraction region inserted in-between, all sharing the same in-plane two dimensional(2D) periodic lattice. Since the natural lattice misfit between elemental Co and Cuis rather small (less than 2 %), we neglect the lattice relaxation at the interfaces andtake all atomic positions as being fixed to the ideal (001)-stacked fcc lattice with thelattice constant equal to the experimental fcc-Cu value of 3 .
61 ˚A. The interactionregion contains the [( N − m /Cu q )/Co m ] multilayered structure and up to 10atomic monolayers (MLs) of Cu on its both sides. These additional Cu MLs aremeant to ensure a smooth transition towards the Cu leads.Schematic representations of selected setups for the interaction region areprovided in Figure 1(b) for varying number of Co repeats (here, N = 2 and N = 3)and individual Co ( m = 4 and m = 6 ML) and Cu ( q = 4 and q = 8 ML) layerthickness. For our investigations we had considered, changing just one variable ata time, N = 1 , . . . , m and q ranging between 4 and 8 MLsthickness. We have determined the longitudinal thermopower occurring under atemperature gradient taken parallel to the growth direction z . Note that, along thisdirection, no periodic boundary conditions are imposed. We furthermore emphasisethat different numbers of repeats effectively mean differently sized finite objects along z ; an increase in N (at a given m and q ) is equivalent to an increase in the thicknessof the interaction region. arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers For each of the configurations, the potentials are determined self-consistently using thescreened KKR-GF method [29], considering spherical potentials in the atomic sphereapproximation (ASA) within the local spin-density approximation in the Vosko, Wilkand Nussair parametrisation [36]. An angular momentum cut-off of l max = 3 wastaken for the Green’s function expansion.In a preliminary step, a separate self-consistent calculation is performed in orderto determine the potential of the two (identical) half-spaces left and right of theinteraction region. A second self-consistent procedure is applied to the interactionregion itself, in which all its potentials are iterated, whereby the outer-most Cupotentials asymptotically match the ones in the leads. This matching is accomplishedby means of the decimation technique [37], in which the leads potentials determinedin the first step provide the appropriate boundary conditions of the heterostructure.Different magnetic couplings between adjacent Co layers, parallel and anti-parallel,were separately considered at each ( N, m, q ) combination. The collinearity of the spinmagnetic moments was the only constraint imposed a priori .As a consequence of the 2D-periodicity of the layered system, the Green’s functioncan be Fourier transformed in a 2D representation with the Bloch vector (cid:126)k (cid:107) as constantof motion and retaining an index i for the position along the growth direction z .Within the KKR-GF scheme, the Green’s function is expressed in terms of the matrix G ij ( (cid:126)k (cid:107) , ε ). This matrix describes the propagation of the electron wave between theatomic sites i and j at positions (cid:126)R i , (cid:126)R j and is labelled, in our adopted representation,by the relativistic quantum numbers Λ = ( κ, µ ), i.e. ( A ) ΛΛ (cid:48) = A ΛΛ (cid:48) [29]. Let usnote here that, since the spin is not a constant of motion we shall use the designationmajority/minority spin rather than up/down ( ↑ / ↓ ). Combining the structural Green’s function matrix calculated for a given 2D-periodicsystem with the matrices M i , M j of the z -component of the relativistic currentoperator at sites i and j enables the calculation of the electronic transmissionprobability between two atomic planes I and J according to the expression [28]: T ( (cid:126)k (cid:107) , E ) = (cid:88) i ∈ I,j ∈ J Tr (cid:104) M i † G ij ( (cid:126)k (cid:107) , ε ) M j G ij † ( (cid:126)k (cid:107) , ε ) (cid:105) , (4)where each 2D vector (cid:126)k (cid:107) can be seen as a conduction channel [35]. By integrating overthe 2D Brillouin zone (2D-BZ) the total transmission probability T ( E ) at energy E is then [35]: T ( E ) = 1 A − BZ (cid:90) − BZ d (cid:126)k (cid:107) T ( (cid:126)k (cid:107) , E ) . (5)In the case of a weak spin-orbit coupling, as it is the case for the light 3d transitionmetals, Popescu et al [27, 28] could show that the transmission through the ”fullyrelativistic resistor” expressed by Equation (4) can be approximated by (cid:101) T k (cid:107) , E ): T ( (cid:126)k (cid:107) , E ) (cid:39) (cid:101) T ( (cid:126)k (cid:107) , E ) = (cid:101) T ↑↑ ( (cid:126)k (cid:107) , E ) + (cid:101) T ↓↓ ( (cid:126)k (cid:107) , E ) ++ (cid:101) T ↑↓ ( (cid:126)k (cid:107) , E ) + (cid:101) T ↓↑ ( (cid:126)k (cid:107) , E ) (6)which provides a spin decomposition essentially equivalent to the Mott two-currentmodel. Equation (6) can be regarded as its generalisation to the relativistic case. In arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers ↑↑ and ↓↓ ) channels, it also includes spin-mixed ones( ↑↓ + ↓↑ ), induced by the spin-orbit coupling. We will use this approximate spindecomposition only for a qualitative discussion in section 3.1.Following Sivan and Imry [38], the Seebeck coefficient S ( T ) can be obtained from T ( E ) through the expression S ( T ) = − eT (cid:90) d E ∂ E f T ( E ) ( E − E F ) (cid:90) d E ∂ E f T ( E ) , (7)where f ≡ f ( E, T, µ ) is the Fermi-Dirac distribution function at energy E ,temperature T , and chemical potential µ , while ∂ E f = ∂f /∂E represents its energyderivative. The denominator in the last equation is related to the temperaturedependent conductance g ( T ) by: g ( T ) = − e h (cid:90) d E ∂ E f T ( E ) . (8)The various parameters involved in the actual evaluation of these quantities werechosen in the following way: For the 2D-BZ integral required for the transmissionprobability, Equation (5), a regular 1000 × (cid:126)k (cid:107) -grid was found necessary to achieveconvergency of T ( E ) over a broad range of energy arguments. For the integralsin Equation (7) and (8), on the other hand, T ( E ) was explicitly calculated on a1 mRy-spaced regular mesh, then interpolated on a denser mesh of 0 . E F were set in such a waythat ∂ E f ( E min / max ) < − , a limit found to be more than sufficient in providingwell-converged results. The formalism employed here rigorously describes elastic