# Thermoelectrical manipulation of nanomagnets

**ABSTRACT** We investigate the interplay between the thermodynamic properties and spin-dependent transport in a mesoscopic device based on a magnetic multilayer (F/f/F), in which two strongly ferromagnetic layers (F) are exchange-coupled through a weakly ferromagnetic spacer (f) with the Curie temperature in the vicinity of room temperature. We show theoretically that the Joule heating produced by the spin-dependent current allows a spin-thermoelectronic control of the ferromagnetic-to-paramagnetic (f/N) transition in the spacer and, thereby, of the relative orientation of the outer F-layers in the device (spin-thermoelectric manipulation of nanomagnets). Supporting experimental evidence of such thermally-controlled switching from parallel to antiparallel magnetization orientations in F/f(N)/F sandwiches is presented. Furthermore, we show theoretically that local Joule heating due to a high concentration of current in a magnetic point contact or a nanopillar can be used to reversibly drive the weakly ferromagnetic spacer through its Curie point and thereby exchange couple and decouple the two strongly ferromagnetic F-layers. For the devices designed to have an antiparallel ground state above the Curie point of the spacer, the associated spin-thermionic parallel to antiparallel switching causes magnetoresistance oscillations whose frequency can be controlled by proper biasing from essentially dc to GHz. We discuss in detail an experimental realization of a device that can operate as a thermomagnetoresistive switch or oscillator.

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**ABSTRACT:**Thermoelectric manipulation of the magnetization of a magnetic layered stack in which a low-Curie temperature magnet is sandwiched between two strong magnets (exchange spring device) is considered. Controllable Joule heating produced by a current flowing in the plane of the magnetic stack (CIP configuration) induces a spatial magnetic and thermal structure along the current flow --- a magneto-thermal-electric domain (soliton). We show that such a structure can experience oscillatory in time dynamics if the magnetic stack is incorporated into an electric circuit in series with an inductor. The excitation of these magneto-thermionic oscillations follow the scenario either of "soft" of "hard" instability: in the latter case oscillations arise if the initial perturbation is large enough. The frequency of the temporal oscillations is of the order of $10^5 \div 10^7 s^{-1}$ for current densities $j\sim 10^6 \div 10^7 A/cm^3$.Low Temperature Physics 01/2012; 38(7). · 0.82 Impact Factor - SourceAvailable from: Anatolii KravetsAnatolii Kravets, Andrey Timoshevskii, Bogdan Yanchitsky, Olga Salyuk, Sergey Yablonovskii, Sebastian Andersson, Vlad Korenivski[Show abstract] [Hide abstract]

**ABSTRACT:**Magneto-structural properties of films of diluted ferromagnetic alloys Ni$_x$Cu$_{1-x}$ in the concentration range $0.7 < x < 1.0$ are studied experimentally. Films deposited by magnetron sputtering show partial phase separation, as evidenced by structural analysis and ferromagnetic resonance measurements. The phase diagram of the Ni$_x$Cu$_{1-x}$ bulk system is obtained using numerical theoretical analysis of the electronic structure, taking into account the inter-atomic exchange interactions. The results confirm the experimentally found partial phase separation, explain it as magnetic in origin, and indicate an additional metastable region connected with the ferromagnetic transition in the system.Journal of Magnetism and Magnetic Materials 01/2012; 324(13). · 2.00 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We show theoretically that a significant spin accumulation can occur in electric point contacts between two ferromagnetic electrodes with different magnetizations. Under appropriate conditions an inverse population of spin-split electronic levels results in stimulated emission of photons in the presence of a resonant electromagnetic field. The intensity of the emitted radiation can be several orders of magnitude higher than in typical semiconductor laser materials for two reasons. (1) The density of conduction electrons in a metal point contact is much larger than in semiconductors. (2) The strength of the coupling between the electron spins and the electromagnetic field that is responsible for the radiative spin-flip transitions is set by the magnetic exchange energy and can therefore be very large, as suggested by Kadigrobov et al. [Europhys. Lett. 67, 948 (2004)].Low Temperature Physics 12/2012; 38(12). · 0.82 Impact Factor

Page 1

arXiv:0904.1156v2 [cond-mat.mes-hall] 8 Jul 2010

Thermoelectrical manipulation of nano-magnets

A. M. Kadigrobov,1,2S. Andersson,3D. Radi´ c,1,4R. I. Shekhter,1M. Jonson,1,5,6and V. Korenivski3

1Department of Physics, University of Gothenburg, SE-412 96 G¨ oteborg, Sweden

2Theoretische Physik III, Ruhr-Universit¨ at Bochum, D-44801 Bochum, Germany

3Nanostructure Physics, Royal Institute of Technology, SE-106 91 Stockholm, Sweden

4Department of Physics, Faculty of Science, University of Zagreb, 1001 Zagreb, Croatia

5School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, UK

6Division of Quantum Phases and Devices, School of Physics, Konkuk University, Seoul 143-701, Korea

(Dated: July 9, 2010)

We investigate the interplay between the thermodynamic properties and spin-dependent transport

in a mesoscopic device based on a magnetic multilayer (F/f/F), in which two strongly ferromag-

netic layers (F) are exchange-coupled through a weakly ferromagnetic spacer (f) with the Curie

temperature in the vicinity of room temperature. We show theoretically that the Joule heating pro-

duced by the spin-dependent current allows a spin-thermo-electronic control of the ferromagnetic-

to-paramagnetic (f/N) transition in the spacer and, thereby, of the relative orientation of the outer

