Diego Jair

Rodriguez Obando

John Jairo

Martinez Molina

Christophe Bérenguer

D J Rodriguez

J J Martinez

C Berenguer

Deterioration modelling of contact surfaces for a friction drive system

INTRODUCTION

The Remaining Useful Life (RUL) of an asset or system is defined as the time left from the current time to the end of the useful life. Prognosis of RUL is a widely studied topic in health management of resources for automated engineered. RUL prognosis has also an important role in the management of energy resources for e.g. autonomous devices. A good overview over existing methods for RUL prognosis is given in [START_REF] Si | Remaining useful life estimation-a review on the statistical data driven approaches[END_REF]. Note that the meaning of "useful life" varies according to the studied engineering field, thus the owner of the asset is the one who defines the sense of useful life and its management.

Deterioration of components and failures affects directly the lifetime of assets, and consequently their usability and/or productivity. On the other hand, energy consumption is another critical variable to manage in relation with the deterioration behavior. For this reason, nowadays manufacturers or end users are becoming increasingly motivated to manage the complete life-cycle of an asset and to optimize the production process and its energy consumption using proactive strategies. Thus, an optimal management between health state of components and use of energy resources is required, which is defined in this paper as a "RUL-aware" operation.

The aim of this work is to develop a mathematical model for the deterioration of the contact surfaces of a friction drive system, which is a type of transmission that uses two circular devices to transfer mechanical power by friction, used e.g. in some electrical bikes with two main components: wheel and DC-motor. The basic model of friction drive transmission relies on the assumption of equal tangential speeds of wheel and motor surfaces. However, the difference between speeds can not be always neglected because this is linked with the efficiency of mechanical transmission that drives the wheel and the production of contact force, which is associated with a transfer of energy that also deteriorates the contact surfaces. Then, the energy transfer is at the same time, the useful phenomenon and the source of deterioration and as such it requires to be properly modeled; thus, allowing an optimal control ensures both a sufficient drive power and an acceptable deterioration. As explained in Section 4, this work can then be seen as a required step towards developing a complete Reliability Adaptive Systems or RAS according to [START_REF] Meyer | Closed-loop control system for the reliability of intelligent mechatronic systems[END_REF].

We consider a special case of a friction drive system (roller-on-tire system) and we develop an analytical physics-based model for the deterioration of the driving contact surfaces. The developed model integrates the deterioration as one of the states of the system, thus it allows: a) to link the deterioration evolution to the inputs and operating modes of the system, b) to assess the behavior of deterioration in critical stages, such as sharp motion and smooth motion and c) to predict analytically an interval of useful lifetime before the total-failure-time of actuator, taking into account stochastic perturbations. The remainder of this paper is organized as follows. Section 2 presents the design and analysis of the deterioration model of the considered friction drive system. Section 2.1 contains the general definition of the particular friction drive which here is called "roller-on-tire" system. In a first step, in Section 2.2, the physics of the system, and in particular the link between deterioration dynamics and energy transfer at the friction surface, is modeled in a deterministic way. In Section 2.3 an useful space state representation is presented. Then, considering either random system parameters or random system inputs and usage, a complete stochastic model for the system deterioration is proposed in Section 2.4.

Section 3 shows the analysis of simulations for a case study. Finally, plans for future work directions are given in Section 4 in the framework of RAS.

FRICTION DRIVE SYSTEM MODELLING

Figure 1 presents the basic principles of the mechanical devices considered in this work: a power transmitted from the source to the load, P L , is higher than the power which actually causes the motion, P * L . This phenomenon is modeled under the general hypothesis that the loss of efficiency in the conversion from energy to motion can be an estimable image of the deterioration of materials in contact.

Roller-on-tire system description

The "roller-on-tire" system is a friction drive composed by a driven device and a motor, as depicted in figure 2, for which it is supposed that: a) both contact surfaces, the rotor of the motor and the external part of the driven device, deteriorate, b) both surface deterioration of the motor and the tire reach eventually a threshold above which the system is considered failed, and c) an inversion of motion sense cannot occur at any time.

Symbols of the equations and units are shown in Table 1.

Analytical physics-based modelling

The mechanical analysis is based on the Dahl model and LuGre model of friction, which are widely studied in the literature of mechanics. A good overview about those models can be read in [START_REF] Geffen | A study of friction models and friction compensation[END_REF] and (Armstrong-Helouvry & Canudas de Wit 1996). These models represent the friction force with the components of the Stribeck friction effect, for which the friction force has an inverse relation with respect to the speed; the Coulomb friction, for which the force remains as a constant with respect to the speed at the beginning of motion; and the viscous friction, for which the friction force is proportional to the speed.

In the present contribution, Stribeck friction effect and Coulomb friction are considered negligible; that is, only the viscous friction is taken into account, due to the fact that it does not exist any possibility of inversion of the motion sense.

Here F c is assumed as a frictional force, tangential to both the motor and the driven devices, which causes a rotational torque over the driven device. A similar approximation of the law of friction by a continuous function of velocity is analysed in [START_REF] Popov | Contact Mechanics and Friction: Physical Principles and Applications[END_REF]. Consequently, the friction force component depicted in Equation (1), depends on the difference ∆ v between the tangential speeds of motor and driven device, v 1 and v 2 respectively, and the coefficient α, which in this paper is called contact quality coefficient and that corresponds to the viscous friction coefficient in the classic context of friction components modelling.

F c = α(v 1 -v 2 ) = α(r 1 ω 1 -r 2 ω 2 ) = α∆ v (1)
The parameter α is classically considered to be constant, but in this paper it is treated as a timevarying parameter. In some sense, this α gives an initial idea about the quality of contact between both rolling devices in each time of the motion. Then, because of the frictional phenomena, the condition of contact depends on the deterioration of materials and varies with time, i.e. α = f (D, t) 0.

Thus both the torque produced by the motor and transferred to the load T L (t) and the torque observed by the driven device T * L (t) depend on F c (t) according to equations ( 2) and (3). In this paper, dependence on time is not written for convenience.