scattering at the interfacesand treats the simultaneous occurrence of spin polarisation and relativistic effects,such as spin orbit coupling, on equal footing. Temperature enters in this approachthrough the Fermi-Dirac distribution function, but temperature-dependent scattering,e.g. by atomic vibrations or spin fluctuations are neglected.Inclusion of atomic displacements at finite temperature in transport propertiescalculations within ab initio methods has been recently accomplished by treating themas static disorder via the coherent potential approximation (CPA) [39]. Alternatively,one could use large 2D supercells and apply a frozen phonon approach averagingover explicit different atomic displacements. We note, however, that S ( T ), being thequotient of two integrals involving the transmission probability T ( E ), any additionaltemperature dependence due to inelastic scattering, appearing both in the numeratorand the denominator, tends to cancel out as long as phonon drag effects can bedisregarded.Accounting for electron scattering by spin fluctuations in various ferromagneticmetals and alloys has been convincingly demonstrated to improve the agreementbetween calculated and experimentally determined temperature dependent resistivity[40, 41]. More recently, Kov´aˇcik et al [26] investigated the effect of static spin disorderon the magneto-thermoelectric phenomena of several nano-structured Co/Cu systems.These authors could show that, while the spin-dependent electron scattering doesindeed influence the spin-caloric transport coefficients at elevated temperatures, the arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers dynamic spinfluctuations influence the Seebeck coefficient is hardly explored. It may be notedthat Piraux et al [3] invoked inelastic spin-dependent electron-magnon scattering toexplain the increase in the MTP at high temperatures observed experimentally inCo/Cu and Fe/Cu thin multilayers. However, as we will show below, an equally largeMTP can be obtained accounting solely for the electronic structure contributions to thethermopower. For systems with a gapped band structure, such as magnetic half-metalsor tunnel junctions, modifications of the electronic properties due to dynamic spinfluctuations at finite temperatures have been addressed via phenomenological models[42, 43] or via the dynamical mean field theory (DMFT) [44]. For systems as large asthose considered here, such an advanced many-particle treatment is computationallynot feasible at present.The importance of these temperature dependent effects notwithstanding, ourprimary focus here is to identify the specific effects on the magneto-thermopowerwhich are intimately connected with the electronic structure and are solely inducedby quantum confinement. As such, properly accounting for the effects discussed aboveis well beyond the purpose of the current investigations, although this is clearly neededin future studies for an improved quantitative agreement with experiment.
3. A single Co layer embedded in Cu(001)
We begin our discussion by presenting results obtained for a single Co layer embeddedin Cu(001), a geometry setup corresponding to N = 1 in the general notationCu[( N − m /Cu q )Co m ]Cu introduced above. We shall first analyse briefly theelectronic structure in the proximity of the Cu/Co interface and illustrate how itspeculiarities are reflected in the transmission probability for a single Co layer of varyingthickness m . The T ( E ) transmission profile for m = 4 will be shown to exhibit apeak in the minority spin channel immediately below the Fermi energy. This peak isintimately connected with a complex formed by a quantum well state (QWS) and ahigh mobility p-band present in the interface layers, with which the QWS hybridises.These findings for the single layer system will be important in understanding thetransport properties of the multilayered Cu[( N − m /Cu q )/Co m ]Cu systems. Electronic structure calculations performed on the Co/Cu systems [14, 18, 23, 24]revealed that the majority spin d-band is completely filled and the energy range atand near the Fermi level is dominated by the 3d minority spin states stemming fromCo. These features are accordingly reproduced by our calculations and reflected inthe spin-resolved density of states (DOS) for the Cu/Co /Cu trilayer system shownin Figure 2(a). Here the DOS is further projected on the Co and Cu atoms in thevicinity of the Cu/Co interface as well as on their angular momentum (s+p)- andd-components.Figure 2(a) evidences that the d-DOS (light blue curve) of both Co and Cu hasan overwhelming contribution to the total DOS of the Co/Cu heterostructure in bothspin channels. We further note the large Co-related contribution in the minority spinchannel in the proximity of the Fermi energy, contrasting the extremely reduced DOS arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers (a) -6 -4 -2 0 2 Energy relative to E F (eV) DO S ( s t a t e s / e v / a t o m ) Cu interface-1 Co interface+1 Co interface Cu interface sum d(s+p) (b) -0.3 -0.1 0.1 0.3 E - E F (eV) T r a n s m i ss i on p r ob a b ilit y majority spinminority spinmixed-spin DOS and transmission probability in
Cu/Co /Cu Figure 2. (a) Spin- and angular momentum resolved local DOS projected onthe Cu and Co atoms in the vicinity of the Cu/Co interface, with the majority(minority) spin channel in the left (right) panels. The (s+p)-DOS are shownas sum (red lines) while the d-DOS (light blue) is seen to have the dominantcontribution to the total DOS (black). (b) Electronic transmission probability inthe Cu/Co /Cu trilayer system. The total transmission (black) is decomposedin its spin-conserving (blue and dark red) and spin-mixing (green) components.Note the pronounced peak in the minority spin transmission at about − . on the Cu sites. Only the first Cu layer near the interface exhibits a slight spinpolarisation, induced by its neighbouring Co atoms.The dominant d-character of the minority spin states close to the Fermi energy isaccordingly reflected in the transport properties. Indeed, the d-states are characterisedby a stronger localisation and a reduced mobility as compared to the s- and p-states.We show in Figure 2(b) the transmission probability T ( E ) for the same Cu/Co /Cutrilayer system, containing a 4 ML thick Co layer. The black curve in this figurerepresents the total transmission, calculated for various energy arguments E usingEquation (5). Applying the spin decomposition leading to the approximate form (6),allows us to identify the spin-conserving and spin-mixing transmission channels in T ( E ). As a result of the rather small spin-orbit coupling, the spin-mixing transmission(green lines) is negligibly small. In spite of a small DOS near E F , the largesttransmission component is the spin-conserving majority one. The minority spintransmission is only about one third of the total. Nevertheless, while (cid:101) T ↑↑ ( E ) is nearlyfeatureless and shows a weak variation with E , it is the (cid:101) T ↓↓ ( E ) component whichmodulates the full transmission profile T ( E ). This different qualitative behaviourarises from the different character of the states involved in the transmission through thevarious channels: nearly exclusively s- and p-states for the majority spin, hybrid s-p- and d-states for the minority spin. The first important conclusion of our investigationscan thus be formulated as follows: Although comparatively small in magnitude, theminority spin transmission is expected to be much more sensitive to the morphologyof the system. Changes in the transport properties caused by geometry modificationscan be traced back nearly exclusively to modifications in the minority spin electronicstructure.While the general characteristics of the transmission profile discussed above were arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers . m = 4 ML. It represents, in fact, a signatureof a QWS arising in the minority spin band of Co. The detailed discussion of theseQWSs makes the subject of the next section. The appearance of QWSs in the Co slab has been found responsible for the non-monotonous behaviour of the magnetic anisotropy energy (MAE), evidenced bothexperimentally [19] and theoretically [20] in the Cu/Co m /Cu trilayers, as well as forthe oscillations in the interlayer exchange coupling occurring in Co/Cu for thin Colayers [17]. In a recent study of the authors [25], the QWSs were also shown to playan important role in inducing an anisotropic MTP in the same systems. In particular,we identified a hybrid complex formed by the QWSs and a p-type band specific tothe Co/Cu interface. These hybrid states provide extremely efficient channels for theminority spin electrons, as evidenced by the aforementioned peak, 0 . E F , inCu/Co /Cu. In the next sections we shall link our findings for the Seebeck coefficientand the MTP precisely to these peculiarities of the electronic structure.The minority spin channel QWSs appearing in the Co slab have been investigatedby calculating the angular momentum and atom projected Bloch spectral function(BSF) A i ( (cid:126)k (cid:107) , E ) [29]: A i ( (cid:126)k (cid:107) , E ) = − πN Im Tr N (cid:88) n,n (cid:48) e i (cid:126)k (cid:107) ( (cid:126)χ n − (cid:126)χ n (cid:48) ) × (cid:90) d r (cid:107) G ( (cid:126)r (cid:107) + (cid:126)R i + (cid:126)χ n , (cid:126)r (cid:107) + (cid:126)R i + (cid:126)χ n (cid:48) ; E ) , (9)where (cid:126)χ n , (cid:126)χ n (cid:48) are translation vectors of the 2D periodic lattice, (cid:126)r (cid:107) = ( r x , r y , (cid:126)R i = (0 , , z i ) represents the z -coordinate of the i th atom. The BSF is a quantity thatcan be regarded as a (cid:126)k (cid:107) -resolved DOS.Figure 3 depicts the minority spin component of A Co ( (cid:126)k (cid:107) , E ) projected on the first[panel (a)] and second [panel (b)] Co atomic layers near the Cu/Co interface, with therespective location of each layer schematically drawn at the side of the figure. Thedifferent frames in each panel follow the variation of the Co layer thickness m MLsin the Cu/Co m /Cu trilayer system. Such E versus (cid:126)k (cid:107) -type plots allow us to identifythe projected band structure in the 2D-BZ, a picture familiar from angle-resolvedphoto-emission experiments.The formation and appearance of a certain QWS will depend on the Co thickness m , alternating between odd and even number of MLs. For a given parity, on the otherhand, the m -dependence is reflected in a variation in the energy position of the QWS.Typical signatures of QWS can be observed as flat bands in the BSF of the interfaceCo layer [panel (a)] near the 2D-BZ centre: (i) around 0 .
15 eV for m = 3 and m = 5and (ii) around − . . m = 4 and m = 6, energy values relative tothe Fermi level. With increased thickness of the Co slab, the QWSs morph into acontinuum, as seen in the right-most frame of Figure 3(a) for m = 10 MLs.The second important aspect revealed by the top panel of Figure 3 is the existenceof a high-mobility p-band for the minority spin carriers, crossing the Fermi energy.This band, evidenced by the red-coloured, S -shaped feature of the BSF in Figure 3(a), arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers (a) Co interface layer Γ X ( k x ,0)-0.9-0.6-0.30.00.30.6 E - E F ( e V ) Γ X ( k x ,0) Γ X ( k x ,0) Γ X ( k x ,0) Γ X ( k x ,0) k → || - DO S ( s t a t e s / e V / a t o m ) m = 3 ML m = 4 ML m = 5 ML m = 6 ML m = 10 ML (b) Co next-to-interface layer Γ X ( k x ,0)-0.9-0.6-0.30.00.30.6 E - E F ( e V ) Γ X ( k x ,0) Γ X ( k x ,0) Γ X ( k x ,0) Γ X ( k x ,0) k → || - DO S ( s t a t e s / e V / a t o m ) m = 3 ML m = 4 ML m = 5 ML m = 6 ML m = 10 ML • m = 3 ML • m = 6 ML (a)(a)(b)(b) Minority spin ~k k -DOS in Cu/Co m /Cu Figure 3.
Minority spin (cid:126)k (cid:107) -resolved DOS projected on the (a) interface and (b)next-to-interface Co atom in the Cu/Co m /Cu trilayer system for various valuesof m , as shown schematically on the right side of the figure. For the interfacelayer (a) typical signatures of QWSs appearing as flat bands above (odd m ) andbelow (even m ) E F for thin Co slabs as well as a p-type band crossing the Fermienergy E F (taken here as reference value) can be recognised. The latter has onlya weak correspondent in the next-to-interface layer (b). Note that the same scalehas been used in both panels. stems from the Cu and Co atoms adjacent to the interface and exhibits no thicknessdependence. In other words, it is an ubiquitous characteristic of the Co/Cu interface.For the case of an even number of MLs m , the QWS forming below the Fermienergy will couple to this p-band, leading to the formation of a p-d hybrid complex.This is precisely the origin of the strong transmission evidenced in the minority spinchannel at − . m = 4 ML Co thick systems. Note that a similarhybrid band complex also appears above the Fermi energy, at 0 .
