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Mechanism of Interlayer Exchange Coupling in Fe/Nb Multilayers

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### Density Functional Theory

### Extension to spin-polarised systems

Mechanism of Interlayer Exchange Coupling in Fe/Nb Multilayers

R. Prasad

Department of Physics

Indian Institute of Technology

Kanpur, India

Collaborators:N. N. Shukla

A. Sen

Phys. Rev. B 70, 014420 (2004)

Introduction

RKKY model

Quantum Well model

Density Functional Theory

Results

Mechanism of the IEC

Conclusions

Exchange Coupling in Multilayers

D

J

Thickness (D)

- Damped oscillations.

S. S. P. Parkin, N. More, and K. P. Roche, Phys. Rev. Lett. 67, 1602 (1991)

RKKY interaction in 3D

Two impurity atoms embedded in a host metal matrix may interact via the RKKY interaction.

Wave length λ=2L/n

L is thickness of the well and n -> energy level

L = Nd ; N -> number of spacer ML, d = interlayer spacing

K=2π/λ = 2πn/Nd

FM

AF

Qiu et al, Phys. Rev. B 46, 8659 (1992)

Mechanism of exchange coupling

- In the RKKY model, the coupling arises from the polarization of electrons in the spacer, while in the QW model it arises from quantum interference effects inside the well.
- Both models predict the same period.
- First-principles calculation can play an important role in elucidating the mechanism of interlayer exchange coupling.

Hohenberg and Kohn, 1964

1. The ground state energy E of an inhomogeneous

interacting electron gas is a unique functional of the

electron density .

2. The total energy E{} takes on its minimum value for the

true electron density.

Exc= exchange-correlation energy

T0 = Kinetic energy of a system with

density without electron-electron interaction

Minimize E subject to the condition

Local density approximation (LDA)

= contribution of exchange and correlation to the

total energy per particle in a homogeneous but

interacting electron gas of density ρ

E

=

s

xc

v

dr

xc

s

Von Barth and Hedin 1972

Rajagopal and Callaway 1973

for uniform spin directions (σ = or )

niσ = Occupation no.

Local spin density approximation (LSDA)

εxc= exchange correlation energy per particle of a

homogeneous, spin-polarized electron gas with density ρ, ρ.

Beyond the LSDAFor higher accuracy, need to go beyond the LSDAGradient expansion approximation (GEA)Kohn and Sham 1965, Herman 1969 For slowly varying densities, the energy functional can be expanded as a Tylor series in terms of gradient of the densityFor real system GEA often is worse than LSDAGeneralized Gradient Approximation (GGA)Ma and Bruckner ; Langreth, Perdew, Wangwhere f is chosen by some set of criteria.Many function have been proposed : Perdew - Wang 1986 (PW86)Becke 1988 (B88)Perdew and Wang 1991 (PW91)

Some methods to solve K-S equations

- Korringa–Kohn–Rostoker (KKR) method
- Linear-Muffin-Tin-Orbitals (LMTO) method
- Augmented-Plane-Wave (APW) method
- Full Potential Linearized Augmented Plane Wave (FLAPW) method
- Pseudo potential method
- Tight-binding method

APW method

Slater (1937) Phys. Rev. 51, 846

ul is the regular solution of

Alm and CG are expansion coefficients, El is a parameter

- APWs are solutions of schrodinger’s equation inside the sphere but only at energy El
- Energy bands (at a fixed k-point) can not be obtained from a single diagonalization.

LAPW method

Andersen (1975) PRB, 12, 3060

The energy derivative, satisfies

Blm are coefficients for energy derivative.

Error of order in Wave function

Error of order in the band energy

Andersen (1975) PRB, 12, 3060

Andersen and Jepsen (1984) PRL, 53, 2571

Partitioning of the unit cell into atomic sphere (I) and interstitial regions (II)

Inside the MT sphere, an eigen state is better described by the solutions of the Schrödinger equation for a spherical potential:

The function satisfies the radial equation:

The only boundary condition: be well defined at

The basis functions can now be constructed as Bloch sums of MTO:

An LMTO basis function in terms of energy and the decay constant may be expressed as:

Here and represent the Bessel and Neumann functions respectively.

Since the energy derivative of vanishes at for it leads to:

In the atomic sphere approximation (ASA), the LMTO’s can be simplified as :

where is given by :

is chosen such that and its energy derivative matches continuously to the tail function at the muffin-tin sphere boundary.

Disadvantages of LMTO-ASA method :

- It neglects the symmetry breaking terms by discarding the non-spherical parts of the electron density.
- The interstitial region is not treated accurately as LMTO replaces the MT spheres by space filling Wigner spheres.

- Provides a way of exploring the coexistence of
- ferromagnetism and superconductivity.
- Strong dependence of the superconducting transition
- temperature on Fe layer thickness.
- Strong exchange coupling which changes in a continuous
- and reversible way by introducing hydrogen in sample.

Computaional Details:

- 1. All calculation are carried out using FLAPW/
- TB-LMTO method within LSDA and GGA.

2. To perform this calculation we constructed

tetragonal supercells.

