Theory of Spin-Dependent Tunneling

Transcription

1 Spring Review 2002 Theory of SpinDependent Tunneling W. H. Butler Center for Materials for Information Technology University of Alabama X.G. Zhang and T. C. Schulthess Oak Ridge National Laboratory J. M. MacLaren Tulane University

2 Tunneling Energy, Wave Function e Energy ikx Tunneling Through Simple Barrier V B ikx re ikx e ± κ x x We have generalized tunneling theory to treat realistic Bloch states and realistic electronic structures for barrier layers and interfaces. te k 2mE = k 2 h 2 mv ( B E) κ = 2 h 2 k 2

3 Systems Investigated Fe ZnSe Fe (also Fe GaAs Fe and Fe Ge Fe) Fe Vacuum Fe Fe Fe Al vacuum Al Fe Fe Al vacuum Fe Co(fcc) vacuum Co(fcc) Co(hcp) vacuum Co(hcp) Fe MgO Fe Experimental Realization: (preprints) Wulfheckel, Klaua, Ullmann, Zavaliche, Kirshner, Urban, Monchesky and Heinrich also Bowen, Cros, Petroff, Fert, Boubeta, CostaKrämer, Anguita, Cebollada, Briones, de Teresa, Morellón, Ibara Güell, Peiró and Cornet Ge Fe

4 Theory and Techniques FirstPrinciples Local Density Approximation to Density Functional Theory was used to calculate the electronic structure of vacuum, insulators, semiconductors, etc. embedded in Fe. Some Interface structures were optimized with a fullpotential plane wave code (VASP). Layer KKR approach with the atomic sphere approximation was used to solve the Schrödinger equation, calculate Green functions, and obtain transmission probabilities. Conductance was calculated using the Landauer formula: G = e2 h T(k ) k

5 Known Limitations Systems are epitaxial (two dimensional periodicity). LDADFT underestimates band gaps. Calculations are for low bias limit.

6 There is a charge redistribution in Fe MgO Fe to properly place the MgO Fermi energy.

7 Calculated DOS for Fe layers in Fe MgO Fe E F E F States near Fermi energy are important for transport. Majority Fermi energy DOS is depressed on the interfacial layer. Minority DOS has a large peak just above the Fermi energy.

8 Calculated DOS for MgO layers in Fe MgO Fe Density of states is very small within gap, especially for interior layer. Minority DOS has a small peak just above the Fermi energy on interfacial layer. LDAGap (.244 H.446H) 5.5eV vs. 7.8eV (exp.)

9 We used the Landauer formula to calculate the tunneling conductance. The tunneling conductance is given by the sum of the transmission probabilities for each channel : G = e2 h T(k ) A channel is a Bloch wave in the iron electrodes for a given value of k. The total number of these channels is proportional to the area of the sample perpendicular to the direction of current flow. k

10 Majority conductance in Fe MgO Fe becomes concentrated near k = 0 as the MgO is made thicker. 4 layers of MgO G=2.4 /Ωµ 2 8 layers of MgO G=3.1x10 3 /Ωµ 2 12 layers of MgO G=5.7x10 6 /Ωµ 2 Majority

11 Simple Theory Predicts a Peak in the Conductance for k =0. Tunneling Through Simple Barrier 1.5 V B Energy, Wave Function e Energy ikx ikx re ikx e ± κ x te 2mE k = k 2 h 2 mv ( B E) κ = 2 h 2 k x G e 2 κ d 2d C k = e 2

12 Tunneling DOS for k = 0 depends strongly on symmetry of Bloch states in Fe. Figures show the density of states (DOS) for electrons incident from the left for the individual bands. One particular majority band ( 1 ) readily enters the MgO and decays slowly inside the MgO. The 1 band is the only one of the 4 bands at the Fermi energy compatible with s angular momentum character. The other bands are nearly pure d in angular momentum character.

13 Bloch State Symmetries

14 The decay rates observed in the tunneling DOS can be obtained from the complex energy bands of MgO. The 1 states decay less rapidly because they have fewer oscillations in the plane of the layers. Gap k becomes imaginary within the gap. ψ k ik a ( r a) = e ψ ( r) k

15 Closer observation of the majority transmission shows oscillations as a function of k. Complex MgO energy bands The decay exponent (k z ) is not always purely imaginary, but is generally complex. Two complex bands interfere strongly when their imaginary parts are equal. This is a common phenomenon Oscillatory dependence on k ; damped oscillatory dependence on thickness. 2 kd I T e krd [1 cos( )]

16 Minority conductance doesn t behave as we would expect. Peak is not at k =0 and k =0 does not dominate as thickness is increased. 4 layers of MgO G=0.46 /Ωµ 2 8 layers of MgO G=1.4x10 5 /Ωµ 2 12 layers of MgO G=5.9x10 10 /Ωµ 2

17 The peak in the minority IFDOS is along k x =0 or k y =0 and is associated with an IFresonance state. However, these electrons cannot tunnel through the MgO.

