Spin Transfer Torque as a Non-Conservative Pseudo-Field

Abstract

In this paper we show that the spin transfer torque can be described by a pseudo magnetic field, proportional to the magnetic moment of the itinerant electrons that enters the Landau-Lifshitz-Gilbert equation in the same way as other external or internal magnetic fields. However, unlike an ordinary magnetic field, which is always conservative in nature, the spin torque induced pseudo field may have both conservative and non-conservative components. We further show that the magnetic moment of itinerant electrons develops an out-of-plane component only at non-equilibrium and this component is responsible for the Slonczewski type switching that acts against the damping and is always non-conservative. On the other hand, the in-plane components of the pseudo field exist both at equilibrium and out-of-equilibrium, and are responsible for the field like term. For tunnel based devices, this term results in lower switching current for anti-parallel (AP) to parallel (P) switching compared to P to AP, even when the torque magnitudes are completely symmetric with voltage.

Full text

Page 1

1
Spin Transfer Torque as a Non-Conservative Pseudo-Field
Sayeef Salahuddin*, Deepanjan Datta and Supriyo Datta
School of Electrical and Computer Engineering and NSF Center for Computational
Nanotechnology (NCN), Purdue University, West Lafayette, IN 47906
*Present address: Electrical Engineering and Computer Science, UC Berkeley, CA-94720
Abstract:
In this paper we show that the spin transfer torque can be described by a pseudo magnetic field,
proportional to the magnetic moment of the itinerant electrons that enters the Landau-Lifshitz-
Gilbert equation in the same way as other external or internal magnetic fields. However, unlike
an ordinary magnetic field, which is always conservative in nature, the spin torque induced
‘pseudo field’ may have both conservative and non-conservative components. We further show
that the magnetic moment of itinerant electrons develops an out-of-plane component only at non-
equilibrium and this component is responsible for the ‘Slonczewski’ type switching that acts
against the damping and is always non-conservative. On the other hand, the in-plane components
of the pseudo field exist both at equilibrium and out-of-equilibrium, and are responsible for the
‘field like’ term. For tunnel based devices, this term results in lower switching current for anti-
parallel (AP) to parallel (P) switching compared to P to AP, even when the torque magnitudes
are completely symmetric with voltage.

Page 2

2
1. Introduction
Spin torque devices [1, 2] that switch the magnetization of small magnets with spin polarized
currents without any external magnetic field, have stirred tremendous interest due to their
potential application as non volatile memory and also as nanoscale microwave oscillators.
Although the concept of spin transfer torque has been demonstrated by a number of experiments
[3, 4], quantitative measurement of spin transfer torque has been achieved only very recently [5,
6, 7]. All these measurements show a significant ‘field-like’ or out-of-plane torque in addition to
the original in-plane torque predicted by Slonczewski [1]. This is very different from metallic
channel based devices where the field like term is minimal. Recent theoretical studies have also
shown the field like term to be significant in tunnel based devices [8, 9, 10, 11]. However, the
details of how this field-like torque can affect the switching behavior is yet to be understood
properly [5, 6, 7, 12, 13].

In this paper we first show that spin torque can be described by a pseudo magnetic field
proportional to the net magnetic moment
of the itinerant electrons, (normalized to the Bohr
magneton ) providing a natural relationship between Slonczewski and field like terms:

(1)

Eqn. (1) is the central result of this paper and is derived in Section 2, starting from the Gilbert
form of the LLG equation and introducing the spin-torque in terms of obtained from non-
equilibrium Green function (NEGF) formalism for the conduction electrons. Note that so
that the pseudo field is in the same direction as and enters Eqn. (1) just like other

Page 3

3
magnetic fields included in. This may seem surprising; since it is well-known that spin-
torque leads to phenomena like coherent precession that do not arise from ordinary magnetic
fields. We show in section 3 that such phenomena can also be understood in terms of Eqn. (1)
once we note that the pseudo-field representing the spin-torque has both a conservative
component like the conventional magnetic fields included in and also a non-conservative
component that makes the curl of overall to be non-zero: ; (Note that
) .
We show that the out-of-plane component of is responsible for the Slonczewski term and
is always non-conservative. On the other hand, the in plane components give the field like term
and can introduce asymmetry in switching currents for opposite polarity in the voltage bias.
Specifically, we shall show that for tunnel based devices, this field like torque can result in a
lower switching voltage for AP to P switching compared to P to AP, even when the torque
magnitudes are completely symmetric with voltage. This can be understood by noting that for
tunneling devices, the in plane component of the pseudo field (responsible for the field like term)
remains conservative even away from equilibrium, and thus acting like an ordinary magnetic
field parallel to the direction of the fixed magnet that helps switching from AP to P while
hindering P to AP transition.

2. Spin-Torque as a Pseudo Field
A typical spin torque device is shown schematically in Fig. 1. Left contact is the fixed
ferromagnet having magnetization along
. Right contact is soft layer and its magnetization
points along
which is free to rotate in easy (z-x) plane. An insulating layer separates the

Page 4

4
ferromagnetic contacts. Following Gilbert’s prescription, we write, the rate of change of the
direction of the magnetization as
(2)
where the spin torque is obtained by integrating the divergence of the spin current carried
by the conduction electrons over the volume of the magnet. Below we will use the non-
equilibrium Green’s function formalism to show that
(3)
where is the magnetic moment of the conduction electrons (normalized to ) and is the
energy splitting of the conduction electrons due to the exchange interaction with the localized
spins that comprise the magnet. Combining Eqns (2) and (3) we obtain our central result stated
earlier in Eqn. (1) with .

Proof of Eqn. 3: We start from the expression for the (2x2) operator representing at site

in a discrete representation (see Eqn. (8.6.3), page 317, [15]) for the conduction electrons
. is the (2x2) correlation matrix at site and the
Hamiltonian [ ] is given by , where is the spin-independent part and
is the spin-dependent part arising from the exchange interaction with the magnet
pointing along, with being a 2x2 identity matrix and representing the Pauli spin
matrices.
The divergence of the spin-current is obtained from the operator
(4)
and substituting for
, we get (note:
is the Levi-Civita antisymmetric tensor)

Page 5

5
(5)
so that the spin-torque is given by
(6)

Defining as the magnetic moment (normalized to) of the
conduction electrons, we obtain
which is the same as
as stated above in Eqn. (3). This completes our proof of Eqn. (1). Note
that this expression for torque is consistent with previous studies [9,10,11].

3. Relation to the standard form
It is shown in the Appendix A that if the conduction electrons are in equilibrium then the spin
density can be written in the form,
(7)
but away from equilibrium, the spin density remarkably develops an additional out-of plane
component that is perpendicular to the magnetization of both magnets:
(8)
so that from Eqn. (3) the spin-torque comes out as .
which has the same form as the standard torque equations used extensively in literature [5, 6, 7].

Our formulation leads to a simple criterion for coherent precession which is considered one of
the hallmarks of spin-torque. To see this we note that one can write Eqn. (1) in the following
form