45 eV. This is howevertoo far to contribute to the integrand of Equation (7).It is easy to see how the morphology of the system may have significant influenceson its electronic structure, and, as a result, on its transport properties. By comparingthe spectral functions of the two different Co atoms in Figure 3(a) and (b) one cansee, for the next-to-interface layer, a significant reduction in the amplitude of the p-band-related BSF below the Fermi energy. Thus, for the transmission channels openedby the QWS-p-band complex it will mean that their weight and importance in the arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers m is odd , on the other hand, theQWS appearing above the Fermi energy couples very weakly, if at all, with the highlymobile p-band. As a consequence, no corresponding high energy transmission peakis expected for odd m . This thickness-triggered filtering of the transmission channelsinvolved in the conduction was found responsible for the non-monotonous behaviourof the Seebeck coefficient in Cu/Co m /Cu trilayers [25].Additional variables come into play in the case of multi-layered systems: Thesize of the Cu spacer placed between the Co layers will modify the way in which thedifferent states will or will not couple across the interfaces, leading to an enhancementor suppression of various transmission channels. Likewise, a varying number of Coand Cu repeats may further complicate the picture. Furthermore, when differentmagnetic alignments between adjacent Co layers are considered, one has to bear inmind that the minority/majority spin channels get swapped. We discuss these aspectsin the next sections, essentially showing that a broad range of values may arise forboth the Seebeck coefficient and the MTP, depending on the different morphologicalparameters.
4. Varying number of Co repeats
In the previous section the discussion focused on electronic structure characteristicsrelated to a single Co layer of varying thickness embedded in Cu(001). We haveanalysed how these may influence directly the transmission probability and, throughit, the various transport properties. The first question to ask is how much andto what extent the knowledge gained so far is transferable to the multilayeredCu[( N − m /Cu q )/Co m ]Cu systems. In this section we discuss results obtainedby modifying the number of Co repeats N while keeping the other parameters, m and q , fixed. For convenience and easier comparison with the results already presented, werestrict the discussion, without losing any generality, to the case m = q = 4 MLs. Themost important conclusions drawn at the end of this section are: (i) the electronicstructure features present in the single-Co layer system transfer to the Co-stackedsystems; (ii) increasing the number of Co repeats reduces the transmission throughthe heterostructure without, however, significantly modifying its E -dependent profile;and (iii) at high temperature, the Seebeck coefficients, both for parallel and anti-parallel alignments, as well as the derived MTP are reaching converged values in N rather fast. The electronic structure calculations performed for the Cu[( N − m /Cu q )/Co m ]Cusystems with N = 1 , . . . , N . Asan illustrative example we show in Figure 4(a) the spin magnetisation profiles for N = 1 (red bullets) and N = 2 (dark blue crosses), that is, one and two Co slabsembedded in Cu, each of a thickness m = 4 MLs. For N = 2 the two Co layers areseparated by a thin Cu spacer ( q = 4 MLs). One can see that there are hardly anydifferences noticeable in the individual spin magnetic moments on the Co atoms inthe two cases, Cu/Co /Cu and Cu/Co Cu Co /Cu. Specifically, for the Co atoms arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers (a) layer along z -direction0.00.30.60.91.21.51.8 µ s p i n ( µ B ) x x Cu Co Cu Co Cu x N =1 N =2 (b) -0.3 -0.1 0.1 0.3 E - E F (eV)0.91.01.1 T r a n s m i ss i on p r ob a b ilit y N =2 N =3 N =4 N =6 N =5 Cu[( N − /Cu )/Co ]Cu – Parallel magnetic configuration Figure 4. (a) Spin magnetisation profile of the Cu/Co Cu Co /Cu system( N = 2, red bullets) with a parallel (P) magnetic configuration as compared tothat of the Cu/Co /Cu trilayer ( N = 1, dark blue crosses). The figure emphasiseson the quasi independence of the spin magnetic moments in the Co layers on thenumber of repeats N . Note that the values corresponding to the Cu atoms aremultiplied by a factor of 10. (b) N -dependence of the electronic transmissionprobability T ( E ) in Cu[( N − /Cu )/Co ]Cu with a P-alignment of the Comagnetic moments. The transmission curves are shown around the Fermi energy E F , taken here as reference in the same range as in Figure 2(b). nearest to Cu, the spin magnetic moments obtained were (in Bohr magnetons µ B ):1 . N = 1), 1 . N = 2, outer Co), and 1 . N = 2, inner Co). In spiteof the very small thickness of the spacer, the individual Co layers obviously displaythe same properties, regardless of N . Analogous results were obtained for the othernumber of repeats; furthermore, also the Cu spacer layers of equal thickness exhibitsimilar characteristics.In other words, these findings imply that all the considerations made in theprevious section regarding the electronic structure of a single Co slab transfer to themulti-layered systems, qualitatively completely and quantitatively to a large extent.This also holds, in particular, for the QWSs formation and the Co/Cu-interface specificp-band in the minority spin channel.Not surprisingly, a similar one-to-one transferability holds only partly in thetransmission probability profiles. These are shown in Figure 4(b) for m = q = 4 MLs, N = 1 , . . . ,
6, and with all the magnetic moments oriented parallel one to another.Note that throughout the next figures we will use the same colour-coding convention,borrowed from the solar spectrum: as the variable under investigation increases,the colours used in the graphical representation change from red to purple. Thegeneral trend that can be recognised from Figure 4(b) is an overall down-scaling of the T ( E ) profiles with increasing N . This is a direct consequence of successively addinginterfaces, that is, electron scattering sources, to the transmission process between theleft and right leads. It is nevertheless obvious that the reduction in transmission is nota uniform function of energy argument. For this reason, the peak at − . N = 1, although still present for all values of N , appears of varying shapeand width. We also point out to the slope of T ( E ) near the Fermi energy, which ischanging sign with N . While being the result of subtle variations in the way differentstates couple, these changes influence the thermoelectric properties of the multilayeredsystem only in the limit of low temperatures. arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers Figure 5(a) displays the dependence of the Seebeck coefficient on the number of Corepeats N in the multilayered Cu[( N − /Cu )/Co ]Cu systems. The two panelscorrespond to results obtained for two magnetic configurations, a parallel (P) and ananti-parallel (AP) alignment. The latter is understood as the succession