T L = F c r 1
(2)

T * L = F c r 2 (3)
The system dynamics using Newton's laws of motion can be written as follows:

ΣT 1 = J 1 ω1 = T m -T L -B 1 ω 1 ω1 = 1 J 1 (K m I -r 1 α(r 1 ω 1 -r 2 ω 2 ) -B 1 ω 1 ) (4)
ΣT 2 = J 2 ω2 = T * L -B 2 ω 2 ω2 = 1 J 2 (r 2 α(r 1 ω 1 -r 2 ω 2 ) -B 2 ω 2 ) (5)
Equations ( 4) and ( 5) can be rewritten in terms of power multiplying them by ω 1 and ω 2 respectively as follows:

J 1 ω1 ω 1 = K m Iω 1 -T L ω 1 -B 1 ω 2 1 (6) J 2 ω2 ω 2 = T * L ω 2 -B 2 ω 2 2 (7)
where terms in equations are shown in Table 2: Taking into account Equations ( 6) and ( 7), power in contact P c can be expressed as follows:

P c = P L -P * L = T L ω 1 -T * L ω 2 P c = F c (r 1 ω 1 -r 2 ω 2 ) = F c ∆ v (8)
Using Equation (1) this takes the form:

P c = α(r 1 ω 1 -r 2 ω 2 ) 2 = α∆ 2 v (9)
We define the dynamic efficiency of the transmission η c (t) as:

η c = P * L /P L = (P L -P c )/P L (10)
which is equivalent to:

η c = F c r 2 ω 2 /F c r 1 ω 1 = r 2 r 1 ω 2 ω 1 = v 2 v 1 (11)
We assume that it is possible to measure the speeds v 1 and v 2 (or equivalently, ω 1 and ω 2 ) in this kind of mechanisms. Thus η can be computed dynamically.

The energy dissipated in the contact E c between the surfaces in roller-on-tire system can be obtained from (9). Thus E c depends on the speed between the surfaces as follows:

E c = t 2 t 1 P c dt = t 2 t 1 α∆ 2 v dt (12)
The dissipated energy could be considered as an image of the heat and material losses produced at the contact level during traction, as considered Similar conclusion were found by [START_REF] Meyer | Closed-loop control system for the reliability of intelligent mechatronic systems[END_REF]. Thus, if heat losses are neglected, deterioration of surfaces D c can be considered as proportional to the dissipated energy in the contact:

D = cE c = c t 2 t 1 P c dt = c t 2 t 1 α∆ 2 v dt ( 13
)
where c is a proportionality factor that relates deteriorated material because of friction with the energy dissipated. That is, here the deterioration is considered as an image of the dissipated energy at the contact level during power transmission. Equation ( 13) shows that the dynamics of the system and the contact surfaces deterioration are linked through α. From Equation (13) and since α = f (D), a nonlinear variation of the deterioration with respect to time according can be obtained:

Ḋ = cα∆ 2 v = H nonlinear (ω 1 , ω 2 , D) (14)
At the same time, α varies depending on the deterioration of materials. It is assumed that the bigger D, the lower α and that D is monotonically increasing and bounded until D max . We define the maximum value of deterioration D max , as the value of D in which the system is considered completely failed. In this paper, it is considered that the evolution of α depends on the evolution of the deterioration D as depicted in Figure 3. Consequently, a first order linear variation of α, with respect to D with slope m > 0, m ∈ R + and initial value b > 0, b ∈ R + , is defined as:

α = -mD + b (15)
From Equations ( 14) and ( 15), we get: Taking into account the term d(t) = c(r 1 ω 1r 2 ω 2 ) 2 , Equation ( 17) can be rewritten as follows:

Ḋ = (-cmD + cb)(r 1 ω 1 -r 2 ω 2 ) 2 (16) Ḋ = -m c∆ 2 v d(t) D + b c∆ 2 v d(t) (17)
Ḋ = -md(t)D + bd(t) (18)
We can compute the maximum value of the deterioration with respect to time D max as:

D max lim t→+∞ D(t) (19)
This can be calculated using Equation ( 18) with Ḋ = 0, thus:

-mD max + b = 0 (20) D max = b/m = α(0)/m (21)
In a complementary way, we define the normalized deterioration as:

D D(t)/D max = (m/α(0))D(t) (22)
From Equation (15), we get the special form:

D(α) = (b -α)/m = (α(0) -α)/m (23) Since ∆ v = v 1 -v 2 = r 1 ω 1 -
r 2 ω 2 and using (4) and ( 5):

∆v = (r 1 /J 1 )(K m I -αr 1 ∆ v -B 1 ω 1 ) -(r 2 /J 2 )(αr 2 ∆ v -B 2 ω 2 ) (24)
Equation ( 24) shows that it is possible to estimate the value of α from the measured speeds (recall that measuring the speeds is considered feasible in this kind of mechanisms), and assuming the other parameters of the systems as constant. Let denote this estimated value as α. Therefore using Equation ( 23), the deterioration D can be estimated by:

D = (α(0) -α)/m (25)
Consequently, using equations ( 21), ( 22) and ( 25) it is obtained the normalized estimation of deterioration D:

D = D/D max = (α(0) -α)/α(0) (26)
Deriving Equation ( 23) with respect to time, and using Equation ( 14), we get: 28) can be rewritten taking into account Equation ( 9):

Ḋ = -(1/m) α (27) α = -mc Ḋ = -mcα∆ 2 v (28) Equation (
α = -mc t 2 t 1 P c dt (29)
Therefore:

α = -mcE c + α(0) (30)
Summarizing, assuming a monotonically increasing of deterioration, the proposed model shows that roller-on-tire system reaches a maximum value of deterioration D max , above which the transmission is consider failed, which depends on the initial value of the contact quality coefficient α(0) and the slope m, which represents the variation of α(0) with respect to deterioration of material D. At the same time the normalized estimation of deterioration D can be obtained from α(0) and its estimation α. The estimated coefficient α can measure the loss of material or the modification of the surface properties at the contact level, i.e. the material that allows the contact (or the existence of mechanical transmission). At the same time α establishes a relation between material of contact and energy of contact.

State-space model for the roller-on-tire system

In accordance with the above, roller-on-tire system is an uncertain system which depends on time-varying unknown-but-bounded uncertain parameters. Taking into account the states:

x = ω 1 ω 2 D , (31)
the control input u = I and the output y, the representation of the system can be expressed as:

ẋ = f (x) + Bu y = Cx (32)
B is an input matrix and C the output matrix. In an explicit form, the representation of the system is:

f (x) =      - 1 J 1 (B 1 + αr 2 1 ) ω 1 + 1 J 1 αr 1 r 2 ω 2 1 J 2 αr 2 r 1 ω 1 - 1 J 2 (B 2 + αr 2 2 ) ω 2 cα(r 1 ω 1 -r 2 ω 2 ) 2      B =    K m J 1 0 0    ; C = 1 0 0 0 1 0 0 0 1 ; u = I (33)
Remark that in this case the non-linearity of the system is due to the introduction of the state D.

Introducing randomness in the model

Randomness in the model can be introduced in two complementary modes: a) internal, to model the uncertainty on the parameters m and b (which are used to relate a with D) for a given system, and the systemto-system variability and b) external, to model the variability of the operating conditions and the usage system, in this case taking into account the input u.

