1

Project no. ICT-257159

Project acronym:

MACALO

PROJECT TITLE: MAGNETO CALORITRONICS

Area: Nanoelectronics Technology (ICT-2009.3.1)

2st Intermediate report

Deliverable EVAL-2

Due date of deliverable: M24 Actual submission date: 1/9/2012

Start date of project: 01/09/2010 Duration: 36 Months

Organization name of lead contractor for this deliverable: UT Del. no. Deliverable

name WP no. Nature Dissemination

level1

Delivery date (proj month)

D5.2 EVAL-2 5 RTD PU 24

1 PU = Public

PP = Restricted to other programme participants (including the Commission Services). RE = Restricted to a group specified by the consortium (including the Commission Services). CO = Confidential, only for members of the consortium (including the Commission Services). Make sure that you are using the correct following label when your project has classified deliverables. EU restricted = Classified with the mention of the classification level restricted "EU Restricted" EU confidential = Classified with the mention of the classification level confidential " EU Confidential " EU secret = Classified with the mention of the classification level secret "EU Secret "

2

Milestones and expected results M-EVAL-i: Ab-initio study of resistivity, Gilbert damping and spin-flip diffusion length for magnetic multilayers (month 12) M-EVAL-ii: Study of the influence of thermal disorder on the resistivity, Gilbert damping and spin-flip diffusion length for Fe, Co, Ni and Heusler alloys (month 24) M-EVAL-iii: Ab-initio study of Gilbert damping enhancement (month 36) Deliverables: (brief description) and month of delivery EVAL-a: Report the results of calculations of the transport properties of magnetic multilayers (month 12) EVAL-b: Report on the influence of thermal disorder on transport parameters (month 24) EVAL-c: Report on analysis of the factors influencing resistivity, Gilbert damping and spin-flip diffusion length for binary alloys (month 36)

3

The Influence of Thermal Disorder on Transport Parameters

1. Introduction

In the field of spintronics, that is driven by room temperature applications, most

interest is naturally focussed on the determination of models relevant to room temperature

(RT) operation. Many of the parameters used in these models were determined in low

temperature experiments and it is not known how different they are at RT.

Magnetocaloritronics is the result of a realization that temperature and temperature gradients

can strongly influence spin dependent transport. One of the main aims of Macalo is to

provide more insight into this temperature dependence and to translate these insights into

improved modelling of spin devices.

First-principles calculations yield material-specific results but are of necessity (almost

always) performed assuming that the temperature T=0. They are then evaluated by comparing

the results with experiments carried out at low-temperatures. In workpackage EVAL, we

address the possibility of calculating the temperature dependence of the parameters of

importance to describe spin-transport in materials that are currently used in spin devices such

as the ferromagnetic metals (FM) Fe, Co and Ni, ferromagnetic alloys such as FexNi1-x or

artificial materials with large perpendicular anisotropy such as Co|Ni magnetic multilayers.

2. Modelling Disorder

The most extensive recent studies of temperature dependent transport are based upon

the so-called Lowest Order Variational Approximation (LOVA) whereby the linearized

Boltzmann equation is solved using first-principles electronic structure calculations to

calculate energy bands, wave functions and related properties and first-principles phonon

calculations to determine the electron-phonon interaction. Such calculations have been

successfully applied to the calculation of the electrical and thermal conductivities of about a

dozen elemental non-magnetic metals.

The complexity of the LOVA calculations makes it difficult to extend them to study

the transport properties of complex materials or even to simple itinerant magnetic materials.

We have instead developed a radically different approach that allows us to flexibly model

different types of disorder and study a broad class of transport properties. It is based upon the

Landauer-Büttiker or scattering formalism of electronic transport whereby crystalline leads

are attached to the left (L) and right (R) sides of a scattering (S) region and we study how

Bloch states in the leads are elastically scattering by different types of disorder in the

4

scattering region [Figure 1]. Our implementation is based upon density functional theory so

we do not depend upon any experimental input for our studies. We use a very efficient

numerical basis set of tight-binding muffin-tin orbitals (TB-MTOs) that when combined with

extensive use of sparse matrix techniques allows us to study scattering regions containing of

order 104 atoms. Recent developments allow us to include spin-orbit coupling and handle

non-collinear magnetic configurations. By calculating the full scattering matrix, we make

contact with a large body of mesoscopic transport theory allowing us to study previously

inaccessible parameters such as the spin-flip diffusion length or the Gilbert damping

parameter.