of Co slabs,each of m MLs thickness (here m = 4), in which the magnetic moments in one slabare opposed to those of its neighbouring Co slabs. As an example, for N = 5 onewould have for the Co slabs the arrangement ( ↑↑↑↑↑ ) for the P-alignment and ( ↑↓↑↓↑ )for AP.As mentioned in the introduction, measurements of the Seebeck coefficient areusually interpreted on the basis of Mott’s formula, Equation (3), which provides adirect link between S ( T ) and the conductivity σ ( E ) of a sample. Its range of validityhas been discussed to some extent in the literature, e.g. by Jonson and Mahan [45],who showed that it gives the correct T → T ( E ) influences the sign and sizeof the Seebeck coefficient at finite T can be understood on the basis of Equation (7).In this equation, a temperature increase effectively extends the integration range, byincreasing the non-zero width of T ( E )( ∂f /∂E ). Because of the ( E − E F ) term, thenumerator may be seen as a centre of mass of T ( E )( ∂f /∂E ) [46]. Consequently,both sign and value of S ( T ) will be sensitive even to small changes in the numerator’sintegrand below or above E F .Finally, we note that, in terms of the transmission probability T ( E ), Mott’sformula can be obtained as the T → S ( T ) being positive (negative) for a negative (positive) slope of T ( E ) near the Fermienergy. In other words, a large transmission below (above) E F will result in a positive(negative) S ( T ). Due to this peculiarity, the Seebeck coefficient measurement is awell-known tool in establishing the nature of carriers, p - or n -type, in semiconductors.The above considerations provide the basis to understand the Seebeck coefficientresults, in conjunction with the transmission probability profiles depicted inFigure 4(b). In both magnetic configurations, S P ( T ) and S AP ( T ) exhibit a non-monotonous behaviour at low temperatures, consistent with the changes in the slopeof T ( E ) near the Fermi energy observed in Figure 4(b). These results are consistentwith Mott’s formula. At increased temperatures, both S P ( T ) and S AP ( T ) becomeand remain positive, with S AP ( T ) much larger (about a factor of four) than S P ( T ).This positive sign is a direct consequence of the enhanced transmission in the range of0 . m = 4 MLs.Figure 5(a) further shows that, above T (cid:39)
100 K, the Seebeck coefficientconverges rather fast with the number of Co repeats N , for both magnetic alignments.Mathematically, the fast convergence of S P / AP ( T ) with N results from its definitionas a quotient of two integrals. The behaviour is clearly different for the T = 0 Kconductance g ( T = 0) which is shown in Figure 5(b) for the same systems andmagnetic configurations. Note that, for clarity, we omitted displaying its temperaturedependence. For both g P ( T ) and g AP ( T ) this was found to be quite weak, a similarresult being reported by Kov´aˇcik et al [26].In order to quantify the magnetic response encountered in the thermoelectric arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers (a)
100 200 300 Temperature (K)0.00.51.01.52.0 S P ( µ V / K )
100 200 300 Temperature (K) 0.02.04.06.08.0 S A P ( µ V / K ) N =2 N =3 N =4 N =5 N =6 • Parallel S P ( T ) Anti-parallel S AP ( T ) • (b) • g ( T = 0) N C ondu c t a n ce g ( T = ) ( e / h ) PAP
Seebeck coefficient and conductance in
Cu[( N − /Cu )/Co ]Cu Figure 5.
Dependence of (a) the Seebeck coefficient S ( T ) and (b) the zerotemperature conductance g ( T = 0) on the number of Co repeats N in theCu[( N − /Cu )/Co ]Cu multilayered systems: (a) S P ( T ) for parallel(P, left panel) and S AP ( T ) for anti-parallel (AP, right panel) alignment ofthe magnetisation in the Co layers. (b) conductance for P (bullets) and AP(diamonds) alignments. Note the different scale used for S P ( T ) and S AP ( T ).
100 200 300 Temperature (K)-20020406080100 M T P ( % ) N -20020406080100 M C ( % ) N =2 N =3 N =4 N =5 N =6 • ( S AP − S P ) /S AP ( g P − g AP ) /g P • Cu[( N − /Cu )/Co ]Cu Figure 6.
Dependence of the magneto-thermopower (MTP/left) and the zerotemperature magneto-conductance (MC/right) on the number of Co repeats N inthe Cu[( N − /Cu )/Co ]Cu multilayered systems. effect, we calculated the MTP ratio according to Equation (2). We note that,regardless of the convention adopted for the denominator, one problem might alwaysarise when plotting a temperature-dependent MTP ratio: Since the Seebeck coefficientmay change sign as a function of T , one will necessarily encounter discontinuities inthe graphical representation of the MTP ratio. Such a situation is indeed observed inFigure 6, where we present the calculated MTP and MC ratios for the multilayeredCu[( N − /Cu )/Co ]Cu systems.As can be seen in this figure, very large and widely spread values for the MTPratio are predicted in the range of low temperatures. These arise in those areas where S AP approaches zero. Extremely large MTP ratios have been purposefully omittedfrom the figure. Although not as fast as S P ( T ) and S AP ( T ), the MTP ratio also arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers N at high temperatures and is in general larger than theMC ratio, a result which is qualitatively consistent with the experimental findings[6, 8, 9]. As was the case for g P / AP ( T = 0), the MC ratio is, in turn, not fullyconverged with N . We note that a quantitative comparison with the experimentaldata is not attempted here. The reported results were obtained either for (111)-grownmultilayers [6] or for nanowires of ∼
10 nm layer thickness [8, 9], much larger thanthe 4 MLs (6-7 ˚A) used in our calculations. The effect of the m and q parameters onthe (magneto)thermoelectric properties of the Co/Cu multilayers is discussed in thefollowing section. We close this section by briefly discussing the interlayer exchange coupling in theinvestigated structures, an electronic structure related issue which is closely connectedto the GMR and MTP effects. The functionality of any GMR device relies on itscapability of switching, under an applied magnetic field, from an anti-parallel toa parallel coupling of the magnetisation in the adjacent ferromagnetic (FM) layersseparated by a non-magnetic (NM) spacer. In the absence of an external magneticfield the ground state magnetic configuration is determined by the interlayer exchangecoupling (IXC). In many FM/NM heterostructures the IXC was found to exhibit anoscillatory behaviour with the thickness of the NM layer.The results we obtained for the IXC in Cu[( N − m /Cu q )/Co m ]Cu withvarying N , m , and q indicate that an anti-parallel magnetic coupling between the Colayers is only favoured in the range of thin Cu spacers, with an even number, q = 4and q = 6, of MLs. This appears to be a common feature for all Co layers, irrespectiveof their own thickness m and the number of repeats N . Our findings are consistentwith previous first principles investigations of the IXC in Co/Cu bilayers, trilayersor superlattices [14, 15, 16, 17]. In particular, we note that all these calculationspredict a more stable parallel alignment for thick Cu spacers in the absence of interfaceroughness [17].