Uncertain/random parameters

In Equation ( 15) b ≥ 0 and m ≥ 0 are considered as unknown parameters, but which belong to a given known interval taking into account real applications. Thus a normal distribution is introduced to represent them:

m ∼ N (m m , σ 2 m ), m m > 0 b ∼ N (b m , σ 2 b ), b m > 0 (34)
where m m and σ m are the mean value and the standard deviation of the slope m respectively, and b m and σ b are the mean value and the standard deviation of b respectively.

As depicted in the shaded area in Figure 4 and according to Equations ( 15) and ( 35), the behavior of α with respect to D constitutes a bi-dimensional space of variation. The solid line inside the shaded area is an example of probable behavior.

Variable operating conditions and usage

The developed model is single input u = I. In this case, the variability of the operating conditions and the usage system is represented as the variation of I.

Consider an input I as a square signal. The times t h , in which I is in high and t l , in which I is in low can vary randomly. Thus an exponential distribution is introduced to represent it:

t l ∼ Exp(1/µ tl ), 0 < µ tl t h ∼ Exp(1/µ th ), 0 < µ tl < µ th (35)
where µ tl and µ th are the mean values of t l and t h respectively.

The deterministic operational mode, the internal mode of randomness and the external mode of randomness are used in a case study analysis in Section 3.

CASE STUDY ANALYSIS

To choose the parameters for simulations in order to validate the usefulness of the model, the contributions of [START_REF] Huang | Friction modeling of a freespinning bicycle wheel[END_REF] and [START_REF] Rodriguez | Active force control for electric assist bicycles[END_REF] were analyzed. In the first article, a similar workbench has a built-in bike wheel to model viscous and dry friction at the wheel bearing. The second contribution is a master-thesis developed in GIPSA-Lab (CNRS-France, Grenoble-INP, University Grenoble Alpes), which analyses the implementation of a real friction drive in an electrical bicycle. Table 3 shows the chosen parameters.

6.36x10 -3 [Kgm 2 /s] B 2 1.76x10 -3 [Kgm 2 /s] J 1 3.47x10 -4 [Kgm 2 ] J 2 0.2 [Kgm 2 ] r 1 0.0315 m r 2 0.35 m K m 0.0477465 [V s/rad]
3.1 Deterministic operational mode

Sharp stage of motion

The system model is simulated with an input step signal I with amplitude = 20A. Figure 5 shows the behavior of the power in the system.

Here, P L the transferred power to the load, and P * L , the power seen by the driven device have a high value only at the beginning, and such as it was supposed P L > P * L . The difference between P L and P * L reflects the loss in contact, primarily at the beginning of motion, i.e. the sharp stage.

Remark that P L and P * L reach a peak in the sharp stage, and after they drop; while P motor , the electrical power of the motor, and P B1 , the power that appears as a result of the viscous damping parameter B 1 reach high and near values. P * L and P B2 are much lower because they depend on ω 2 which is lower than ω 1 . Here, it is possible to see that there is a considerable influence of the viscous damping parameter B 1 in the consumption of energy, i.e. the motor must provide more power to drive its own viscous damping parameter B 1 .

Figure 6 shows the behavior of the difference of power for given values of α with respect to time. Experimentally, it was found a maximum peak of this difference, i.e. the minimum efficiency η when α is equal to 5.25. It means that the increasing of α tends to give lower losses but it do not necessarily produces the highest peak of losses. These dynamic changes reflect the importance to estimate and possibly to control the value of α. Figure 7: Deterioration for given values of α for sharp stage of motion. The higher the ∆ v , the higher the deterioration of material.

Figure 7 shows the behavior of the instantaneous D for a given parameter c = 1 for the same given values of α than the figure 6. It is evident that D is always increasing and that the highest values are related with power losses. It is also remarkable that the highest consumption of contact energy (material of contact) happen in the first seconds for every value of α, i.e. the sharp stage of motion when also ∆ v is higher.

Smooth stage of motion

Given values c=1, m=0.01 and b=10 are determined arbitrarily in the framework of possible implementation and taking into account previous simulations in order to introduce the variation of α according to Equation (15). Figure 8 shows the behavior of states v 1 , v 2 and D. Here is possible to see the behavior of deterioration at the smooth stage of motion, i.e. when ∆ v is lower, until the total failure. Deterioration reaches the D max value at 300 hours of use for the given scenario. Finally, the efficiency η of the mechanical power transmission is evaluated in the sharp stage and in the failure stage as depicted in Figure 9. Remark that in the sharp stage the variation of η is sharp also. On the other hand, η begins to variate before of the total failure and it has a slow variation.

Variable behavior of input

A square signal is introduced to represent several cycles of use. In this case t h and t l are the times in which the signal is in high and low respectively. One use cycle is the time in which the contact material deteriorate, i.e. when ∆ v > 0.

Figure 10 shows the states v 1 , v 2 and D with an fixed input of t h = 10000s and t l = 10s chosen arbitrarily. As a result, v 1 continues responding to the input signal, but not v 2 . Here is shown that the rolleron-tire system reaches the D max near to 96.7h. It means that with multiple cycles of activation, an earlier D max is reached. Figure 11 shows the behavior of α for the same scenario.

Stochastic operational mode

Internal mode of randomness

According to Section 2.4 we can introduce randomness in the model. Firstly, a fixed step input is introduced to the system, which represents a cyclical behavior, where t h and t l are constant.

Figure 12 compared with an scenario with fixed values of t h = 10000s and t l = 10s. This simulation allows inferring that if it is possible to know the values of m and b since the beginning then is could be possible have a good estimation of time of failure.

External mode of randomness

Figure 13 shows D for 100 simulations. Here t h and t l are chosen randomly using an exponential distribution with mean values of t h = 10000s and t l = 10s respectively and parameters b = 10 and m = 0.01. The different behaviors are compared with an scenario with fixed values of t h = 10000s and t l = 10s.

This kind of external mode of randomness is similar to real operation conditions, for which the t h and In some systems, for example in the friction-based ones, the failure is imminent because of deterioration of materials in contact, and the main goal is trying to optimize the use of functioning resources until the failure. In other words, planned actions which tend to initiate changes of behavior of the component, rather than reacting to failures are needed. To achieve that, to lower work load or wear by selecting appropriate optimal system configurations of components is required according to [START_REF] Meyer | Closed-loop control system for the reliability of intelligent mechatronic systems[END_REF]. Systems which possess this capability are called Reliability-Adaptive Systems or RAS by [START_REF] Rakowsky | Modelling reliability-adaptive multisystem operation[END_REF]. In other words a reliability control can allow to manage the deterioration of components like in the self-optimizing mechatronic systems, which are, according to (Gausemeier, Rammig, Schäfer, & Sextro 2014), a class of intelligent technical systems able to autonomously adapt their behavior if user requirements or operating conditions change.