F-layers in the device (spin-thermo-electric manipulation of nanomagnets). Supporting experimental

evidence of such thermally controlled switching from parallel to antiparallel magnetization orien-

tations in F/f(N)/F sandwiches is presented. Furthermore, we show theoretically that local Joule

heating due to a high concentration of current in a magnetic point contact or a nanopillar can be used

to reversibly drive the weakly ferromagnetic spacer through its Curie point and thereby exchange

couple and decouple the two strongly ferromagnetic F-layers. For the devices designed to have an

antiparallel ground state above the Curie point of the spacer, the associated spin-thermionic parallel-

to-antiparallel switching causes magneto-resistance oscillations whose frequency can be controlled

by proper biasing from essentially DC to GHz. We discuss in detail an experimental realization of

a device that can operate as a thermo-magneto-resistive switch or oscillator.

I.INTRODUCTION

The problem of how to manipulate magnetic states

on the nanometer scale is central to applied magneto-

electronics. The torque effect1,2, which is based on the

exchange interaction between spin-polarized electrons in-

jected into a ferromagnet and its magnetization, is one of

the key phenomena leading to current-induced magnetic

switching. Current-induced precession and switching of

the orientation of magnetic moments due to this effect

have been observed in many experiments3–12.

Current-induced switching is, however, limited by the

necessity to work with high current densities. A natural

solution to this problem is to use electrical point contacts

(PCs). Here the current density is high only near the PC,

where it can reach13,14values ∼ 109A/cm2. Since almost

all the voltage drop occurs over the PC the characteristic

energy transferred to the electronic system is comparable

to the exchange energy in magnetic materials if the bias

voltage Vbias ∼ 0.1 V, which is easily reached in exper-

iments. At the same time the energy transfer leads to

local heating of the PC region, where the local tempera-

ture can be accurately controlled by the bias voltage.

Electrical manipulation of nanomagnetic conductors

by such controlled Joule heating of a PC is a new prin-

ciple for current-induced magnetic switching. In this pa-

per we discuss one possible implementation of this prin-

ciple by considering a thermoelectrical magnetic switch-

ing effect. The effect is caused by a non-linear interac-

tion between spin-dependent electron transport and the

magnetic sub-system of the conductor due to the Joule

heating effect. We predict that a magnetic PC with a

particular design can provide both voltage-controlled fast

switching and smooth changes of the magnetization di-

rection in nanometer-size regions of the magnetic mate-

rial. We also predict temporal oscillations of the magne-

tization direction (accompanied by electrical oscillations)

under an applied DC voltage. These phenomena are po-

tentially useful for microelectronic applications such as

memory devices and voltage controlled oscillators.

II.EQUILIBRIUM MAGNETIZATION

DISTRIBUTION

The system under consideration has three ferromag-

netic layers coupled to a non-magnetic conductor as

sketched in Fig. 1. We assume that the Curie temper-

ature T(1)

c

of region 1 is lower than the Curie temper-

atures T(0,2)

c

of regions 0 and 2; in region 2 there is a

magnetic field directed opposite to the magnetization of

the region, which can be an external field, the fringing

field from layer 0, or a combination of the two. We re-

quire this magnetostatic field to be weak enough so that

at low temperatures T the magnetization of layer 2 is

kept parallel to the magnetization of layer 0 due to the

exchange interaction between them via region 1 (we as-

sume the magnetization direction of layer 0 to be fixed).

In the absence of an external field and if the tempera-

ture is above the Curie point, T > T(1)

the proposed F/f(N)/F tri-layer is similar to the antipar-

allel spin-flop ‘free layers’ widely used in memory device

c , the spacer of

Page 2

2

FIG. 1:

three ferromagnetic layers (0, 1, 2) coupled to a non-magnetic

conductor (3); the right arrow indicates the presence of a mag-

netic field H, which is antiparallel to the stack magnetization.

Orientation of the magnetic moments in a stack of

FIG. 2: Sketch of the spatial dependence of the orientation of

the magnetic moments in the stack of Fig. 1 at a temperature

T higher than the temperature T(or)

orientation becomes unstable, but lower than the Curie tem-

perature T(1)

c

of layer (1).

c

, at which the parallel

applications15.

As T approaches T(1)

ment of layer 1 decreases and the exchange coupling be-

tween layers 0 and 2 weakens. This results in an inhomo-

geneous distribution of the stack magnetization, where

the distribution that minimizes the free energy of the

system is given by Euler’s equation (see, e.g., Ref. 16):

c

from below the magnetic mo-

d

dx

?

α(x)M2(x)dθ

dx

?

−β

2M2sin2θ+HM

2

sinθ = 0. (1)

Here the x-axis is perpendicular to the layer planes of the

stack, the z-axis is directed along the magnetization di-

rection in region 0; the magnetization direction depends

only on x and the vector M rotates in-plane (that is in

the yz-plane)16; θ(x) is the angle between the magnetic

moment? M(x) at point x and the z-axis (in the yz-plane)

and M(x) = |? M(x)|. In the case under consideration

α(x) = α1, β(x) = β1 for 0 ≤ x ≤ L1 and α(x) = α2,

β(x) = β2 for L1 < x ≤ L2; here α1,2 ∼ I1,2/aM2

where a is the lattice spacing, I1,2∼ kBT(1,2)

are the exchange energies and magnetic moments of re-

gions 1 and 2; β1(β2) is a dimensionless measure of the

anisotropy energy of region 1 (region 2); kBis the Boltz-

mann constant. Below we assume the lengths L1,2of re-

gions 1 and 2 to be shorter than the domain wall lengths

in these regions l1,2=?α1,2/β1,2.