Figure 1. Sketch of the configuration used in the Landauer-Büttiker transport formulation to calculate the two terminal conductance. A (shaded) scattering region (S) is sandwiched by left- (L) and right-hand (R) leads which have full translational symmetry and are partitioned into principal layers perpendicular to the transport direction. The scattering region contains N principal layers but the structure and chemical composition are in principle arbitrary.

3. Temperature dependent resistivity of non-magnetic metals: Cu

We illustrate our procedure with a calculation of the temperature dependence of the

resistivity of the non-magnetic metal (NM) Cu that results from the electron-phonon

interaction. The basic physical assumption is that the dominant scattering of electrons by

phonons is elastic and that there is a separation of electronic and ionic time scales, the

conventional Born-Oppenheimer approximation that should hold well for a material like

copper. The left hand panel of Figure 2 shows the phonon dispersion relationship calculated

from first principles using the linear response formalism. The red dots are experimentally

measured points, the continuous lines are the results of calculations within the generalized

gradient approximation (GGA). The vibration frequencies we calculate are lower than those

observed. We can understand this because the GGA yields an equilibrium lattice constant of

3.67 Å that is larger than the experimental value of 3.60 Å. Using the local density

approximation (LDA) results in an equilibrium lattice constant 3.55 Å that is smaller than

5

experiment and vibration frequencies that are higher than those observed. When we thermally

occupy the phonon spectrum, the lower (higher) vibration-frequency spectra result in larger

(small) root mean square (rms) vibration amplitudes that give rise to more (less) scattering

than observed experimentally; the experimentally determined resistivity values (above 200 K)

lie between the two calculated curves (right hand panel of Figure 2). The overall agreement is

good. We emphasize that though we make many approximations, our theory does not contain

any adjustable parameters. The discrepancies seen at low temperature result from the

unconstrained way we populate the phonon modes which overestimates the scattering in this

limit. We are currently studying how to improve our description of the low temperature limit.

0 100 200 300 400 500

Temperature (K)

0

1

2

3

Res

istiv

ity �!"#�

cm

)

0 100 200 300 400 5000

1

2

3

GGA a0=3.67 ÅLDA a0=3.55 ÅExpt. a0=3.60 Å

Figure 2. Left panel: Phonon dispersion of fcc Cu calculated from first principles using the linear response formalism. Right panel: Resistivities determined as a function of temperature using the phonon dispersion relation calculated within the generalized gradient (GGA) and local density approximations (LDA). 4. Temperature dependent spin-diffusion length and resistivity of Pd and Pt

The spatial decay of a non-equilibrium magnetization introduced by injecting a spin-

polarized current from a magnetic into a non-magnetic material is described in the diffusive

limit by the spin-flip diffusion length, lsf . Experimental techniques used to measure the spin

Hall conductivity and spin Hall angle in Pt that depend on the value of lsf have focussed

attention on discrepancies between the values of lsf determined for Pt under different

experimental conditions, in particular at low and high temperatures. We can evaluate lsf in our

scattering formalism by studying how a polarized current injected through an FM|NM

interface decays as a function of the distance from the interface when spin-orbit coupling and

temperature-dependent lattice disorder are taken into account. It is interesting in this context

to study not only the 5d transition metal element Pt but also the 4d element Pd which has the

same electronic and crystal structure as Pt but a weaker spin-orbit interaction by virtue of

being lighter.

� X W X K � L0

10

20

30

� (m

eV)

fcc Cu

6

We begin by modelling the temperature-dependent lattice disorder of Pd and Pt as we did for

Cu, by calculating the first-principles phonon dispersion relations within the GGA for Pd and

LDA with spin-orbit coupling (SOC) for Pt. These are shown in Figure 3. In the case of Pd,

we find an equilibrium lattice constant of 3.98 Å, which is larger than the experimental value

3.89 Å so the calculated phonon frequencies are lower than those observed. In the case of Pt,

LDA+SOC yields the same lattice constant (3.92 Å) as the experimental value. This leads to

a better agreement between our calculated phonon dispersion and the experimental

measurements.

