ABSTRACT
Material templates used in spintronics include various metallic magnetic materials that show a direct magnetoresistive effect or metallic junctions with tunneling magnetoresistance. Also, nonmagnetic semiconductors are recognized as a material template for spintronic devices since the spin-polarized electrons injected from metallic ferromagnetic contact may have a long spin lifetime enabling spin transfer to another ferromagnetic electrode where the spin information can be read out. The conductivity mismatch between a ferromagnetic metal contact and a semiconductor active region degrades the efficiency of spin injection. There are two basic ways to avoid the conductivity mismatch: One is to use tunnel contacts, another is to replace ferromagnetic metals with semiconductor alloys containing transition-metal atoms such as Fe, Mn, and Co providing they show room-temperature ferromagnetism [1,2]. Ferromagnetic semiconductors would benefit device performance as the all-semiconductor structure can be grown lattice matched to the other part of the device, reducing the defect density, and increasing the lifetime of non-equilibrium carriers and the spin relaxation time.Ferromagnetism in diluted magnetic semiconductors is the subject of intense theoretical and experimental studies with respect Wide Bandgap Semiconductor Spintronics Vladimir Litvinov Copyright © 2016 Pan Stanford Publishing Pte. Ltd. ISBN 978-981-4669-70-2 (Hardcover), 978-981-4669-71-9 (eBook) www.panstanford.com
to their various applications in spintronics. The central point of the study is the nature of magnetic interaction between electrons and localized spins and also between localized magnetic moments. In this chapter, we discuss several mechanisms of indirect exchange interaction between magnetic atoms in metals and semiconductors. The details of this interaction are important as they determine the ferromagnetic critical temperature and help in the engineering of high-temperature ferromagnetic semiconductors. 5.1 Direct Exchange InteractionA Hamiltonian of two electrons coupled by Coulomb interaction has the form1 1 2 2 1 2= ( ) + ( ) + ( , ),H H H Ur r r r (5.1)where H1,2 are one-particle energy operators and U(r1, r2) is Coulomb interaction. Basis wave functions are eigenfunctions of one-particle parts of the Hamiltonian: j1(r) and j2(r). The wave function for the pair of non-interacting electrons (r1, r2) can be expressed through products of one-electron functions j1(r) and j2(r). Composing the products one has to account for electron spin (s1 = s2 = 1__2 ) and write two-electron wave functions corresponding to total spins S = 0 (antiparallel spins in the pair) and S = 1 (parallel spins). Swapping the coordinates of electrons 1 and 2 changes the wave function by a factor (–1)S: (r1, r2) = (–1)S (r2, r1), so, the wave function should be symmetric for S = 0 and anti-symmetric for S = 1: 1 2 1 1 2 2 1 2 2 1( , ) = ( ) ( ) ± ( ) ( )S j j j jr r r r r r (5.2)The total energy of the two electrons is given as1 2= + + ,SE E E ED (5.3)where E1,2 are one-particle energies of non-interacting electrons (eigenvalues of H1,2, respectively), and DES is the first-order perturbation correction induced by Coulomb interaction: * 1 2 1 2 1 2 1 2 = ( , ) ( , ) ( , ) S S SE U d dD r r r r r r r r (5.4)
Making use of Eq. (5.2) to calculate DES one obtains DΕ0 = C + Jex, DΕ1 = C – Jex where the direct Coulomb shift C and the exchange correction Jex are given below:
2 21 1 2 2 1 2 1 2* *ex 1 2 2 1 1 2 1 1 2 2 1 2 = | ( )| | ( )| ( , ) = ( ) ( ) ( , ) ( ) ( ) .C U d dJ U d dj jj j j j r r r r r rr r r r r r r r (5.5)Exchange corrections ±Jex depend on the total spin of the electrons. The total spin vector S = s1 + s2 squared gives the relation
3 ,– = 01 4( ) = [ ( + 1) – ( + 1) – ( + 1)] = .12 , = 14 S
S S s s s s S
s s (5.7) Exchange energy corrections can be expressed with the help of (5.7) as ex ex 1 2 ev1± = – + 2( ) .2J J s s (5.8)These are eigenvalues of the Hamiltonian ex ex 1 21= – + 2 .2H J s s (5.9)The Hamiltonian (5.9) was obtained by Dirac and then its spin-dependent part was used by Heisenberg in the theory of ferromagnetism in solids. In magnetic materials the Heisenberg model H = – Jex s1s2 describes the direct exchange interaction between localized spins s1 and s2 and favors the parallel spin orientation if Jex > 0. Exchange interaction between atoms takes place only if their electron wave functions overlap, so direct Heisenberg interaction is adequate for magnetic materials where the spin moments of the host atoms are placed at a distance of the
order of the lattice constant. This is not the case in magnetically doped metals, semiconductors, or magnetic alloys where the content of magnetic atoms is a small fraction of that of the host atoms. Still, the coupling between distant spins exists as it is mediated by the host: free electrons in metals and degenerate semiconductors, or nonmagnetic atoms in dielectrics. This type of coupling is called an indirect exchange interaction.In this chapter, we discuss some features of indirect exchange in metals and III-V narrow and wide bandgap semiconductors as well as the picture of ferromagnetic phase transition in magnetically doped semiconductors. 5.2 Indirect Exchange InteractionIn metals and semiconductors, indirect exchange appears as a result of coupling between an impurity spin and a free s-electron spin. The localized spin originates from the unfilled d-shell of a transition metal atom or the f-shell of a rare earth element, so the coupling is referred as s-d or s-f interaction. As the inner unfilled orbitals are more localized than the valence electron s-states, the s-d interaction occurs in close proximity to a magnetic impurity and can be well approximated by a contact interaction. The full one-electron Hamiltonian is given as 20
( ) = ( ) + ( ),( ) = + ( ), ( ) = – ( – ),2 e sd
H H H
p J H V H
m n
r r r
r r r S r R iσ (5.10) where p = –i∂⁄∂r, = {x, y, z} are the Pauli matrices, m0 is the free electron mass, V(r) is the electron energy in the lattice periodic potential, J is the coupling constant of an isotropic s-d interaction, n = N/V is the density of the host atoms, and Ri is the impurity position in the lattice. In the effective mass approximation, the free electron Hamiltonian can be rewritten as He(r) = p2⁄2m, where the effective mass m includes effects of the periodic field V(r) and gives the correct energy dispersion in the vicinity of the edge of the conduction band. This approximation works well in metals and degenerate semiconductors if energy corrections from remote energy bands are negligible.