the stack one may solve Eq. (1) in regions 1 and 2 to

1,2,

c

and M1,2

In order to find the magnetization distribution inside

get θ1(x) and θ2(x), respectively, and then match these

solutions at the magnetization interface x = L1. Inte-

grating Eq. (1) with respect to x in the limits L1− δ ≤

x ≤ L1+ δ, δ → 0 one gets the matching condition as

follows:

α2M2

2

dθ2(x)

dx

???

x=L1= α1M2

1

dθ1(x)

dx

???

x=L1;

θ2(L1) = θ1(L1).(2)

The boundary condition at the ferromagnetic interface

x = 0 between layers 0 and 1 follows from the require-

ment that the direction of the magnetization in layer 0 is

fixed along the z-axis (i.e., θ(x) = 0 in this layer):

θ1(0) = 0. (3)

At the ”free” end of the ferromagnetic sample the

boundary condition for the magnetization

d? M(x)/dx = 0 (see, e.g. Ref. 17), so that

? M(x) is

dθ2(x)

dx

??

x=L1+L2= 0.(4)

Solving Eq. (1) in regions 1 and 2 under the assump-

tion L1,2≪ l1,2and with the boundary conditions (2) -

(4) one finds the magnetization in region 1 to be inho-

mogeneous,

θ1(x) = Θ(L1)x

L1

+ O(L1

l1

); 0 ≤ x ≤ L1,(5)

while due to the boundary condition (4) the magnetic

moments in region 2 are approximately parallel, to within

corrections of order α1M2

1(T)L2/α2M2

2(T)L1≪ 1, i.e.

θ2(x) = θ2(L1+ L2) −H sinθ2(L1+ L2)

8α2M2

(L1+ L2− x)2

(6)

where L1 ≤ x ≤ L1+ L2. Using the above boundary

conditions one finds that θ2(L1) ≈ θ2(L1+ L2) ≡ Θ is

determined by the equation

Θ = D(H,T)sinΘ,T < T(1)

T ≥ T(1)

c

Θ = ±π,

c

(7)

where

D(H,T) =L1L2HM2(T)

4α1M2

1(T)

.(8)

In Eq. (8) M1(T) = M(0)

are the magnetic moments of region 1 and 2, respectively;

the parameter D(H,T) is the ratio between the magnetic

energy and the energy of the stack volume for the in-

homogeneous distribution of the magnetization. As the

second term inside the brackets in Eq. (6) is negligibly

1

?

(T(1)

c

− T)/T(1)

c

and M2(T)

Page 3

3

small, the magnetization tilt angle Θ in region 2 becomes

independent of position and is simply given as a function

of H and T by Eq. (7).

By inspection of Eq. (7) one finds that it has either one

or several roots in the interval −π ≤ Θ ≤ π depending

on the value of the parameter D(H,T).

At low temperatures the exchange energy prevails,

the parameter D(H,T) < 1 and Eq. (7) has only one

root, Θ = 0. Hence a parallel orientation of all mag-

netic moments in the stack is thermodynamically sta-

ble. However, at temperature T(or)

D(T(or)

c

,H) = 1, two new roots Θ = ±|θmin| ?= 0 appear.

The parallel magnetization corresponding to Θ = 0 is

now unstable18and the direction of the magnetization in

region 2 tilts as indicated in Fig. 2. Using Eq.(8) one finds

the critical temperature of this orientation transition to

be equal to

c

< T(1)

c

for which

T(or)

c

= T(1)

c

?

1 −

δT

T(1)

c

?

,

δT

T(1)

c

=L1L2HM2

4α1M2

1(0)≡ D0.

(9)

theThe tilt increases further with T until at T = T(1)

exchange coupling between layers 0 and 2 vanishes and

their magnetic moments become antiparallel.

c

Thermally assisted exchange decoupling in F/f/F

multilayers

To demonstrate the properties of the tri-layer mate-

rial system proposed above we have suitably alloyed Ni

and Cu to obtain a spacer with a Tc just higher than

room temperature (RT). The alloying was done by co-

sputtering Ni and Cu at room temperature and base pres-

sure 10−8torr on to a 90x10 mm long Si substrate in such

a way as to obtain a variation in the concentration of Ni

and Cu along the substrate. By cutting the substrate

into smaller samples along the compositional gradient, a

series of samples were obtained, each having a different

Curie temperature.

One of the multi-layer compositions chosen was NiFe

8/CoFe 2/NiCu 30/ CoFe 5 [nm] where the NiFe layer is

used to lower the coercive field of the bottom layer (HC0)

in order to separate it from the switching of the top layer

(HC2). A magnetometer equipped with a sample heater

was used to measure the magnetization loop as the tem-

perature was varied between 25◦C and 130◦C. The re-

sults for a Ni concentration of ∼70% are shown in Fig. 3.

The strongly ferromagnetic outer layers are essentially

exchange-decoupled at T > 100◦C (F/Paramagnetic/F

state), as evidenced by the two distinct magnetization

transitions at approximately 15 and 45 Oe in Fig. 3. As

the temperature is reduced to RT, the switching field of

the soft layer increases and the originally sharp M-H

transition becomes significantly skewed. This confirms

the theoretical result, expressed by Eqs. (5) and (6) for

Θ(H,T), that the magnetic state of the sandwich is of

the spring-ferromagnet type19. The lowering of temper-

ature leads at the same time to a lower switching field of

the magnetically hard layer, which is due to the stronger

effective magnetic torque on the top layer in the coupled

F0/f/F2 state. This thermally-controlled interlayer ex-

change coupling is perfectly reversible on thermal cycling

within the given temperature range.