Figure 3. Phonon dispersion of fcc Pd (left panel) and of Pt (right panel) calculated from first principles using the linear response formalism and the Generalized Gradient Approximation. For a chosen value of temperature T, we calculate the resistance R of a length L of scattering

region as a function of L. When this is linear, we are in the diffusive regime and can define

the resistivity ρ, for which R=ρL. ρ(T) calculated in this way is shown in the upper panels of

Figure 4 for Pd (left panel) and Pt (right panel) where a comparison is made to the results of

LOVA calculations [S.Y. Savrasov and D.Y. Savrasov, PRB54, 16487 (1996)] and to

experimental values cited by these authors. As was the case for Cu, the agreement between

calculated and measured resistivities is very good.

0

10

20

� (µ�

cm

) Expt.Savrasov

fcc Pd

0 100 200 300 400 500Temperature (K)

0

20

40

l sf (n

m)

Morota

0

20

40

� (µ�

cm

) KimuraAndoMosendzMorotaLiu

fcc Pt

0 100 200 300 400 500Temperature (K)

0

5

10

15

l sf (n

m)

Figure 4. Phonon dispersion of fcc Pd (left panel) and Pt (right panel) calculated from first principles using the linear response formalism and the Generalized Gradient Approximation.

� X W X K � L0

10

20

30

� (m

eV)

fcc Pd � X W X K � L0

10

20�

(meV

)

fcc Pt

7

In the lower panels of Figure 4, we show the values of lsf extracted from injecting spin-

polarized currents into the NM metal. Our calculated values of lsf are proportional to the

conductivity which is in turn proportional to 1/T (ρ is proportional to T). This is in agreement

with the Elliott-Yafet model which was derived from considerations of the splitting of

degenerate bands by spin-orbit interaction. Unlike the Elliott-Yafet model, we take account

of the splitting of energy bands throughout the Brillouin zone and are able to calculate

absolute values of lsf reliably [1]. The values we calculate for lsf at RT support a number of

measurements (shown as symbols in Figure 4) but do not provide an explanation for the very

low room temperature value suggested by the Cornell group (Liu et al. in Figure 4); we

believe these low values may be determined by inhomogeneity of the system studied in

experiment where interfaces may play a dominant role.

5. Temperature dependent resistivity and Gilbert damping in Fe, Co and Ni

In an exploratory study of the effect of the electron-phonon interaction on

magnetization relaxation for Fe, Co, and Ni [2], we adopted a Debye like model of lattice

disorder in which the root mean square displacement of the atoms Δ is given by

Δ2 = | u |2 = 92

MkBΘD

T 2

ΘD2

x

ex −1dx

0

ΘD

T∫ + 14

⎛

⎝⎜⎜

⎞

⎠⎟⎟

in terms of a Debye temperature ΘD and M is the mass of atoms. We displaced the atoms

(and their potentials) in the scattering region rigidly and randomly from their ideal lattice

positions with a Gaussian distribution characterized by Δ (or ΘD which can be obtained from

measured phonon spectra). This “frozen thermal lattice disorder” approach successfully

reproduces the nonmonotonic damping behaviour observed in ferromagnetic resonance

measurements and yields reasonable quantitative agreement between calculated and

experimental values; see Figure 5.

We model the spin disorder in an analogous fashion by assuming a Gaussian

distribution of polar angles θ of the magnetic moments with respect to the global quantization

axis, together with a uniform random distribution in the azimuthal angle ϕ. The rms value of

θ can be fixed empirically by requiring the total magnetization of the system to match the

experimental M(T). The results of this study show that a simple relaxation time does not

uniquely characterize disorder; the red curve in the top right panel of Figure 5 shows that

amounts of lattice disorder and spin disorder that yield the same resistivity values result in

8

different damping values. This finding underlines the need to develop a unified treatment of

thermally induced disorder.