In the framework of PHM modern approaches the proactive actions are focused in system-level and component-level. Then, combination of experienced-based, data-driven and model-based approaches [START_REF] Byington | Prognostic enhancements to gas turbine diagnostic system[END_REF], seems to be an appropriate way to improve the efficiency of management proactive actions according to [START_REF] Muller | Formalisation of a new prognosis model for supporting proactive maintenance implementation on industrial system[END_REF]. In this context, common goals for the proactive actions are a) the reduction of the probability of unexpected component failures in a reliable way and/or b) the feasible implementation on-line.

In this sense, component-level proactive strategies continue being explored (See for example [START_REF] Nguyen | Remaining useful life estimation of stochastically deteriorating feedback control systems with a random environment and impact of prognostic result on the maintenance process[END_REF]). Physics-based models can explain the phenomena of deterioration in a deterministic way, which can be useful to manage the useful lifetime of a component in control loop to get systems type RAS with feasible implementation. This is specially useful in embedded systems due the computational restrictions.

In this paper, we developed a deterioration model linking the system control input and the deterioration behavior, which can be seen as a first step towards the development of a comprehensive reliability adaptive system. The next step is using the information and the features of the model to estimate on-line the deterioration and the model parameters, to synthesize a reliability adaptive control law in order to manage the RAS taking into account jointly the different point of views of reliability, performance and energy efficiency.

CONCLUSIONS

The proposed model allows analytically to predict an interval of useful lifetime before the total-failure-time of actuator, including stochastic perturbations. The model takes into account the variation of the contact quality coefficient with respect to the deterioration of the contact-surface material, and this deterioration with respect to the difference of the tangential speeds. Results show that the model represents properly and dynamically the deterioration of surfaces depending of the difference of tangential speeds, i.e. the more the difference (sharp motion stage), the more deterioration. In the smooth motion stage, the estimation of deterioration depends on the initial estimation of the contact quality coefficient.

The estimation of contact quality coefficient is supported on the assumptions that the relative speed is known at every time-instant. Results show that if it is possible to know the value of contact quality coefficient at every time-instant, then it could be possible to have a good estimation of the failure time. It could be useful, for example to estimate the failure time with an acceptable level of uncertainty. Additionally, despite of the unknown input conditions, the deterioration behavior follows central tendency. The precision only depends on the size of the uncertainties on the system parameters and future system usages. Therefore, it could be feasible, for example to esti-mate the failure time with an acceptable level of uncertainty.

This model can be probed in close loop scenarios, with the advantage of probing and simulating control techniques on reliability and fault tolerant control areas for a basic mechanical system. This is particularly useful in learning and researching about RAS systems. Indeed, since such a model would link the control and the reliability of the system, it opens the door to develop a control system which allows the device to adapt autonomously its behavior if user requirements or operating conditions change.

Figure 1 :

 1 Figure 1: Losses of power in contact

Figure 2 :

 2 Figure 2: Basic roller-on-tire motor system

Figure 3 :

 3 Figure 3: α as a function of D

Figure 4 :

 4 Figure 4: α as a function of D

Figure 5 :Figure 6 :

 56 Figure 5: Behavior of power in roller-on-tire system

Figure 8 :Figure 9 :

 89 Figure 8: Deterioration D, tangential speed of motor v 1 and tangential speed of wheel v 2 for a step input. It shows the failure of system at 300h.

Figure 10 :Figure 11 :

 1011 Figure 10: Deterioration D, tangential speed of motor v 1 and tangential speed of wheel v 2 for a square input with t h = 10000s and t l = 10s. It shows the failure of system at 96.7h.

-

  Figure 12: Internal mode of randomness. Blue lines: Deterioration D, for 100 random performances. m and b are chosen randomly using a normal distribution. Red line: simulation of performance for fixed values of t h = 10000s and t l = 10s.

-

  Figure13: External mode of randomness. Blue lines: Deterioration D, for 100 random performances. t h and t l are chosen randomly using an exponential distribution with mean values of t h = 10000s t l = 10s respectively. Red line: simulation of performance for fixed values of t h = 10000s and t l = 10s.

Table 1 :

 1 Parameters, units and symbols

	Symbol Units	Physical meaning
	v 1	[m/s]	Tangential speed of the motor
	v 2	[m/s]	Tangential speed of the driven device
	ω 1	[rad/s]	Angular speed of the motor
	ω 2	[rad/s]	Angular speed of the driven device
	ω1	[rad/s 2 ]	Angular acceleration of the motor
	ω2	[rad/s 2 ]	Angular acceleration of the driven device
	r 1	[m]	External radius of the motor
	r 2	[m]	External radius of the driven device
	B 1	[Kgm 2 /s] Viscous damping coefficient of the motor
	B 2	[Kgm 2 /s] Viscous damping coefficient of the driven device
	J 1	[Kgm 2 ]	Moment of inertia of the motor
	J 2	[Kgm 2 ]	Moment of inertia of the driven device
	T m	[N m]	Torque of the motor
	T L T * L	[N m] [N m]	Load torque seen by the motor Source torque observed by the driven device
	K m	[V s/rad] Motor back electromotive force constant
	I	[A]	Electrical current of the motor
	α	[N s/m]	Contact quality coefficient

Table 2 :

 2 Terms of power equations Term Physical meaning Symbol J 1 ω1 ω 1 Total power in the motor P T M K m Iω 1 Electric power in the motor P motor T L ω 1 Power caused by viscous damping B 1 P B1 J 2 ω2 ω 2 Total power for the driven device Power caused by viscous damping B 2 P B2

		Power transferred to the load	P L
	B 1 ω 2 1
	T * L ω 2	Power caused by contact	P T W P * L
		observed by the driven device
	B 2 ω 2 2

Table 3 :

 3 Chosen parameters

	Parameter Value	Unit
	B 1


Satadeep Bhattacharjee

Surendra Singh

Dawei Wang

Michel Viret

Laurent Bellaiche

Prediction of novel interface-driven spintronic effects

Keywords: numbers: 72.25.-b, 72.10.-d, 85.75.-d

The recently-proposed coupling between the angular momentum density and magnetic moment [A. Raeliarijaona et al, Phys. Rev. Lett. 110, 137205 (2013)] is shown here to result in the prediction of (i) novel spin currents generated by an electrical current and (ii) new electrical currents induced by a spin current in systems possessing specific interfaces between two different materials. Some of these spin (electrical) currents can be reversed near the interface by reversing the applied electrical (spin) current. Similarities and differences between these novel spintronic effects and the well-known spin Hall and inverse spin Hall effects are also discussed.