?60?40?200204060

?1

0

1

H [Oe]

M / Ms

HC0

HC2

55°C

130°C

25°C

?M

(T)

FIG. 3:

90/Ni80Fe20 8/Co90Fe10 2/Ni70Cu30 30/Co90Fe10 5/Ta 10

[nm] as the temperature is varied from 25◦C to 130◦C. HC0

and HC2 are the coercive fields of the bottom and top mag-

netic layers, respectively.

Magnetization loop for a sample of SiO2/Cu

We further demonstrate an exchange-biased magnetic

tri-layer of the generic composition AF/F0/f/F2, where

the spacer separating the outer ferromagnetic layers (F)

is a low-Curie temperature diluted ferromagnetic alloy (f)

and one of the F0 layers is exchange-pinned by an antifer-

romagnet (AF). In addition to the tri-layer a Cu spacer

and a reference layer, pinned by an AF, have been added

on top of the stack in order to measure the current-in-

plane giant magnetoresistance (GMR). The specific stack

composition chosen was Si/SiO2/NiFe 3/MnIr 15/CoFe

2/ Ni70Cu30 30/CoFe 2/NiFe 10/CoFe 2 /Cu 7/CoFe

4/NiFe 3/MnIr 15/Ta 5 [nm]. The sample was deposited

at room temperature in a magnetic field of 350 Oe, then

annealed at 300◦C for 20 minutes, and field cooled to

RT in ∼ 800 Oe.The NiCu spacer was co-sputtered

while rotating the substrate holder, such that the final

concentration was 70% Ni and 30% Cu having the TC

suitably above RT. Fig.

4 shows how the interlayer

exchange field Hex of this sample varies with tempera-

ture.Hex shown in the main panel of Fig.

fined as the mid point switching field of the soft F2-

layer (∼ 18, ∼ 32, and ∼ 47 Oe for 100◦C, 60◦C, and

25◦C, respectively; see inset), which reflects the strength

of the interlayer exchange coupling through the spacer

undergoing a ferromagnetic-paramagnetic transition in

this temperature range. To explain why this is so, we

need to consider the difference in effective magnetic thick-

ness between the top and bottom pinned ferromagnetic

layers.The effective magnetic thickness for the bot-

4 is de-

Page 4

4

tom pinned CoFe/NiCu/CoFe/NiFe/CoFe layers is ap-

proximately three times larger than for the top pinned

CoFe/NiFe. From the inset to Fig 4, the temperature

variation of the exchange pinning for the top pinned

CoFe/NiFe is 20 Oe or 0.3 Oe/K. If we were to assume

that the bottom pinned CoFe/NiCu/CoFe/NiFe/CoFe

layers are coupled and reverse as one layer, and that the

variation in exchange field is caused solely by the weaken-

ing pinning at the bottom MnIr interface, then we would

expect an exchange field three times smaller than for the

top pinned CoFe/NiFe. With a three times smaller ex-

change field at RT the expected temperature variation

would be 7 Oe or 0.1 Oe/K, which clearly is much lower

than the observed change of 25 Oe (from 45 Oe to 20 Oe)

and therefore the measured de-pinning of the switching

layer is predominantly due to a softening of the exchange

spring.

We have separately measured the strength of the ex-

change pinning at the bottom MnIr surface. For CoFe

ferromagnetic layers 2-4 nm thick, the pinning strength

at RT is 500 Oe or more. At 130 C, at which the spacer

is paramagnetic and fully decoupled from the underlying

MnIr/CoFe bilayer, the pinning strength is still above 100

Oe. We therefore conclude that the dominating effect in

question is the weakening exchange spring in the spacer.

This demonstrates the principle of the thermionic spin-

valve proposed, where the P to AP switching is controlled

by temperature. The AF-pinned implementation of the

spin-thermionic valve presented should be highly relevant

for application.

50 100

T [°C]

150200

15

30

45

60

Hex [Oe]

0 50 100

14

15

H [Oe]

R [?]

100 C

60 C

25 C

o

o

o

FIG. 4:

ature T.

Si/SiO2/Ni80Fe20 3/ Mn80Ir20 15/Co90Fe10 2/ Ni70Cu30

30/Co90Fe10

2/Ni80Fe2010/Co90Fe10

4/Ni80Fe20 3/Mn80Ir20 15/Ta 5 [nm]. Inset: Current-in-plane

GMR at T=25, 60 and 100◦C.

Interlayer exchange field Hex versus temper-

The composition of the complete stack is

2 /Cu7/Co90Fe10

As is obvious from the above analysis the dependence

of the magnetization direction on temperature allows

electrical manipulations of it by Joule heating with an

applied current flowing through the stack. In the next

section we find connection between the magnetization di-

rection and the current-voltage characteristics (IVC) of

such a spin-thermionic valve.

III.THERMOELECTRIC MANIPULATION OF

THE MAGNETIZATION DIRECTION.

A.Current-voltage characteristics of the stack

under Joule heating.