Figure 5. Left panels: Gilbert damping (black) and resistivity (red) for bcc Fe, hcp Co, and fcc Ni calculated as function of the rms atomic displacements measured in units of the corresponding lattice constants, a. The error bars reflect the configuration spread. Experimental resistivities are used to label a number of resistivity values with a temperature (blue). Right panels: Because we do not know what the temperature dependence of the magnetization was in the corresponding experiments, we plot the Gilbert damping frequency λ rather than α as a function of resistivity for bcc Fe, hcp Co, and fcc Ni. Calculated results are shown as lines: for frozen thermal lattice disorder as black solid lines and for frozen spin disorder as a red dashed line (Fe only). Symbols are experimental damping values. We do this for lattice disorder by calculating the phonon spectrum of Fe in the same way as

we did above for Cu, Pd and Pt. To treat spin disorder, we extract distance-dependent

exchange interaction parameters from the total energies of (periodic) spin spirals calculated

self-consistently as a function of the spin-spiral pitch. These parameters are used to calculate

the magnon spectrum of Fe which is compared to experiment in Figure 6. For both phonons

and magnons, temperature is introduced in the form of temperature-dependent population of

the corresponding modes. This allows us to construct scattering regions with temperature

dependent lattice and spin disorder and carry out the scattering calculations. The resistivities

resulting from only phonons, only magnons, and both are shown in Figure 7, with the

experimental data for comparison. The resistivity obtained from having simultaneously

9

phonons and magnons agrees quite well with the experimental measurements and is

substantially larger than the sum of the resistivities from phonons and magnons calculated

separately.

� H N � P N0

100

200

300

400

� (m

eV)

Fe(12%Si)@ RT

Pure Fe@ 10 K

Figure 6: Comparison of magnon spectrum of Fe calculated from first-principles (line) and measured for pure Fe (blue circles) and for Fe doped with Si (purple circles).

0 100 200 300 400 500 600Temperature (K)

0

10

20

30

40

Elec

trica

l res

istiv

ity l

(µ1

cm

)

lexplph+mglmglph

lexp

lph+mg

lph

lmg

Figure 7: Temperature-dependent resistivity of Fe calculated with only phonons (purple upward-pointing triangles), only magnons (blue right-pointing triangles) and both (red diamonds). The experimental data are shown for comparison (black circles).

6. Temperature-dependent magnetoelectric properties of half-metallic NiMnSb Half-metallic Heusler alloys such as NiMnSb are very attractive materials in

spintronics because the electronic structure at the Fermi level is metallic or semiconducting

depending on the spin. Because of their low Gilbert damping, Heusler alloys have been

10

proposed to improve the performance of spin-torque oscillators. Two nodes in the Macalo

network are focusing on using NiMnSb in spin-torque devices.

Introducing spin-orbit coupling will in principle destroy the half-metallic

ferromagnetic property since it mixes the metallic and semiconductor spin channels. To more

quantitatively determine how it influences the transport and magnetization dissipation, we

calculate the temperature dependence of the resistivity, current polarization and Gilbert

damping of NiMnSb, modelling intrinsic temperature effects using the frozen thermal lattice

and spin disorder schemes discussed above. A Debye temperature of ΘD=312 K obtained

from the experimental phonon spectrum at room temperature [C. N. Borca et al., Appl. Phys.

Lett. 77, 88 (2000)] was used to describe the lattice disorder. The rms value of θ was chosen

to make the total magnetization of NiMnSb system match the experimental M(T) [C.

Hordequin et al., JMMM 162, 75 (1996)]. Since the Mn atoms account for most of the

magnetism (3.7 µB out of 4 µB per unit cell), we rotate only the Mn magnetic moments in the

calculation, checking that the rotation of other atoms does not change any results reported

here.

0 10 20 30 40 50

L (nm)0

5

10

15

AR (f1

m2 )

Lattice and spin disorderLattice disorder

300 K

0 10 20 30 40 50

L (nm)0

0.2

0.4

0.6

0.8

1

J m/J

Lattice disorderLattice and spin disorder

300 K

Figure 8. Left panel: Total resistance as a function of the length L of the scattering region at 300 K. The lines are linear fits leading to resistivity values ρ=22.6±0.6 µΩ cm with lattice disorder (black dashed line) and ρ=24.4±0.7 µΩ cm with lattice and spin disorder (red solid line). Right panel: Fractional spin-polarized current densities Jσ/J. Spin disorder increases the resistivity of the majority channel but decreases that of the minority channel by introducing spin-flip scattering. Therefore, although it does not much increase the total resistivity, it decreases the current polarization from 0.941±0.004 to 0.829±0.014. Figure 8 shows that spin disorder has a relatively small effect on the resistivity of

NiMnSb. Where spin disorder is important is - unsurprisingly - in determining the transport

polarization that is seen to decrease quite quickly with temperature in Figure 9; tracking the

magnetism of NiMnSb whose Curie temperature of ~730 K corresponds to disordering of

(robust) local Mn moments. Figure 10 shows that spin disorder also does not modify the

11

temperature dependence of the Gilbert damping constant significantly. The value calculated

with intrinsic disorder only is at the bottom of the range of reported measured values.