5. Multilayers of varying Co and Cu thickness
In this section we shall study how thickness changes of the individual ferro- andnon-magnetic components (the Co and Cu layers) affect the (magneto)thermoelectricproperties of the heterostructure. Particular attention will be given to the QWS-p-band hybrid states and their anticipated evolution with the morphology of the system,as suggested by the findings discussed in section 3.As demonstrated in Figure 2(b), for the systems investigated here the transmissionis highly spin-conserving. We have also shown that the QWS-p-hybrids are onlypresent in the minority spin band and are characterised by a strong localisation atthe Co/Cu interface. An increase of either Co or Cu layer thickness is expectedto modify the transmission probability profiles through their influence on the thetransmission channels opened by these states, diminishing their amplitude and weight.In particular, by removing the large contributions to T ( E ) below E F , a correspondingchange in sign and increase in absolute value is expected for S ( T ).For the results to be presented in the following we keep a fixed number of repeats N = 4 in the Cu[( N − m /Cu q )/Co m ]Cu system. We will start our discussion byfocusing on the changes induced in the transmission profiles by the modifications arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers m and q of the Co and Cu layers and then we will derive thecorresponding transport properties. The very general expectations formulated abovewill be compared with the results provided by the actual calculations. We will showthat the values obtained for both the Seebeck coefficient and the MTP span a verybroad range, depending on the particular ( m, q ) combination. Thus, we concludethat, although the expected trends based on ’educated guesses’ are generally fulfilled,in most cases explicit calculations are needed in order to make accurate predictions[26, 46]. When discussing the minority-spin BSFs of a single Co layer in section 3 we haveemphasised on two important aspects: (i) the appearance of QWSs and (ii) theexistence of an interface-related, high-mobility p-band. The positions of the formerare obviously thickness dependent but they may hybridise with the latter. As a result,strong transmission channels for the minority spin carriers are opened.As shown in Figure 3, this p-band is ever present, regardless of the Co thickness.The energy position of the QWSs, on the other hand, will change as the Co layerbecomes thicker: by a larger extent when m switches from even to odd and only bya smaller amount for m → m + 2 k (identical parity). Eventually, the QWSs morphinto a continuum as the thickness of the Co layers further increases. This evolutionof the QWS-p-band hybrids with m must be accordingly reflected in the transmissionchannels opened by these states.The results displayed in Figure 7 represent a convincing proof that thisis indeed the case. Here we show the calculated transmission profiles forCu[3(Co m /Cu q )/Co m ]Cu in the parallel (P) magnetic alignment as a function of either m at fixed q or vice-versa. For the clarity of the picture, the data for odd numberof MLs, otherwise in line with the expected trends, have been omitted. From left toright, the different panels of Figure 7 show the changes in T ( E ) for (a) q = 4 andvarying m , (b) q = 8 and varying m , and (c) m = 8 and varying q . Note that thevarying (fixed) quantity in the figure is denoted by dark red (light blue) colours.Not surprisingly, the thickness dependence of the QWS-p band complexesdiscussed above has a significant influence on the transmission probability profiles.The transmission channels connected to these states do follow the expected shifts inposition and intensity. As seen in Figure 7(a), the increase of the Co layers thicknessfrom m = 4 to m = 6 and then to m = 8 MLs causes a dramatic drop in T ( E )below E F . A strong reduction in transmission can also be observed for the highenergy peak at 0 . q values [Figure 7(a) with q = 4 MLs and (b) with q = 8 MLs] makes clear that thesmoothening of the transmission profiles is mainly caused by the variations in the Cothickness, independent of the Cu spacer thickness. It is the direct consequence of thecorresponding changes in the electronic structure landscape evidenced by the BSFs inFigure 3.Indeed, one might conclude from Figure 7(a) and (b) that the Cu spacer only playsthe role of a ’propagation medium’ of varying size, without too much of an influenceon the main features of the transmission profile. Such an interpretation is apparentlysupported also by the results displayed in Figure 7(c) for different transmission curvesat fixed m = 8 MLs and varying Cu thickness q . It is only true, however, for thicker Colayers, in which case the interface-related effects have a smaller weight. In the range arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers (a) varying m ; q = 4 (b) varying m ; q = 8 -0.3 -0.1 0.1 0.3 E - E F (eV)0.80.91.01.1 T r a n s m i ss i on p r ob a b ilit y -0.3 -0.1 0.1 0.3 E - E F (eV) m =4 m =6 m =8 m =4 m =6 m =8 (c) m = 8 ; varying q -0.3 -0.1 0.1 0.3 E - E F (eV) 0.80.91.01.1 T r a n s m i ss i on p r ob a b ilit y q =4 q =6 q =8 Transmission probability in
Cu[3(Co m /Cu q )/Co m ]Cu – Parallel magnetic alignment Figure 7.