If the stack is Joule heated by a current J its temper-

ature T(V ) is determined by the heat-balance condition

JV = Q(T), J = V/R(Θ),(10)

and Eq. (7), which determines the temperature depen-

dence of Θ(T(V )). Here Q(T) is the heat flux from the

stack and R(Θ) is the stack resistance. In the vicinity of

the Curie temperature T(1)

c

Eq. (7) can be re-written as

Θ =

?

±π,T ≥ T(1)

T < T(1)

c

D0

T(1)

c

T(1)

c

−TsinΘ,

c

,

(11)

(here D0is defined in Eq.(9)).

Equations (10) and (11) define the current-voltage

characteristics (IVC) of the stack, J = G(Θ(V ))V , G =

R−1, in a parametric form which can be re-written as

J =

?

?

Q(T(1)

c )

?

?

G(θ)(1 −¯Dsinθ

θ

)

V =

Q(T(1)

c )

R(θ)(1 −¯Dsinθ

θ

). (12)

The parameter θ is defined in the interval −π ≤ θ ≤ π,

¯D = D0T

Q

dQ

dT

??

T=T(1)

c

≈ D0,

and in order to derive Eq. (12) we used the expansion

Q(T) = Q(T(1)

It follows from Section II that the stack resistance is

R(0) in the entire temperature range T(V ) < T(or)

R(π) in the range T(V ) > T(1)

IVC branches J = G(0)V and J = G(π)V are linear for,

respectively,

c ) + Q′

T(T(1)

c )(T − T(1)

c ) [Q′

T≡ dQ/dT].

c

and

c . This implies that the

V < V1=

?

R(0)Q(T(or)

c

)(13)

(0 − a in Fig. 5) and

V > Vc=

?

R(π)Q(T(1)

c ) (14)

(b − b′in Fig. 5). If V1≤ V ≤ Vcthe stack temperature

is T(or)

c

≤ T(V ) ≤ T(1)

c , and the direction of the magne-

Page 5

5

tization in region 2 changes with a change of V ; hence

the IVC is non-linear there. Below we find the condi-

tions under which this branch of the IVC has a negative

differential conductance.

Differentiating Eq. (12) with respect to V one finds

dJ

dV

= R(Θ)[G?θ)(1 −¯Dsinθ/θ?]′

[R(θ)(1 −¯Dsinθ/θ)]′

???

θ=Θ(V )

(15)

where [...]′means the derivative of the bracketed quan-

tity with respect to the angle θ, and Θ(V ) is found from

the second equation in Eq. (12). From this result it fol-

lows that the differential conductance Gd(V ) ≡ dJ/dV is

negative if

d

dΘ

(1 −¯DsinΘ/Θ)

R(Θ)

< 0.

For a stack resistance of the form

R(Θ) = R+(1 − rcosΘ) (16)

where

r =R−

R+;R±=R(π) ± R(0)

2

(17)

one finds that the differential conductance dJ/dV < 0 if

D0<

3r

1 + 2r

(18)

Hence the IVC of the stack is N-shaped as shown in

Fig. 5.

We note here that the modulus of the negative differ-

ential conductance may be large even in the case that the

magnetoresistance is small. Using Eq.(15) at r ≪ 1 one

finds the differential conductance Gdiff as

Gdiff≡dJ

dV

= −R−1(0)1 − D0/3r

1 + D0/3r

(19)

which is negative provided D0 < 3r, the modulus of

Gdiff being of the order of R−1(0).

Here and below we consider the case that the elec-

tric current flowing through the sample is lower than

the torque critical current and hence the torque effect

is absent20

As the IVC curve J(V ) is N-shaped the thermoelec-

trical manipulation of the relative orientation of layers

0 and 2 may be of two different types depending on the

ratio between the resistance of the stack and resistance

of the circuit in which it is incorporated

In the voltage-bias regime which corresponds to the

case that the resistance of the stack is much larger than

resistance of the rest of the circuit, the voltage drop

across the stack preserves the given value which is ap-

proximately equal to the bias voltage and hence there

is only one value of the current (one point on the IVC)

J = Jbias corresponding to the bias voltage Vbias (see

FIG. 5:

netic stack of Fig. 1 calculated for R(Θ) = R+ − R−cosΘ,

R−/R+ = 0.2, D0 = 0.2; Jc = Vc/R(π). The branches 0 − a

and b − b′of the IVC correspond to parallel and antiparal-

lel orientations of the stack magnetization, respectively (the

parts a − a′and 0 − b are unstable); the branch a − b cor-

responds to the inhomogeneous magnetization distribution

shown in Fig. 2.

Current-voltage characteristics (IVC) of the mag-

FIG. 6: The angle Θ, which describes the tilt of the direction

of the magnetization in layer 2 with respect to that in layer

1 (see Fig. 2), as a function of voltage in the voltage-biased

regime (top) and current in the current-biased regime (bot-

tom). Both curves were calculated for R(Θ) = R+−R−cosΘ,

R−/R+ = 0.2, D0 = 0.2; Jc = Vc/R(π).

Page 6

6

Fig.5) In this case the relative orientation of the magne-

tization of layers 0 and 2 can be changed smoothly from

being parallel to anti-parallel by varying the bias voltage

through the interval V1≤ Vbias≤ Vc. This corresponds

to moving along the a−b branch of the IVC. The depen-

dence of the magnetization direction Θ on the voltage

drop across the stack is shown in Fig. 6.