0 100 200 300 400Temperature (K)

0

10

20

30

40

Resis

tivity

(µ1

cm

)

Hordequin et al.Otto et al.Wang et al.Lattice disorderLattice and spin disorder

0 100 200 300 400

Temperature (K)

0.7

0.8

0.9

1

Cur

rent

pol

ariz

atio

n

0 0.2 0.4Ms-M(T) (µB)

0.7

0.8

0.9

1

Figure 9. Resistivity (left panel) and current polarization (right panel) of NiMnSb as a function of temperature. Inset: With spin disorder, current polarization shows a linear dependence on Ms-M(T). The green straight line is a guide to the eye.

0 10 20 30 40 50

L (nm)0

0.05

0.1

0.15

G/aM

sA (n

m)

Lattice disorderLattice and spin disorder

300 K

0 100 200 300 400

Temperature (K)0.002

0.003

0.004

0.005

0.006

0.007

_

Lattice disorderLattice and spin disorder

Figure 10. Left panel: Gilbert damping as a function of the length L of the scattering region. The linear fitting yields a parameter α=0.00334±0.00029 with lattice disorder and α=0.00333±0.00035 with lattice and spin disorder. The experimental data are 0.00333—0.00474 [Andreas Riegler, Ph.D. thesis, Universitat Wurzburg, 2011]. Right panel: Gilbert damping of NiMnSb as a function of temperature. Spin disorder has very little effect on α. The green bar shows the experimental values.

6. Summary and Plan We have developed a novel scheme for calculating transport properties of magnetic materials such as the resistivity, transport polarization, spin-flip diffusion length and Gilbert damping parameter and illustrated it with applications to materials of current interest. To be able to address crystalline materials, we introduced thermally induced lattice and spin disorder which yielded such promising results that we extended these empirical schemes by calculating phonon and magnon spectra from first principles. This scheme has been successfully applied to the non-magnetic materials Cu, Pd and Pt and a first application to crystalline Fe was presented. We intend to extend these studies in a number of ways: for example, to more complex materials and to the low temperature regime.

12

References to recent publications:

1. Unified First-Principles Study of Gilbert Damping, Spin-Flip Diffusion, and Resistivity in Transition Metal Alloys Anton A. Starikov, Paul J. Kelly, Arne Brataas, Yaroslav Tserkovnyak, and Gerrit E.W. Bauer Phys. Rev. Lett. 105, 236601 (2010).

2. First-principles calculations of magnetization relaxation in pure Fe, Co, and Ni with frozen thermal lattice disorder Yi Liu, Anton A. Starikov, Zhe Yuan, and Paul J. Kelly Phys. Rev. B 84, 014412 (2011).

3. Ab Initio Calculation of the Gilbert Damping Parameter via the Linear Response Formalism

H. Ebert, S. Mankovsky, and D. Ködderitzsch, and P.J. Kelly Phys. Rev. Lett. 107, 066603 (2011).

4. Spin pumping and spin transfer

A. Brataas, Y. Tserkovnyak, G.E.W. Bauer, and P.J. Kelly, in Spin Current: Vol.17 of Semiconductor Science and Technology, edited by S. Maekawa, S.O. Valenzuela, E. Saitoh and T. Kimura, Oxford Science Publications. (Oxford, 2012), p87-135.

5. Spin-orbit-coupling induced domain-wall resistance in diffusive ferromagnets Zhe Yuan, Yi Liu, Anton A. Starikov, Paul J. Kelly, and Arne Brataas Phys. Rev. Lett. submitted (2012).

6. First-principles calculation of a temperature dependent spin-flip diffusion length for Pd and Pt. Yi Liu, Anton A. Starikov, Zhe Yuan, and Paul J. Kelly in preparation.

7. Temperature dependent transport properties of the ferromagnetic elements Fe, Co, and Ni from first-

principles. Yi Liu, Zhe Yuan, Anton A. Starikov, and Paul J. Kelly in preparation.