Spin transport electronics, commonly termed spintronics, is attracting a lot of attention for fundamental purposes as well as for its potential applications in electronic technologies [1][2][3]. Particularly striking examples of spintronics are the spin Hall effect (SHE) and inverse spin Hall effects (ISHE) that were, interestingly, both first predicted [4,5] before being observed [6][7][8][9]. In case of the SHE, an electric current generates a transverse spin current involving specific component of the magnetic moments, with the spin current reversing its direction when the electric current is reversed. On the other hand, it is a transverse electric current that is created and controllable by a spin-current for the ISHE.

One may wonder if there are novel spintronic effects that remain to be discovered. For instance, it is legitimate to explore if new interface-driven spintronic phenomena may occur, once realizing that a variety of spectacular and unusual features have recently been discovered near or at the interface between two different materials. Examples of such features, among many others, are interface-mediated conduction [10], superconductivity [11], multiferroic effect [12], and improper ferroelectricity [13].

In particular, it is worth pursuing if the recently proposed coupling [14] between the angular momentum density [START_REF] Jackson | Classical Electrodynamics[END_REF] and magnetic moments can guide the (hypothetical) discovery of new spintronic phenomena, since this coupling not only allowed to re-derive in a rather straightforward fashion the anomalous Hall effect (which is another Hall effect for which the transverse conductivity depends on the system's magnetization [START_REF] Karplus | [END_REF][17][18]) but also resulted in the prediction of a novel Hall effect [19]. Note that, in addition to the anomalous Hall effect, this coupling also explained [14] why magnetic vortices can be controlled by the cross product between the electric field and the magnetic field, led to the so-called spin-current model [20,21] in multiferroics, and yielded the prediction of a novel anisotropic effect [14], which further emphasizes its usefulness in tackling a variety of complex problems involving electromagnetism in materials.

The goal of this manuscript is to demonstrate that, indeed, novel spintronic effects emerge near the interface between two materials from this recently proposed coupling [14].

For that, let us first recall that the coupling between the angular momentum density and magnetic moments leads to the following energy for a conduction electron [14,19]:

E = - a 2 r ¯× (E ¯× H ¯) • m ¯, ( 1
)
where a is a material-dependent constant, r ¯is the position vector of the electron and m ¯is its magnetic moment. E ¯and H ¯are, respectively, the electric and magnetic fields experienced by the electron. Note that, similar to the well-known potential energy associated with a uniform electric field, E is position-dependent while difference in energy and resulting force (i.e., minus the gradient of E with respect to r ¯) are independent of position for uniform fields.

Let us now use Eq. ( 1) to tackle three different cases.

Case 1 : The subscript "C1" will be adopted in the following to denote the energy and fields appearing in Eq. (1) for Case 1, which corresponds to the application of an electric field along a specific direction (to be chosen as the x-axis) to a system having an interface (between two materials) lying in an xy plane. In that situation (denoting the unit vectors along the x-, y-and z-axes by x, ŷ and ẑ), the electric field E ¯C1 is the sum of the external electric field E a x and a local electric field E lz ẑ that is oriented along the z-axis. This local electric field occurs at the interface (see Fig. 1) and originates from the gradient of the potential across the interface. We thus have

E ¯C1 = E a x + E lz ẑ . (2)
Regarding the magnetic field, H ¯C1 , we note that electrons having a magnetic moment m ¯and moving with velocity v ¯possess an electric dipole moment given by [23,[START_REF] Panoksy | Classical Electricity and Magnetism 2nd edition[END_REF]:

d ¯= 1 c 2 v ¯× m ¯. ( 3
)
where c is the speed of light. This dipole interacts with the electric field given in Eq. (2) to produce an energy of the form

E int = -d ¯• E ¯C1 = - 1 c 2 (v ¯× m ¯) • E ¯C1 = - 1 c 2 (E ¯C1 × v ¯) • m ¯. ( 4
)
This latter form can be thought as the Zeeman energy resulting from the interaction between the magnetic moment and a magnetic field given by:

H ¯C1 = 1 µc 2 (E ¯C1 × v ¯) , (5)
where µ is the permeability of the medium.

Moreover, the velocity of the electron v ¯is directly proportional in magnitude, but opposite in sign, to the applied electric field. This is because this velocity is directly proportional but opposite in direction to the current density, the latter being simply related to the applied field via the conductivity of the material. As a result, one can write

v ¯= - σ ne E a x , (6)
where σ is the conductivity, n is the density of electrons and e is the magnitude of the electronic charge. This proportionality between the velocity and the applied electric field combined with Eq. ( 2), allows us to rewrite Eq. ( 5) as

H ¯C1 = - 1 µc 2 σ ne E a E lz ŷ . (7)
Using Eqs. ( 2) and (7) in Eq. ( 1) we obtain

E C1 = aσEaE lz 2neµc 2 {-xE a m y + y[E lz m z + E a m x ] -zE lz m y } (8)
where (x, y, z) and (m x , m y , m z ) represent the Cartesian components, respectively, of the position vector and magnetic moment of the electron. It is important to realize that, unlike local electric fields associated with impurities, E lz can be considered as a constant here across the (001) interface (the actual spatial variation of the field in the transition layers may depend on the quality of the interface, impurities, and other details of the transition region. To avoid getting into these unnecessary details we take the field at the interface to be constant, as consistent with the discontinuity in the potential across the interface).

Electrons near this interface will thus experience a force whose z-component F C1,z is minus the derivative of E C1 with respect to z:

F C1,z = aσ 2neµc 2 E a E 2 lz m y (9)
Consider now what happens when this force is incorporated into the Drude model [START_REF] Ashcroft | Solid State Physics[END_REF] near the interface:

dp z dt = -eE lz + aσ 2neµc 2 E a E 2 lz m y - p z τ , (10)
where p z is the z-component of the momentum of the electron, e is the magnitude of the electronic charge, m is the mass of the electron, and τ is the mean time between two successive electronic collisions.