Dependence of the electronic transmission probability T ( E ) inCu[3(Co m /Cu q )/Co m ]Cu (number of repeats N = 4) on the thickness m and q of the Co and Cu layers: (a) fixed Cu thickness q = 4 MLs, varying Co thickness m ; (b) fixed Cu thickness q = 8 MLs, varying Co thickness m ; and (c) fixedCo thickness m = 8 MLs, varying Cu thickness q . The magnetic configurationcorresponds to a P-alignment of the Co magnetic moments. Transmission profilesfor odd number of MLs ( m and q ) were skipped for the sake of clarity. of thin Co layers (small m values) the spatial separation of the interfaces within the non-spin-polarised spacer will also affect the transmission profiles, albeit in a moresubtle way and on a smaller scale. This can be seen by comparing the two curveslabelled m = 4 in the two panels (a) and (b) of Figure 7.To summarise, significant changes in the transmission profiles occur when thethickness of the Co layers is varied. We could establish a direct connection betweenthese variations and the modifications in the electronic structure. In turn, a thicknessincrease of the Cu leads to less spectacular changes in T ( E ). Despite the difference inthe magnitude of the two effects, we will show, in the next section, that the Seebeckcoefficient as well as the MTP are equally sensitive to both m and q variations. Thermopower measurements on Ni and Fe-Ni films [47] have shown that even at a20 nm thickness of the sample, the Seebeck coefficient is about half the value measuredfor bulk. This is a general characteristic of nano-structured metallic systems and thetransition from thin films to bulk can be understood as resulting from the reducedweight of the interface-related transmission channels. With an increased thickness ofthe film, the s- and p-states will have an enhanced contribution to the transmissionabove the Fermi energy. Specific to the currently investigated systems, both S P ( T ) and S AP ( T ) turn negative for larger m and q values, with significantly increased absolutevalues.This behaviour is illustrated in Figure 8(a) where we show the Seebeck coefficientsfor the parallel (P, left) and anti-parallel (AP, right) magnetic alignments of theCu[3(Co m /Cu q )/Co m ]Cu multilayered system with a fixed Cu spacer thickness q =4 MLs (top) and q = 8 MLs (bottom), for varying Co thickness m . The mostspectacular result, anticipated from the changes in the transmission profiles, is thechange in sign obtained for both S P ( T ) and S AP ( T ) when m increases. Note that arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers (a) -4.0-3.0-2.0-1.00.01.02.0 S P ( µ V / K ) -12.0-8.0-4.00.04.08.0 S A P ( µ V / K ) m =4 m =5 m =6 m =7 m =8 q = 4 ML
100 200 300Temperature (K)-4.0-3.0-2.0-1.00.01.02.0 S P ( µ V / K )
100 200 300Temperature (K) -12.0-8.0-4.00.04.08.0 S A P ( µ V / K ) m =7 m =4 m =5 m =6 m =8 q = 8 ML • Parallel S P ( T ) Anti-parallel S AP ( T ) • (b) • g ( T = 0) C ondu c t a n ce g ( T = ) ( e / h ) PAP q = 4 ML m (ML)0.00.20.40.60.81.01.2 C ondu c t a n ce g ( T = ) ( e / h ) PAP q = 8 ML Seebeck coefficient and conductance in
Cu[3(Co m /Cu q )/Co m ]Cu Figure 8.
Dependence of (a) the Seebeck coefficient S ( T ) and (b) the zerotemperature conductance g ( T = 0) on the Co layer thickness m (in MLs) in theCu[3(Co m /Cu q )/Co m ]Cu multilayered systems for q = 4 (top) and q = 8 MLs(bottom). (a) S P ( T ) for parallel (P, left panel) and S AP ( T ) for anti-parallel (AP,right panel) alignment of the magnetisation in the Co layers. (b) conductance forP (bullets) and AP (diamonds) alignments. Note that, while a different scale isused for S P ( T ) and S AP ( T ), each of them remains unchanged when varying Cuthickness q . the same colour-coding convention (from red to purple for increasing m ) is used inthis figure as introduced above. The m -dependence of S ( T ) patterns, both in P-and AP-alignment, show remarkable similarities for the two q values, indicating theless important role played by the Cu spacer in governing the (magneto)thermoelectricproperties of the investigated systems. Notable differences can only be seen for m = 5and m = 7 MLs Co. In these systems the Seebeck coefficient has a small absolute valueand fluctuating sign, as seen, for example in the S P ( T ) corresponding to ( m, q ) = (5 , , g ( T = 0). In contrast to the Seebeckcoefficient, the conductance is seen to exhibit much less fluctuations with m and q . Thereason for this different behaviour lies, once again, in the actual energy range wherethe transmission probability is changing with m . As we could see, this essentially takesplace ± . g ( T = 0).Quantum confinement effects do manifest, however, also in the conductance: A clearseparation in the m dependence for even and odd values is evidenced in Figure 8(b).This originates from the parity dependence of the standing waves formed inside theCo layers by the interaction of the interface states at either sides. arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers
100 200 300 Temperature (K)-80-4004080120160200 M T P ( % )
100 200 300 Temperature (K) 100 200 300 Temperature (K) m =4 m =5 m =6 m =7 m =8 m =4 m =5 m =8 m =6 m =7 m =4 m =5 m =6 m =7 m =8 q = 6 ML q = 4 ML q = 8 ML m (ML) -80-4004080120160200 M C ( % ) q =4 q =6 q =8 • ( S AP − S P ) /S AP ( g P − g AP ) /g P • Magneto-thermopower and -conductance in
Cu[3(Co m /Cu q )/Co m ]Cu Figure 9.
