Magnetotransport in nanostructures

Universidade Federal do Rio de Janeiro Instituto de Física

José d’Albuquerque e Castro

PAN AMERICAN ADVANCED STUDIES INSTITUTE Ultrafast and Ultrasmall; New Frontiers and AMO Physics

March 30 - April 11, 2008

Nanostructures

•  Structure and composition: nanometer scale ⇒  ultra fine films and multilayered structures ⇒  quantum wires and dots ⇒  granular systems etc.

•  Main interest

⇒  distinct physical properties ⇒  confinement effects (quantum interference) ⇒  possibility of controlling their physical properties ⇒  technological applications

•  1988: Albert Fert/Peter Grünberg (Nobel Prize 2007)

Giant magnetoresistance

AF FM

H=0 H

two currents

AF FM

Source of the effect: spin dependent scattering

Giant magnetoresistance

Giant magnetoresistance

•  Source of the effect: spin dependent scattering

•  It may occur in both regimes: ⇒ diffusive ⇒ ballistic

•  Diffusive regime: the usual approach is based on the Boltzmann formalism

⇒ R. E. Camley and J. Barnás, PRL 63, 664 (1989) ⇒ R. Q. Hood and L. M. Falicov, PRB 46, 8287 (1992)

Boltzmann theory

Semiclassical theory of transport

⇒ Bloch states

•  crystalline system: H0

translational symmetry ⇒

Boltzmann theory

Semiclassical theory of transport

⇒ Bloch states

•  Wannier states

•  crystalline system: H0

translational symmetry ⇒

•  for slowly varying potential V

•  external potential V( r )

Wannier ⇒

⇒ fn( r ,t) = envelope function

NB: interband transitions n → n’ have been neglected

•  example

⇒

Ga As

•  semiclassical approximation (correspondence principle)

•  semiclassical equations of motion

with

the wave packet follows the classical trajectory determined by the corresponding classical Hamiltonian

•  trajectories in phase space

r

koccupied empty

•  validity

V(x)

λ

Δx a0

λ >> Δx >> a0

•  trajectories in phase space

r

koccupied empty

•  validity

V(x)

λ

Δx a0

λ >> Δx >> a0

•  trajectories in phase space

r

koccupied empty

•  validity

V(x)

λ

Δx a0

λ >> Δx >> a0

•  distribution function:

density of occupied states in the phase space at time t

⇒  equilibrium (V=0) distribution:

⇒  electric current:

• equation for the distribution function

• Boltzmann equation

⇒  Liouville theorem

• relaxation time approximation

τ = relaxation time

•  Ohm’s law

(cubic symmetry)

⇒  electron gas: σ = ne2τ /m

•  Conductance A

L

W

L

•  ballistic regime

linear dimensions << mean free path

⇒  finite conductance !

W

ballistic conductor

B. J. van Wees et al.

Giant magnetoresistance

•  Boltzmann formalism: distribution function

I II

z

•  Important point: no interference between and

Giant magnetoresistance

•  Ballistic regime (λ >> L): Landauer formalism

€

ε = (µ1−µ2) /evoltage drop

M = # channels between µ1 and µ2

T

I1+ I2

+

ε I1- €

I1+ =

2eh

M µ1−µ2[ ]

€

I2+ =

2eh

MT µ1−µ2[ ]

⇒

€

G =I

µ1−µ2( ) /e=2e2

h

MT

€

I1− =

2eh

M (1−T) µ1−µ2[ ]

Giant magnetoresistance

two current model

€

Rα = Gα↑ + Gα

↓( )−1

translational symmetry

α = FM, AF

Magnetotransport in multilayers

Magnetotransport in multilayers

•  How could the magnetoresistance ratio be enhanced?

•  Would it be possible to have in such systems an insulating antiferromagnetic configuration ( )?

•  Could interference effects lead to such situation?

Usual situation

EF uniform spacer

ferromagnetic band structure

Uniform spacer

FM

AF

Modulated spacer

EF

Modulated spacer

EF

Modulated spacer

EF

transmission bands

transmission gaps

transmission bands

Ferromagnetic configuration

Antiferromagnetic configuration

Transmission coefficients

“Enhanced magnetoresistance effect in layered systems” M. S. Ferreira, J. dA.C., R. B. Muniz and Murielle Villeret,

Appl. Phys. Lett. 75, 2307 (1999)

Modulated spacer

Interesting features:

⇒ huge magnetoresitance ratio

⇒ spin filtering effect

⇒ Could be used as a logical gate

Modulated spacer

•  Challenge: to find real materials which could be used to fabricate such a device