In the current-bias regime, on the other hand, which

corresponds to the case that the resistance of the stack

is much smaller than the resistance of the circuit, the

current in the circuit J is kept at a given value which

is mainly determined by the bias voltage and the circuit

resistance (being nearly independent of the stack resis-

tance). As this takes place, the voltage drop across the

stack V differs from the bias voltage Vbias, being deter-

mined by the equation J(V ) = J. As the IVC is N-

shaped, the stack may now be in a bistable state: if the

current is between points a and b there are three possible

values of the voltage drop across the stack at one fixed

value of the current (see Fig. 5). The states of the stack

with the lowest and the highest voltages across it are sta-

ble while the state of the stack with the middle value of

the voltage drop is unstable. Therefore, a change of the

current results in a hysteresis loop as shown in Fig. 6:

an increase of the current along the 0 − a′branch of the

IVC leaves the magnetization directions in the stack par-

allel (Θ = 0) up to point a, where the voltage drop V

across the stack jumps to the right branch b − b′, the

jump being accompanied by a fast switching of the stack

magnetization from the parallel to the antiparallel orien-

tation (Θ = ±π). A decrease of the current along the

b′−0 IVC branch keeps the stack magnetization antipar-

allel up to point b, where the voltage jumps to the left

0 − a′branch of the IVC and the magnetization of the

stack comes back to the parallel orientation (Θ = 0).

In the next Section we will show that this scenario for

a thermal-electrical manipulation of the magnetization

direction is valid for small values of the inductance in the

electrical circuit. If the inductance exceeds some critical

value the above steady state solution becomes unstable

and spontaneous oscillations appear in the values of the

current, voltage drop across the stack, temperature, and

direction of the magnetization.

B.Self-excited electrical, thermal and directional

magnetic oscillations.

1. Current perpendicular to layer planes (CPP)

Consider now a situation in the bias voltage regime

where the magnetic stack under investigation is con-

nected in series with an inductance L and biased by a

DC voltage Vbias, as described by the equivalent circuit

in Fig. 7. The thermal and electrical processes in this

FIG. 7: Equivalent circuit for a Joule-heated magnetic stack

of the type shown in Fig. 1. A resistance R(V ) = J(t)/V (t),

biased by a fixed DC voltage Vbias, is connected in series with

an inductance L; V (t) is the voltage drop over the stack and

J(t) is the total current.

system are governed by the set of equations

CVdT

dt

= J2R(Θ)−Q(T);LdJ

dt+JR(Θ) = Vbias, (20)

where CV is the heat capacity. The relaxation of the

magnetic moment to its thermodynamically equilibrium

direction is assumed to be the fastest process in the prob-

lem, which implies that the magnetization direction cor-

responds to the equilibrium state of the stack at the

given temperature T(t). In other words, the tilt angle,

Θ = Θ(T(t)), adiabatically follows the time-evolution of

the temperature and hence its temperature dependence

is given by Eq. (7).

A time dependent variation of the temperature is ac-

companied by a variation of the magnetization angle

Θ(T(t)) and hence by a change in the voltage drop across

the stack via the dependence of the magneto-resistance

on this angle, R = R(Θ).

The system of equations Eq.(20) has one time-

independent solution (¯T(Vbias),¯J(Vbias)) which is deter-

mined by the equations

J2R(Θ(T)) = Q(T),JR(Θ(T)) = Vbias

(21)

This solution is identical to the solution of Eqs.(7,10)

that determines the N-shaped IVC shown in Fig.5 with

a change J →¯J and V → Vbias.

In order to investigate the stability of this time-

independent solution we write the temperature, current

and the angle as a sum of two terms,

T =¯T(Vbias) + T1(t);

J =¯J(Vbias) + J1(t);

Θ =¯Θ(Vbias) + θ1(t),(22)

where T1, J1and θ1each is a small correction. Insert-

ing Eq.(22) into Eq.(20) and Eq. (7) one easily finds that

the time-independent solution Eq.(21) is always stable

at any value of the inductance L if the bias voltage Vbias

corresponds to a branch of the IVC with a positive differ-

Page 7

7

FIG. 8: Spontaneous oscillations of the current J(t) and the

voltage drop V (t) over the stack calculated for R−/R+ = 0.2,

D0 = 0.2 and (L−Lcr)/Lcr = 0.013; Jc = Vc/R(π). J(t) and

V (t) develop from the initial state towards the limit cycle

(thick solid line) along which they execute a periodic motion.

The thin line is the stationary IVC of the stack. The bottom

figure shows the limit cycle along which Θ(t) and V (t) execute

a periodic motion.

ential resistance (branches 0-a and b-b’ in Fig.5). If the

bias voltage Vbiascorresponds to the branch with a neg-

ative differential resistance (V1< Vbias< Vc, see Fig.5)

the solution of the set of linearized equations is T1 =

T(0)

1

exp{γt}, J1 = J(0)

1

exp{γt} and θ1 = θ(0)

where T(0)

1

and θ(0)

1

are any initial values close to

the steady-state of the system, and

1

exp{γt}

1, J(0)

γ =

¯R

2L

?L − Lc

Lc

±

??L − Lc

Lc

?2

− 4|Rd|

¯R

L

Lc

?

(23)

where

Lc=

CV

|d(GQ)/dT|

???

T=T(V )

(24)

and Rd= dV/dJ,¯R = R(¯Θ) is the differential resistance.