- 3. Lattice parameters : a = b = 3.067 Å , and
- c = 3.067 Å to 12.269 Å.
- 4. The exchange coupling J is calculated by taking
- the energy difference
- where d is the thickness of the spacer layer.

d

IEC is FM for Nb

Thickness less

Than 14.0 Å.

Phys. Rev. Lett. 68, 3252 (1992)

Period = 6.0 Å.

Phys. Rev. B 70, 014420 (2004)

Phys. Rev. B 70, 014420 (2004)

Fe magnetic moments reduced 25% of bulk value (Expt. 40%)

The calculated oscillatory interlayer exchange coupling (solid circles) as a function of the number of Nb spacer layers in the Fe3Nbn (n=1-16) multilayer system. The solid line is the fitted plot.

T1 = 4.14 ML (6.3 Å)

T2 = 5.05 ML (7.7 Å)

T3 = 2.86 ML (4.4 Å)

T4 = 20.28 ML (31.1 Å)

Cross sections of the Fermi surface of Nb in the (100) plane. Гlabels the center of the Brillouin zone, N indicates the center of each face of the dodecahedron and H labels the corners of the four-fold symmetry on the zone boundary.

Higher harmonics and the “Vernier” periods

T4 in terms of T3

T3 = 2.86 ML

T4 = 20.43 ML (from T3)

T4 = 20.28 ML (Calculated)

The interlayer exchange coupling in Fe/Nb multilayers as a function of Fe magnetic moment parameterized by α. A nonlinear fit shows the RKKY character up to α =0 .6, and a non-RKKY behaviour at higher α values. The asterisks represent Ex of the 16-atom supercell; triangles and diamonds short-period amplitudes; circles and squares long-period ones. We carry out the exercise initially for a 54-atom Fe/Cr supercell and obtain a similar behaviour as outlined in Harrison’s paper [ Phys. Rev. Lett. 71 3870 (1993)]

Nb moments are more ferromagnetically aligned away from the interface as Fe thickness is increased.

Biasing in moments is due to non-RKKY terms.

J. Mathon et al, Phys. Rev. B 59, 6344 (1999)

The calculated bulk energy bands of Nb and bcc Fe ( ↑ and ↓) along the [100] direction.

Confinement of electrons QW state

The calculated bulk energy bands of Nb and bcc Fe ( ↑ and ↓) along the [110] direction

Only majority-spin states in Nb exhibit quantum well character at the Fermi level since the minority-spin Δ2 and Σ1 states couple with the corresponding states in Fe.

Oscillations in the Density of states at the Fermi level

QW Periods 4.6 Å and 6.1 Å (see Fig. 7)

Interlayer coupling periods 4.4 Å and 6.3 Å (see Fig. 1)

║

Spin-polarized QW states interlayer magnetic coupling

QW state gets wider as Nb thickness is increased.

Oscillations in the density of states at the Fermi level, EF, with the Nb spacer thickness, caused by the quantum well states in Fe3Nbn (n = 1--16) heterostructures.

QW state gets wider as Nb thickness is increased.

This gives

for QW period = 4.6 Å

for QW period = 6.1 Å

B

FM

NM

FM

d

Phase Accumulation Model (PAM)

The total phase accumulation must be an integral multiple of 2π.

This is nothing more than the problem of a particle in a box of width d, with and embodying the wave function matching condition at the boundaries of the box

The condition for a quantized state of quantum well (QW) of width d, is

, n being the number of nodes in the wave function within the NM

layer perpendicular to the surface.

For multilayers, we may write

, where I stands for interface

This gives,

, for QW states at the Fermi level

We know,

since

(2) – (1) yields:

Phase Accumulation Model (Contd.)

The quantization condition for existence of QW state:

ΦI can be approximated as

EU and EL represent the upper and lower energies of the potential well

where

m* is the electron effective mass; V0 is a constant offset of the periodic potential; 2U is the energy gap at the zone boundary.

The thickness dependence of the QW energies in Fe/Nb multilayers generated by Eq. (12) of the phase accumulation model with respect to (a) Δ2 and (b) Σ1 bands.

A fit to the self-consistently calculated Δ2 band of Nb yields U = 2.05 eV,

V0 = -9.85 eV and m* = 1.08me, where me is the electron mass. On the other

hand, upon fitting Σ1 of Nb, we have m* = 1.05me, U = 1.66 eV and V0 = -5.5 eV.

The quantization condition for existence of QW state:

ΦI can be approximated as

EU and EL represent the upper and lower energies of the potential well

yieldsG = 24.65 eV, U = 4.93 eV

- First principles calculations agree reasonably well with the experimental results.
- Long period appears to be the “Vernier” period in favor of QW mechanism.
- Analysis of IEC by artificially changing Fe magnetic moments supports QW mechanism.
- DOS at EF shows oscillation with periods 4.6Å and 6.1Å .
- QW period of 4.6Å yields KF/KBZ = 0.33 in agreement to the corresponding value of 0.30 from bulk bands topology in [100] direction.
- Oscillations in induced magnetic moments in Nb shows a ferromagnetic bias as the thickness of Fe layer is increased.
- The phase accumulation model provides a reasonable quantitative description in favor of QW mechanism.

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