Exchange coupling and GMR in Magnetic Multilayers

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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)
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Plan of the talk

• Introduction
• RKKY model
• Quantum Well model
• Density Functional Theory
• Results
• Mechanism of the IEC
• Conclusions

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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)
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Giant Magnetoresistance (GMR)
.
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RKKY model
RKKY interaction in 3D
Two impurity atoms embedded in a host metal
matrix may interact via the RKKY interaction.
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RKKY interaction in layered systems
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RKKY Theory for fcc lattices
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Quantum well model
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Quantum Well (QW) model
Wave length ?2L/n
L is thickness of the well and n -gt energy level
L Nd N -gt number of spacer ML, d interlayer
spacing
K2p/? 2pn/Nd
FM
AF
Qiu et al, Phys. Rev. B 46, 8659 (1992)
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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.

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Density Functional Theory

• 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

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Kohn-Sham Equation
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 ?
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Extension to spin-polarised systems

• Von Barth and Hedin 1972
• Rajagopal and Callaway 1973
• for uniform spin directions (s ? or ? )
• nis Occupation no.
• Local spin density approximation (LSDA)
• exc exchange correlation energy per particle of
  a
• homogeneous, spin-polarized electron gas with
  density ??, ??.

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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)
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Some methods to solve K-S equations

1. KorringaKohnRostoker (KKR) method
2. Linear-Muffin-Tin-Orbitals (LMTO) method
3. Augmented-Plane-Wave (APW) method
4. Full Potential Linearized Augmented Plane Wave
   (FLAPW) method
5. Pseudo potential method
6. Tight-binding method

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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 schrodingers equation
  inside the sphere but only at energy El
• Energy bands (at a fixed k-point) can not be
  obtained from a single diagonalization.

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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
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LMTO Method
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
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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
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In the atomic sphere approximation (ASA), the
LMTOs 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

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

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Fe/Nb Multilayers

• 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.

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Fe/Nb Multilayers
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
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FLAPW
IEC converges faster than total energy.
Phys. Rev. B 70, 014420 (2004)
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FLAPW
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)
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FLAPW
Phys. Rev. B 70, 014420 (2004)
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TB-LMTO
Fe magnetic moments reduced ? 25 of bulk value
(Expt. ? 40)
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Mechanism of exchange coupling ?
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TB-LMTO
The calculated oscillatory interlayer exchange
coupling (solid circles) as a function of the
number of Nb spacer layers in the Fe3Nbn
(n1-16) 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)
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Cross sections of the Fermi surface of Nb in the
(100) plane. ? labels 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.
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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)
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The interlayer exchange coupling in Fe/Nb
multilayers as a function of Fe magnetic moment
parameterized by a. A nonlinear fit shows the
RKKY character up to a 0 .6, and a non-RKKY
behaviour at higher a 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
Harrisons paper Phys. Rev. Lett. 71 3870
(1993)

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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)
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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 ?2 and S1 states couple with the
corresponding states in Fe.
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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.
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QW Periods ? 4.6 Å and 6.1 Å
QW state gets wider as Nb thickness is increased.
This gives
for QW period 4.6 Å
for QW period 6.1 Å
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Phase Accumulation Model (PAM)
The total phase accumulation must be an integral
multiple of 2p. 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
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Phase Accumulation Model (Contd.)
The quantization condition for existence of QW
state
FI 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) ?2
and (b) S1 bands.
A fit to the self-consistently calculated ?2 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 S1 of Nb, we have m
1.05me, U 1.66 eV and V0 -5.5 eV.
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Phase Accumulation Model
The quantization condition for existence of QW
state
FI can be approximated as
EU and EL represent the upper and lower energies
of the potential well
yields G 24.65 eV, U 4.93 eV
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Conclusions

• 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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Thank You
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