Multiplying this equation by -eτ n I m , where n I is the density of electrons near the interface leads to:

j z,I + τ dj z,I dt = n I e 2 τ m E lz - an I στ 2nmµc 2 E a E 2 lz m y , (11)
where j z,I = -en I m p z is the z-component of the current density near the interface. The right-hand side of Eq. ( 11) (which has the dimension of a current density) has therefore two parts. The first part is given by n I e 2 τ m E lz , "simply" arises from the existence of the local electric field inherent to the interface, and is experienced by any electron independently of the sign of its m y . On the other hand, the second part has a global sign that depends on the sign of m y : in the case that the coefficient a is positive, electrons having a positive m y will generate a negative z-component of the current density near the interface while electrons possessing a negative y-component of their magnetic moment will result in a positive (opposite) contribution to j z,I when the applied field E a is along the +x-axis (the opposite situation holds if a is negative). As a result, the (001) interface tends to act as a "filter" to deflect electrons with a definite sign for the y-component of their magnetic moment along a specific side of the interface and a spin current exists along the z-axis, provided that the magnitude of the second term of Equation ( 11) is not negligible in front of the first term [26]. Interestingly, motions that are precisely opposite to the ones we just described (related to the second term of Eq. ( 11)) will occur when the applied field is reversed from the +x to -x direction, because Eq. ( 9) is linearly dependent on E a . Such field-controllable motion of electrons with specific magnetic moment, shown schematically in Figs. 1, resembles the spin Hall effect (SHE) [4-9, 27, 30-32]. Note, however, that the SHE is different from the effect described here because (1) SHE can occur in bulk systems while the effect predicted here is an interface-driven phenomenon; and (2), unlike the new effect, SHE does not require the coupling between angular momentum density and magnetic moments for occurring. For instance, in case of the extrinsic SHE [START_REF] Dyakonov | [END_REF], which is typically explained in terms of the so-called skew-scattering and side-jump mechanisms involving impurities [START_REF] Karplus | [END_REF][33][START_REF] Mott | The theory of atomic collisions[END_REF][START_REF] Goldberger | Collision Theory[END_REF][START_REF] Smit | [END_REF][37][38][39], (i) one can just consider the Zeeman interaction energy between the magnetic moments and the magnetic field given by Eq. ( 5); (ii) realize that the local field around an impurity is inhomogeneous, unlike the case considered here; and (iii) finally obtain a force along the z-axis by taking the derivative of this Zeeman energy with respect to z, viz., -σ 2neµc 2 E a ∂E lz ∂z m y . This force naturally explains the extrinsic SHE in bulk systems, since one can easily demonstrate that ∂E lz ∂z has a definite sign around an impurity (e.g., by considering a central potential whose gradient provides the local electric field).

Case 2 : Let us now consider Case 2, which is similar to Case 1 with the important exception that the interface now lies in a (011) plane. Using the subscript "C2" to denote appropriate quantities, the electric and magnetic fields of Eq. ( 2) and ( 5) in this case become

E ¯C2 = E a x + E ly ŷ + E lz ẑ (12)
and

H ¯C2 = 1 µc 2 σ ne E a (E ly ẑ -E lz ŷ) . (13)
Note that the local electric field in this case acquires a y-component, E ly (equal in magnitude to E lz ) since we are now dealing with a (011) interface. Then the energy associated with the coupling between the angular momentum density and magnetic moments can be written as:

E C2 = aσE a 2neµc 2 {xE a (E ly m z -E lz m y )} (14)
+ aσE a 2neµc 2 {y[{(E ly ) 2 + (E lz ) 2 }m z + E a E lz m x ]} - aσE a 2neµc 2 {z[{(E ly ) 2 + (E lz ) 2 }m y + E a E ly m x ]} .
The y-and z-components of the force, F C2,y and F C2,z , associated with this energy near the interface are then:

F C2,y = - aσ 2neµc 2 E a E 2 ly + E 2 lz m z - aσ 2neµc 2 E 2 a E lz m x , (15a)
F C2,z = aσ 2neµc 2 E a E 2 ly + E 2 lz m y + aσ 2neµc 2 E 2 a E ly m x . (15b)
Incorporating these two components into the Drude model [START_REF] Ashcroft | Solid State Physics[END_REF] near the interface yields:

dp y dt = -eE ly - aσ 2neµc 2 E a E 2 ly + E 2 lz m z - aσ 2neµc 2 E 2 a E lz m x - p y τ (16a) dp z dt = -eE lz + aσ 2neµc 2 E a E 2 ly + E 2 lz m y + aσ 2neµc 2 E 2 a E ly m x - p z τ . (16b)
Multiplying these equations by -eτ n I m results in:

j y,I + τ dj y,I dt = n I e 2 τ m E ly + an I στ 2nmµc 2 E a E 2 ly + E 2 lz m z + an I στ 2nmµc 2 E 2 a E lz m x (17a) j z,I + τ dj z,I dt = n I e 2 τ m E lz - an I στ 2nmµc 2 E a E 2 ly + E 2 lz m y - an I στ 2nmµc 2 E 2 a E ly m x , (17b)
where j y,I and j z,I are the y-and z-components of the current density near the interface.

The first terms on the right-hand side of Eq. ( 17a) and (17b) indicate that the local electric field existing near the (011) interface induces a current density that is oriented along the [011] direction, independently of the sign and magnitude of the Cartesian components of the magnetic moments of the electrons. If we assume that a is positive, the second terms on the right-hand side of Eq. ( 17a) and (17b) tell us that that the electrons with positive (negative) m z tends to generate an additional positive (negative) y-component of the current density near the interface while those with positive (negative) m y induces an additional negative (positive) z-component of the current density near the interface when the electric field is applied along the +x-axis, with these motions reversing when the applied electric is reversed. In other words, as in Case 1, the presence of the interface leads to spin currents that are controllable by the applied electric field. What is new in Case 2 with respect to Case 1 is the presence of the third terms on the right-hand side of Eq. ( 17a) and (17b). These third terms indicate that, if a is positive, electrons with positive m x will generate another current density (near the interface) that is along +ŷ -ẑ, that is, along a direction that is parallel to the interface, while electrons with negative m x will generate an opposite additional current density. Interestingly, this "in-plane" spin-current involving the

x component of the magnetic moments can not be reversed when the applied electric field is reversed, because the third terms of Eq. ( 17a) and (17b) are proportional to the square of the applied electric field. This new (interface-driven) phenomenon, different in nature from a spin Hall-like effect, is summarized in Fig. 2 [26].

