Magnetism and Spin Tunneling in Nanostructures
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ELASTIC AND INELASTIC TUNNELING, MODEL The model that we will consider below includes a Hamiltonian for non-interacting conducting spin-split electrons 7N0, electron-phonon interaction 7/ep, and exchange interaction with localized d, electrons 7/W, the later giving rise to the electron-magnon interaction. Impurities will be described by a short-range confining potential Vi, / = o + Wep + W.+ Wi = 1ZVi(r-ni)
,
(1)
ni
where r stands for the coordinate of the electron and ni denotes the impurity sites. The non-interacting part of the Hamiltonian W-/describes electrons in the ferromagnetic 7 electrodes and insulating barrier according to the Schr6dinger equation
(Woo00 - h.&o = E,
(2)
where oo -(h 2 /2m,)V 2 + U, is the single-particle Hamiltonian with U(r) the potential energy, h(r) the exchange energy (= 0 inside the barrier), a stands for the Pauli matrices; indices a=1, 2, and 3 mark the quantities for left terminal, barrier, and right terminal, respectively (No is the expression in brackets). We shall also use the following notations to clearly distinguish between left and right terminal: p = k, and k = k 3 . Solution to this problem in the limit of a thick barrier provides us with the basis functions for electrons in the terminals and barrier to be used in Bardeen's tunneling Hamiltonian approach.9 ' 10 We assume that all many-body interactions in the electrodes are included in the effective parameters of (2). To fully characterize tunneling we add to Bardeen's direct tunneling term 7/o the contributions from 7 and p:
S=
(3)
WTo + 7/x + 'To= aka rkalpa + h.c., p,ka
Tioaka
-h _ 2/(2Tn 2) dA (/ka V7pa - V/ka kpa); (4) 4 = Tj,'p(n) [(S' (Sn3))(rttlpt rtkI,,) + Snrtlpt S-r ~ip,] + h.c.,. Tk•n kt rnlp ktpj]i- + Sn an,k,p
7P
Z
TkQP (q)rktalpa(bqa
- bt qa) + h.c.
(5)
aank,p
Here the surface E lies somewhere in the barrier and separates the electrodes, we have subtracted an average spin Sn3 - (Sn) in each of electrodes as part of the exchange potential, the exchange vertex is TJ - Jn exp(-Kw), and the phonon vertex is related to the deformation potential D in the usual way [TCP(q) - %Dq(h/2Mwq)1/2 exp(-nw)], where M is the atomic mass, q is the phonon momentum, n marks the lattice sites, and the vertices contain the square root of the barrier transparency.1 " 11 The operators la and ra annihilate electrons with spin a on the left and right electrodes, respectively. Two more things to note: (i) the summations over p and k always include the densities of initial gLa and final gRb states, that makes both exchange and phonon contributions spin-dependent, (ii) when the magnetic moments on the electrodes are at a mutual angle 0, one has to express the operator r w.r.t. the lab system and then use it in 7WT (5). 118
10 The tunnel current will be calculated within the linear response formalism as
I(V, t) = -eJ dt'([dNL(t)/dt, 7/r(t')])o,
(6)
where NL(t) = Epa lta(t)lpa(t) is the operator of the number of electrons on the left terminal in the interaction representation, ( )o stands for the average over Wo,
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