18 Transmission and Conductance for AntiParallel Alignment of the Femoments on opposite sides of the MgO Barrier. G=0.23 /Ωµ 2 G=1.3x10 5 /Ωµ 2 G=9.7x10 9 /Ωµ 2

19 The large magnetoconductance arises from the majority 1 state.

20 The Magnetoresistance for Fe MgO Fe is predicted to be quite large, especially for thick barriers. Bowen et al. report 60% MR for FeCo MgO Fe

21 Recent experiments indicate that an FeO layer may form during deposition of MgO on Fe. Side view 2.26 Å 1 st MgO layer 0.19 Å "FeO" layer d 12 =1.66 Å d bulk =1.43 Å Top view a 0 (bccfe)= Å 1.85 Å a 0 (bccfe)= Å 2.03 (15) Å thanks to: H. L. Meyerheim, R Popescu, J. Kirschner N. Jedrecy, M. SauvageSimkin B. Heinrich, and R Pinchaux FeO layer

22 Interface Chemistry Affects Band Offsets and Decay Rates 2 Fe MgO Fe Fe FeO MgO FeO Fe 1.5 Fe FeO MgO Fe Fe FeO MgO Fe 1 E f Fe FeO MgO FeO Fe Fe MgO Fe 0.5 (k z) Band in MgO Energy (Hartrees) Decay rates are lower with FeO layer.

23 Interface Chemistry Affects Band Offsets and Decay Rates With FeO Layer on left No FeO Layer Energy (Hartees) E F Fe MgO E F Fe Layer Number FeO layer at one interface will give tilted barrier.

24 Fe MgO Fe vs. Fe FeO MgO Fe [100] For a barrier consisting of 8 atomic layers of MgO, the calculated TMR decreases from 4600% to 74% on inserting an FeO layer. Fe MgO FeO Fe

25 The change in TMR is due to a decrease in the majority channel conductance by a factor of 30. FeMgOFe Conductance FeFeOMgOFe Conductance in 1/Ωm 2 in 1/Ωm 2 3.1x x x10 7 2x10 6

26 and to an increase of the antiparallel conductance in one channel. FeMgOFe Conductance in 1/Ωm 2 FeFeOMgOFe Conductance in 1/Ωm 2 3.3x x x x10 7

27 Density of States The decrease in the parallel majority conductance appears to be an interfacial effect that occurs at the FeFeO and FeO MgO interfaces e05 1e06 Metal DOS Fe MgO Fe Layer Number Density of States e05 1e06 Fe FeO MgO Fe Metal DOS Layer Number The k =0 conductance flows primarily through the oxygen sites.

28 The surprising increase in conductance for the antiparallel configuration in the minority(feoside) channel appears to arise from intefacial resonance effects. Density of States for Down to Up at peak in transmission DOS (States/Hartree) e05 Fe Minority FeO MgO Fe Majority 1e06 1e Layer Number

29 Tunneling through vacuum is simple for free electrons. d Metal E V Metal Tk e 2 κ d ( ) 2m κ = 2 V E k h ( ) 2 Conductance decreases exponentially with thickness. Electrons with zero transverse momentum are most likely to get through the barrier.

30 The decay rate for Bloch electrons depends on their symmetry even in vacuum. Oscillations in the wave functions parallel to the interface increase the decay rate perpendicular to the interface 2 2 φ φ m x y κ = 2 ( V E ) h φ φ

31 n' k ' 2 Julliere/MacDonald Formula 2 e 2 G = t( n, k; n' k ') δ ( Enk EF ) δ ( En' k ' EF ) h nk t( n, k; n' k ') t( n, k) t( n', k ') 2 e G t( n, k) δ ( E E ) t( n', k ') δ ( E E ) h G nk t N t N L L R R G 2PLPR G( AP) 1 P P P = t N t N t N t N L R nk F n' k ' F n' k ' G G G

32 The separability hypothesis that leads to a Jullierelike formula can be tested. If Julliere is valid, all points should fall on the line. Figures show T(AP) and Sqrt(T T ) for different values of k. The assumption that T(AP) = Sqrt(T T ) does not appear to be very accurate. Sqrt(G G )/G = 5, 16, 6 for 4, 8, 12 layers of MgO

33 General Results for SpinDependent Tunneling Bloch states of different symmetry decay at different rates within the barrier. The decay rate is determined by the complex energy bands of the same symmetry in the barrier. There may be quantum interference between the decaying states in the barrier. This leads to an oscillatory dependence of the tunneling current on k and a damped oscillatory dependence on barrier thickness. Interfacial resonance states can allow particular Bloch states to tunnel efficiently through the barrier. Oscillations in the wave function (e.g. dstates) parallel to the interface increase its rate of decay in the barrier.

34 General Results Continued The reason that majority electrons dominate the conductance is that in the common magnetic transition metals and alloys, the majority channel tends to have more slike electrons at the Fermi energy. These tunnel more readily than dstates which have additional oscillations in the plane of the interface causing faster decay perpendicular to the interface. Interface chemistry can affect band offsets, decay rates and barrier shape. A single FeO layer significantly decreases the TMR for 8 MgO layer barrier. Our results do not support models that relate the magnetoconductance to the polarization. Reference: Butler, Zhang, Schulthess, and MacLaren: PRB 63, and (2001) see also Mathon et al. Phys. Rev. B 63, (2001)

35 Present and Future Work Interfacial Structure is Critical Effects of interface chemistry preliminary results indicate that O in final Fe layer greatly reduces majority tunneling current and reduces MR. Effects of disorder especially interfacial disorder We have developed techniques for treating the disorder within the coherentpotential approximation calculation of reduction in specular transmission is straightforward calculation of effect of diffuse scattering is more difficult