Case 3 : Let us now consider the situation (for which the subscript "C3" will be used in the following) of a spin current in which electrons having a negative m z move along the +y axis with a velocity v y ŷ while precisely the opposite motion occurs for electrons having a positive m z . For simplicity, we do not include here the possibility that these spin electrons may also have a definite x-or y-components of their magnetic moments. As shown in Fig. 3, an interface lying in a yz plane is considered in this system, which results in the formation of local electric field aligned along the x-axis at this interface:

E ¯C3 = E lx x. ( 18
)
Because of the existence of the spin current, the electrons having a negative z-component of their magnetic moments will experience a magnetic field of the form

H ¯C3,↓ = 1 µc 2 E ¯C3 × v y ŷ = 1 µc 2 v y E lx ẑ , (19)
while electrons possessing a positive m z will feel precisely the opposite magnetic field

H ¯C3,↑ = - 1 µc 2 E ¯C3 × v y ŷ = - 1 µc 2 v y E lx ẑ . ( 20
)
Inserting Eqs. ( 18) and ( 19) into Eq. ( 1) gives for electrons with negative m z ,

E C3 = - av y 2µc 2 xE 2 lx |m z | , (21)
where |m z | is the absolute value of the z-component of the magnetic moment. Interestingly, the electrons having a positive m z also possess the same energy, E C3 (this can be shown either by inserting Eqs. ( 18) and ( 20) into Eq. ( 1) or by realizing that Eq. ( 1) is invariant under a simultaneous change of sign of the magnetic field and magnetic moment). As a result, all electrons experience the same force along the x-axis near the interface:

F C3,x = av y 2µc 2 E 2 lx |m z | . (22)
Consider this force in the Drude model [START_REF] Ashcroft | Solid State Physics[END_REF] near the interface therefore gives:

dp x dt = -eE lx + av y 2µc 2 E 2 lx |m z | - p x τ , (23)
Multiplying this latter equation by -eτ n I m leads to:

j x,I + τ dj x,I dt = n I e 2 τ m E lx - an I ev y τ 2mµc 2 E 2 lx |m z | , (24)
where j x,I = -en I m p x is the x-component of the current density near the interface. Equation ( 24) therefore tells us that, in addition to n I e 2 τ m E lx that is valid for any electron, electrons with both positive and negative m z will generate an additional negative x-component of current density in the vicinity of the interface if a is positive (otherwise this additional xcomponent is positive) thereby creating an additional electrical current that is transverse to the spin current. This additional transverse electrical current, unlike n I e 2 τ m E lx , can be reversed by switching the direction of the spin current since Eq. ( 22) depends linearly on the spin-current velocity v y . All these effects, shown schematically in Figs. 3, are reminiscent of the so-called inverse spin Hall effect (ISHE) [4,5]. However, unlike the traditional ISHE, the effect predicted here is interface-driven and originates from the coupling between angular momentum density and magnetic moments.

Let us also indicate what happens in the steady state regime. In that situation, dj x,I dt =0 and j x,I will vanish as a result of the formation of another electric field along the x-axis, arising from the transfer of charge associated with the electrons that have already crossed the interface and that will oppose the further motion of electrons along the x-axis (as, e.g., similar to the cases of the regular Hall effect or p-n junction [START_REF] Ashcroft | Solid State Physics[END_REF]). When denoting such new electric field as E opp,x , Eq. ( 24) will then lead to

E opp,x = -E lx + av y 2eµc 2 E 2 lx |m z | , (25)
Interestingly, Eq. ( 25) therefore tells us that E opp,x (and the resulting associated voltage existing inside the system along the x-axis) will change in magnitude when reverting v y , which can be practically used to experimentally demonstrate the existence of the (novel) second term of Eq. ( 24).

Note that this interface-driven formation of a voltage from the application of a spin current is similar in nature from the recently observed spin-to-charge conversion at the interface between non-magnetic materials [START_REF] Rojas Sanchez | [END_REF]. However, these two effects are technically different because the latter phenomenon (which is termed the inverse Rashba-Edelstein effect [29]) involves the application of a spin current that is along (rather than perpendicular to) the normal of the interface and the creation of an electrical current that is perpendicular (rather than parallel) to the normal of the interface. In fact, it is interesting to realize that the experiment of Ref. [START_REF] Rojas Sanchez | [END_REF] precisely corresponds to the reciprocal situation of our 'Case 1': In Ref. [START_REF] Rojas Sanchez | [END_REF], a spin current flowing along the normal of the interface (i.e., the z-axis) and involving spins oriented along an axis that lies inside the interface plane (e.g., spins having y-components) leads to the creation of an electric voltage that is along the third Cartesian axis (e.g., the x-axis), while our Case 1 is about the formation of a spin current oriented along the z-axis (which is parallel to the normal of the interface) of spins having y-components from the application of an electric field being along the x-axis.

Finally, one can easily demonstrate, that the case that differs from"Case 3" only by having an interface along a (110) (rather than a (100) plane), will also result in the creation of an additional electric current along the normal of the interface (in the non-steady state situation), which can be reversed by reversing the spin current.

In summary, we have demonstrated that the coupling between angular momentum density and magnetic moments predicts novel spintronic effects near interfaces between two different materials. We hope that these novel effects, summarized in Figs. 1-3, will stimulate experimental work for their conformation. Note that, as hinted in Ref. [19] and in the supplemental material of Ref. [14], the a coefficient appearing in Eq. ( 1) requires spin-orbit interactions to be non-zero. As a result, the phenomena predicted here are more likely to be observed in systems with strong spin-orbit interactions. It may also be worth investigating if theories involving Berry-phase curvature [30][31][32] can also explain the novel phenomena predicted here. This appears to be a promising scenario especially when we recall that the coefficient a of Eq.( 1) has been recently found [19] to be directly related to Berry-phase curvature [START_REF] Berry | Proc. R. Soc. London[END_REF] in case of the anomalous Hall effect.

  FIG. 2: (Color online) Schematics of the predicted effect associated with Case 2 for a system possessing a (011) interface between two materials. Panel (a) and (b) differ by the direction of the applied electric field E ¯a (i.e., parallel or antiparallel to the x-axis). They show the corresponding motion of all components of the magnetic moments near the interface (associated with the second and third terms of Eqs. (17a) and (17b)). Electrons are represented by dots while the components of their magnetic moments are shown by solid lines going though the dots. The direction of the force (associated with Eqs. (15a) and (15b)) experienced by these electrons is displayed via dashed lines.

We thank ARO Grant No. W911NF-12-1-0085 for personnel support. Office of Basic Energy Sciences, under contract ER-46612, ONR Grants No. N00014-11-1-0384 and N00014-12-1-1034, and NSF Grant No. DMR-1066158 are also acknowledged for discussions with scientists sponsored by these grants. The authors thank Vincent Cross and Huaxiang Fu for insightful discussions.


Soraya Sangiao

Michel Viret

Electrical detection of internal dynamical properties of domain walls

Keywords: number(s): 76.50.+g, 75.78.Cd, 75.78.Fg

We report here on the electrical detection of the ferromagnetic resonance in a notched Py stripe using the spin rectification effect. A lock-in detection technique is used to measure the rectified voltage generated when applying a radio-frequency field. The multiple peaks associated with the presence of a domain wall, observed during field scans, are properly identified with the help of dynamical micromagnetic simulations. It is found that at any frequency below 6 GHz some specific areas of the inhomogeneous magnetization within the domain wall can find a resonant condition and generate a voltage by rectification of the induction currents in the circuit.