As is seen from Eq.(23) the steady-state solution

Eq.(21) is stable only if the inductance L ≤ Lc; if the

inductance exceeds the critical value Eq.(24) the system

looses its stability and a limit cycle appears in the plane

(J,T) (see, e.g.,22). This corresponds to the appearance

of self-excited, non-linear and periodic temporal oscil-

lations of the temperature T = T(t) and the current

J = J(t), which are accompanied by oscillations of the

voltage drop across the stack˜V (t) = J(t) and the the

magnetization direction Θ(t) = Θ(T(t)). For the case

that (L − Lc)/Lc ≪ 1 the system executes nearly har-

monic oscillations around the steady state (see Eq.(22))

with the frequency ω = Imγ(L = Lc), that is the tem-

perature T, the current J, the magnetization direction Θ

and the voltage drop across the stack V (t) = R(Θ(t))J(t)

execute a periodic motion with the frequency

ω =

?¯RRd

Lc

(25)

With a further increase of the inductance the size of

the limit cycle grows, the amplitude of the oscillations

increases and the oscillations become anharmonic, the

period of the oscillations therewith decreases with an in-

crease of the inductance L.

In order to investigate the time evolution of the voltage

drop across the stack and the current in more details

it is convenient to introduce an auxiliary voltage drop

˜V (t) and a current J0(t) related to each other through

Eqs. (10) and (11). Hence we define

˜V (t) =

?

R(T(t))Q(T(t)));J0=˜V (t)/R(T(t)), (26)

where R(T) = R(Θ(T)). Comparing these expressions

with Eq. (10) one sees that at any moment t Eq. (26)

gives the stationary IVC of the stack, J0= J0(˜V ), defined

by Eq. (12) (changing J → J0and V →˜V ), see Fig. (5).

Differentiating

˜V (t) with respect to t and using

Eqs. (20) and (26) one finds that the dynamical evolution

of the system is governed by the equations

τ0d˜V

dt

=J2− J2

2J0(˜V )

J˜V

J0(˜V )= Vbias

0(˜V )

LdJ

dt+

(27)

where

τ0=

CV

(QR)′

T

???

T=T(˜V ).

As follows from the second equation in Eq. (26), at

any moment t the voltage drop over the stack V (t) =

R(T(t))J(t) is coupled with˜V (t) by the following rela-

tion:

V =

J

J0(˜V )

˜V .

The coupled equations (20) have only one steady-state

Page 8

8

FIG. 9: Spontaneous oscillations of the current J(t), the volt-

age drop V (t), the magnetization direction angle Θ(t), and the

temperature T(t) corresponding to motion along the limit cy-

cle shown in Fig.8. Calculation parameters are R−/R+ = 0.2,

D0 = 0.2 and (L − Lcr)/Lcr = 0.3 × 10−4; Jc = Vc/R(π).

solution J = J0(Vbias) where J0(V ) is the IVC shown

in Fig.5 (see Eqs. (26). However, in the interval V1 ≤

Vbias≤ Vcthis solution is unstable with respect to small

perturbations if L > Lcr. As a result periodic oscilla-

tions of the current J(t) and˜V (t) appear spontaneously,

with J(t),˜V (t) eventually reaching a limit cycle. The

limiting cycle in the J-V plane is shown in Fig. 8. The

stack temperature T = T(t), the magnetization direction

Θ(t) = Θ(T(t)), follow these electrical oscillations adia-

batically according to the relations Q(T(t)) =˜V (t)J0(t)

(here J0(t) ≡ J0(˜V (t))) and Θ(t) = Θ(T(t)) (see Eq. (7))

as shown in Fig. 9.

The character of the oscillations changes drastically in

the limit L ≫ Lcr. In this case the current and the volt-

age slowly move along the branches 0 − a and b − b

the IVC at the rate˙J/J ≈ R+/L, quickly switching be-

tween these branches at the points a and b with the rate

∼ 1/τ0(see Fig. 10). Therefore, in this case the stack pe-

riodically switches between the parallel and antiparallel

magnetic states (see Fig. 11).

′of

2.Current in the layer planes (CIP).

If the electric current flows in the plane of the lay-

ers (CIP) of the stack the torque effect is insufficient or

absent1,21while the magneto-thermal-electric oscillations

under consideration may take place. In this case the to-

tal current flowing through the cross-section of the layers

may be presented as

JCIP=?R−1(Θ) + R−1

0

?V(28)

where R(Θ) and R0are the magneto-resistance and the

angle-independent resistance of the stack in the CIP set

of the experiment.

In a CIP configuration the stack is Joule heated by

both the angle-dependent and the angle-independent cur-

FIG. 10: Spontaneous oscillations of the current J(t) and the

voltage drop˜V (t) calculated for R−/R+ = 0.2, D0 = 0.2 and

(L−Lcr)/Lcr = 535; Jc = Vc/R(π). The time development of

J(t) and˜V (t) follows one or the other of the dashed lines to-

wards the limit cycle (thick solid line) depending on whether

the initial state is inside or outside the limit cycle. The bot-

tom figure shows how the current oscillations develop if the

initial state is inside the limit cycle. The stationary IVC of

the stack is shown as a thin solid line.

FIG. 11: Spontaneous oscillations of the magnetization direc-

tion angle Θ(t) calculated for R−/R+ = 0.2, D0 = 0.2 and

(L − Lcr)/Lcr = 535; Jc = Vc/R(π).

Page 9

9

rents and hence Eq.(10) should be re-written as follows:

JCIPV = Q(T), J = V/Reff(Θ),(29)

where

Reff(Θ) =

R(Θ)R0

R(Θ) + R0

(30)

Using Eq.(15) and Eq.(28) one finds that the pres-

ence of the angle-independent current in the stack modi-

fies the condition of the negative differential conductance

dJCIP/dV : it is negative if

¯D <

3r

(1 + 2r)R0+ (1 − r)2(1 − 4r)R+

?