Dependence of the magneto-thermopower (MTP/left) and the zerotemperature magneto-conductance (MC/right) on the Co layer thickness m (inMLs) in the Cu[3(Co m /Cu )/Co m ]Cu multilayered systems. Coming back to the Seebeck coefficient calculated for the two magneticalignments, P and AP, we note that, regardless of the explicit m and q values, largedifferences are predicted between S AP ( T ) and S P ( T ). Since, on the other hand, thesedifferences are not independent of T , a rather broad range of values can be expectedfor the MTP. The temperature dependent MTP ratios are shown in Figure 9 (leftpanel) for various Cu spacer thickness q and compared with the zero temperature MC(right panel) on an identical scale.As can be seen in this figure, the range of attained MTP values is much broaderthan that of the corresponding MC. The latter exhibits slight fluctuations with m , butit remains in an interval of 40-60 %, rather independent of the Cu spacer thickness q .Note that, as was the case for the N -varying systems, exceedingly large values of theMTP ratios, caused by S P ( T ) approaching zero, are not displayed. For temperatureshigher than 100 K, the MTP is obviously larger than the MC, essentially any MTPratio between 40 and 100 % being accessible by an appropriate ( m , q ) selection.While this is merely of theoretical interest, as not any ( m , q )-combination isnecessarily attainable experimentally or energetically stable, the results point out toan important aspect. As far as the electronic structure contribution to the transportproperties is concerned, using a thermal gradient rather than an electric field couldindeed be more advantageous in order to gain a large magnetic sensitivity in a magneticread-out device. In order to complete our discussion, we analyse how the change on the Cuspacer size, at a fixed Co thickness, influences the transport properties of theCu[3(Co m /Cu q )/Co m ]Cu multilayered systems. As illustrated by the transmissionprofiles shown in Figure 7(c) for fixed m = 8 MLs and varying q , the role of the Cuthickness in modelling the transmission is minor: Once a basic shape in the energydependence of the transmission is set by the given Co thickness m , i.e., by filteringand smoothening the QWS-related channels, no significant changes occur in T ( E ) as q increases. arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers (a) -4.0-3.0-2.0-1.00.01.02.03.0 S P ( µ V / K ) -12.0-8.0-4.00.04.08.0 S A P ( µ V / K ) q =4 q =5 q =6 q =7 q =8 m = 4 ML
100 200 300Temperature (K)-4.0-3.0-2.0-1.00.01.02.03.0 S P ( µ V / K )
100 200 300Temperature (K) -12.0-8.0-4.00.04.08.0 S A P ( µ V / K ) q =4 q =5 q =6 q =7 q =8 m = 8 ML • Parallel S P ( T ) Anti-parallel S AP ( T ) • (b) • g ( T = 0) C ondu c t a n ce g ( T = ) ( e / h ) PAP m = 4 ML q (ML)0.00.20.40.60.81.01.2 C ondu c t a n ce g ( T = ) ( e / h ) PAP m = 8 ML Seebeck coefficient and conductance in
Cu[3(Co m /Cu q )/Co m ]Cu Figure 10.
Same as Figure 8 but showing the variation of S P ( T ), S AP ( T ), and g ( T = 0) in Cu[3(Co m /Cu q )/Co m ]Cu with the Cu layer thickness q for m = 4(top) and m = 8 (bottom). Note that in panel (a) the same scale is used column-wise, that is, when comparing S P / AP ( T ) for varying Co thickness m , while thescales are different for S P ( T ) and S AP ( T ) at equal m . This is accordingly reflected in the temperature dependence of the Seebeckcoefficient for both P- and AP-alignments shown in Figure 10(a). The S P ( T ) and S AP ( T ) curves corresponding to various q values are grouped together over the wholetemperature range. The notable exception is the m = 4 MLs system in P-alignment,where the formed QWS-p-band complexes couple stronger across the Cu spacer. Incontrast, for m = 4 MLs in the AP-alignment the S AP ( T ) values of the differentspacer thickness are fairly close to another because the electron scattering is essentiallyspin-conserving. For the AP magnetic configuration this corresponds to an effectivespacer between the Co layers larger than the actual, physical one. For a Co thickness m = 8 MLs, equivalent to an absence of the QWS-p-band enabled channels, thevariations with q of both S P ( T ) and S AP ( T ) is even more reduced. This behaviouris consistent with the experimental findings of Shi et al [4] in Co/Cu multilayers ofcomparable thickness but in a CIP geometry.The corresponding zero temperature conductance results shown in Figure 10(b)follow a similar characteristic of a rather weak dependence on q at a fixed m . We notein particular the complete absence of any oscillations in g ( T = 0) between odd andeven q , as was the case of varying m .In spite of the much smaller spread over varying q of both S P ( T ) and S AP ( T ) atfixed m = 8 MLs, the MTP ratio still exhibits a broad range of values, as shown inthe left panel of Figure 11. Above (cid:39)
100 K, however, the temperature dependence arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers
100 200 300 Temperature (K)020406080 M T P ( % ) M C ( % ) q =4 q =5 q =6 q =7 q =8 • ( S AP − S P ) /S AP ( g P − g AP ) /g P • Cu[3(Co /Cu q )/Co ]Cu Figure 11.
Dependence of the MTP (left panel) and the zero temper-ature MC (right panel) on the number of Cu layer thickness q in theCu[3(Co m /Cu q )/Co m ]Cu multilayered systems. of the MTP for a fixed q is much weaker, even for small values of the Cu spacerthickness. In contrast to the MTP, the MC ratio (right panel) is almost independentof q , showing small fluctuations around 50 %. This proves, once again, that the MTPoffers in principle a much larger sensitivity to small changes in the electronic structurethan the MC. It also implies that the reproducibility of independent experimentaldata might turn into a problematic issue. In addition, we note that the analysis ofthe results of this section, encompassing different ( m, q ) combinations, does not leadto any obvious correlation between the MC and the MTP, nor does enable us to makea definite statements about one configuration being better suited than another for anenhanced MTP.
6. Conclusions
In summary, we have presented results of ab initio calculations of the magneto-thermoelectric properties for a series of Co/Cu multilayered systems embedded inCu(001) with the general formula Cu[( N − m /Cu q )/Co m ]Cu. Our investigationsfocused on the influence the various morphological parameters — number of repeats N ,layer thickness m and q — have upon the underlying electronic structure and, throughthe induced modifications, on the various transport properties of the heterostructures.While adopting a spin-polarised fully relativistic formalism, we have neverthelessfound that the electronic transmission in the Co/Cu multilayers is to a large extentspin-conserving. For thin Co layers ( m ≤ m occur, leading toa large sensitivity of the Seebeck coefficient and the magnetothermopower (MTP) tothe thickness of the Co layers. The other geometrical parameters, N and q , have amuch smaller influence on the transport properties. We need to emphasise, however, arge morphological sensitivity of the magnetothermopower in Co/Cu multilayers Acknowledgments
This work was supported by the German Research Foundation (
DeutscheForschungsgemeinschaft – DFG ) within the Priority Program 1538 ”Spin CaloricTransport (SpinCaT)”. The authors gratefully acknowledge the computing timegranted by the John von Neumann Institute for Computing (NIC) and providedon the supercomputer JUROPA at J¨ulich Supercomputing Centre (JSC). Additionalcomputer facilities have been offered by the Center for Computational Sciences andSimulation (CCSS) at the University Duisburg-Essen.
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