R0−(1−r)2R+

?

(31)

As is seen from here, an IVC with a negative differential

resistance is possible if R0> (1 − r)2R+(see Eq.[17] for

definitions of R±and r).

The time evolution of the system is described by the

set of equations Eq.(20) in which one needs to change

J → JCIP and R(Θ) → Reff(Θ).

this change, the temporal evolution of the system in

a CIP configuration is the same as when the current

flows perpendicular to the stack layers: if the bias volt-

age corresponds to the negative differential conductance

dJCIP/dV < 0 and the inductance exceeds the critical

value

Therefore, under

Lc=

CV

|d(GeffQ)/dT|

???

T=T(V )

(32)

where Geff = R−1

rent JCIV, voltage drop over the stack V , the tempera-

ture T and the angle Θ(T(V )) arise in the system, the

maximal frequency of which being

eff, self-excited oscillations of the cur-

ω =

?|dV/dJCIP|Reff(T(V ))

Lcr

???

V =Vbias

(33)

if (L − Lcr)/Lcr≪ 1

Below we present estimations of the critical inductance

and the oscillation frequency which are valid for both

the above mentioned CPP and CIP configurations of the

experiment.

Using equations Eq.(21) and Eq.(24) one may esti-

mate the order of magnitude of the critical inductance

and the oscillation frequency as Lc ≈ Tcv/j2d and

ω ≈ ρj2/Tcvwhere cvis the heat capacity per unit vol-

ume, ρ is the resistivity, and d is a characteristic size of

the stack. For point contact devices with typical values

of d ∼ 10−6÷ 10−5cm, cv∼ 1 J/cm3K, ρ ∼ 10−5Ωcm,

j ∼ 108A/cm2and assuming that cooling of the device

can provide the sample temperature T ≈ T(1)

one finds the characteristic values of the critical induc-

tance and the oscillation frequency as Lcr≈ 10−8÷10−7H

and ω ≈ 1GHz

c

∼ 102K

IV.CONCLUSIONS.

The experimental implementation of the new princi-

ple proposed in this paper for the electrical manipula-

tion of nanomagnetic conductors by means of a controlled

Joule heating of a point contact appears to be quite fea-

sible. This conclusion is supported both by theoretical

considerations and preliminary experimental results, as

discussed in the main body of the paper. Hence we ex-

pect the new spin-thermo-electronic oscillators that we

propose to be realizable in the laboratory. We envision

F0/f/F2 valves where two strongly ferromagnetic regions

(Tc∼ 1000 K) are connected through a weakly ferromag-

netic spacer (Tc ≪ 1000 K). The Curie temperature of

the spacer would be variable on the scale of room temper-

ature, chosen during fabrication to optimize the device

performance. For example, doping Ni-Fe with ∼ 10% of

Mo brings the Tc from ∼ 1000 K to 300-400 K. Alter-

natively, alloying Ni with Cu yields a spacer with a Tc

just above Room Temperature (at RT or below RT, if

needed). If a sufficient current density is created in the

nano-tri-layer to raise the temperature to just above the

Tc of the spacer, the magnetic subsystem undergoes a

transition from the F0/f/F2 state to an F0/N/F2 state,

the latter being similar to conventional spin-valves (N for

nonmagnetic, paramagnetic in this case). Such a transi-

tion should result in a large resistance change, of the same

magnitude as the “giant magnetoresistance” (GMR) for

the particular material composition of the valve.

Local heating (up to 1000 K over 10-50 nm) can read-

ily be produced using, e.g., point contacts in the thermal

regime, with very modest global circuit currents and es-

sentially no global heating13. Heat is known to propagate

through nm-sized objects on the ns time scale, which can

be scaled with size to the sub-ns regime. When voltage-

biased to generate a temperature near Tc(f), such a

F0/f/F2 device would oscillate between the two magnetic

states, resulting in current oscillations of a frequency that

can be tuned by means of connecting a variable induc-

tance in series with the device. Spin rotation frequencies

may be tuned from the GHz-range down to quasi-DC (or

DC as soon as the inductance is smaller than the critical

value). For F0/f/F2 structures geometrically designed

in the style of the spin-flop free layer of today’s magne-

toresistive random access memory (MRAM), the dipolar

coupling between the two strongly ferromagnetic layers

would make the anti-parallel state (F0 ↑ /N/F2 ↓ ) the

magnetic ground state above Tc(f). The thermal transi-

tion in the f-layer would then drive a full 180-degree spin-

flop of the valve. The proposed spin-thermo-electronic

valve can be implemented in CPP as well as CIP geome-

try, which should make it possible to achieve MR signals

of 10.

In conclusion, we have shown that Joule heating of the

magnetic stack sketched in Fig. 1 allows the relative ori-

entation of the magnetization of the two ferromagnetic

layers 0 and 2 to be electrically manipulated.

on this principle, we have proposed a novel spin-thermo-

Based

Page 10

10

electronic oscillator concept and discussed how it can be

implemented experimentally.

Acknowledgement. Financial support from the Swedish

VR and SSF, the European Commission (FP7-ICT-2007-

C; proj no 225955 STELE) and the Korean WCU pro-

gramme funded by MEST through KOSEF (R31-2008-

000-10057-0) is gratefully acknowledged.

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18The temperature T(or)

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< T(1)

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(Oxford, University Press, 2000).

c

is the critical temperature

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