I. INTRODUCTION

The new generation of electronic devices, so-called spintronic devices [1][2][3], combine standard electronics with spindependent effects and use the electron spin, rather than its charge, to carry information. The trend towards miniaturization in electronics has led to the need for understanding and controlling the magnetization dynamics of nanostructures. Present day lithographic techniques allow the patterning of magnetic nanostructures with controlled submicron size and shape. Classical inductive techniques do not reach the sensitivity allowing us to study the magnetization dynamics in individual magnetic nanostructures, but electrical transport has been shown to reach this goal [4,5]. Thus, the dynamics of the motion of domain walls (DWs) has been recently extensively studied both theoretically [6,7] and experimentally [4,8]. It is typically found that DWs move at typical velocities of tens of metres per second corresponding to typical frequencies below 0.5 GHz [9]. So far, most reported works concern the rigid motion of the walls within some artificially induced potential. The aim of the present paper is to detect the dynamics of the internal magnetic texture of a unique immobile DW. It corresponds to frequencies higher than 1 GHz, in the range of those used for usual ferromagnetic resonance (FMR).

For this purpose we fabricate Ni 80 Fe 20 (Py) stripes with a notch along one of the edges that serves as an artificial pinning site for a DW. A micron sized stripline is also lithographied very close to the Py stripe in order to generate the excitation microwave field. We detect the DC voltage signal generated by rectification of the induction current in the conducting structure [10,11]. This voltage is caused by the spin-rectification effect [12,13] that couples the induced rf current in the circuit containing the magnetic element to any change of resistance at the same frequency. In our samples the latter arises from the anisotropic magnetoresistance (AMR) that oscillates due to the precession of magnetization. Py stripes are attractive systems because they exhibit a sizable AMR contribution and have nearly zero magnetocrystalline anisotropy and magnetostriction, hence low damping. To further enhance the sensitivity of our measurements we modulate the rf field (at ω rf ) by a square signal and measure the lock-in voltage generated across the stripe at the modulation * michel.viret@cea.fr frequency. The voltage picked up in our measurements arises from the current induced at ω rf by the varying flux in the stripe rectified by mixing with the AMR contribution at ω rf . Thus, the line shape of the lock-in voltage we measure depends on the phase difference between the rf magnetization and the rf induced current leading to a measured signal composed of a linear combination of symmetric and antisymmetric Lorentz line shapes [3].

II. EXPERIMENT

10 μm long, 1 μm wide, and 20 nm thick Py stripes with a 600 nm wide notch are deposited on kapton substrates. The stripes are positioned with electron beam lithography 500 nm away from the current line of an rf antenna, previously defined by optical lithography (see Fig. 1). It is worth mentioning that we grow the samples on kapton, which is a flexible substrate, to use them also in break junction experiments, but kapton is not required to reproduce our results. The last step in sample fabrication consists in defining two (Au) voltage contacts on each stripe, designed in a loop shape so that the induced current in the stripe is maximized. The sample is placed between the poles of a rotating electromagnet that provides DC magnetic fields up to 1.2 T at any in-plane angle. The voltage measurements are performed at room temperature, and an rf signal, square-wave modulated at 1.98 kHz, is sent to the antenna at frequencies up to 40 GHz and at 0 dBm.

III. RESULTS AND DISCUSSION

We show in Fig. 2(b) the rectified voltage measured while sweeping the field at a constant frequency of 9 GHz, for different angles between the (in-plane) static magnetic field and the longitudinal axis of the stripe. A clear peak is measured at fields around 0.1 T for which the stripe is saturated, this is the FMR. Its exact position depends on the angle between the magnetization and the stripe direction (i.e., that of the induced current flow) but the line shape does not change. This is expected as, at a fixed frequency, the phase difference between the rf magnetization and the rf induced current remains unchanged. On the other hand, the voltage signal amplitude varies progressively and disappears at angles close to the transverse in-plane direction. In this configuration and for a 0.1 T field, the magnetization lays perpendicular to the stripe which makes the rectified voltage vanish [3]. Indeed, in such a configuration, the AMR gives a signal at twice the rf frequency which cannot be rectified with the induced current. The measured peaks are therefore induced by the uniform FMR mode of the stripe.

Interestingly, when the frequency is lowered, a signal starts to appear near zero field as the sample is no longer homogeneously magnetized. This is shown in Fig. 3(a) for frequencies between 1 and 4 GHz where it can be seen that the signal is the highest for 3 GHz. Moreover, the sample's resistance is also measured using a small current at 7 Hz driven through the sample while measuring the voltage at the same frequency. Figure 3(b) shows the sample resistive variation with the magnetic field applied at 90 degrees with respect to the stripe axis. The curve is globally bell shaped as the AMR progressively decreases when the angle between the magnetization and the stripe increases towards 90 degrees. A slight opening is also visible, both at positive and negative low fields, between approximately 10 and 15 mT. Interestingly, 10 mT corresponds to the field at which a peak is observed in the rectified voltage coming back from positive saturation (and symmetrically -10 mT when coming from negative saturation). Such a curve opening in the magnetoresistance often indicates the appearance of an inhomogeneous magnetic state. In the particular case of our structure with an almost transverse applied field, it corresponds to the nucleation of a domain wall in the notch area [14,15]. The presence of the DW increases the total stripe resistance by around 0.6 . For this reason, hereinafter we call this peak the nucleation peak. As visible in Fig. 3 resistivity, a quantity equivalent to a resistive susceptibility near zero frequency. Indeed, it corresponds to the rectified voltage one would measure at low frequency in the presence of a constant current. Globally, their shapes are rather similar, but the rectified voltage presents extra features including a peak at -35 mT as well as the nucleation peak at 10 mT. It is also important to notice that the "saturation" peak at -12 mT appears before the actual saturation given by the peak in the dR/dH curve. It is hard to understand the exact origin of these extra features but it is obvious that most of them appear when the magnetization is inhomogeneous, i.e., when a DW is present. Experimentally, it is also important to note that these extra peaks are only found for frequencies below 6 GHz. We believe this indicates that for these lower frequencies, the varying internal field near the notch provides regions in space where a resonant condition is met. In the following, we concentrate on the rectified voltage measurements performed at 3 GHz, at angles close to the transverse direction, which are representative of the DW effects on the RF susceptibility.

Figure 4(a) presents the measured rectified voltages at different applied angles in a range of 75 to 105 degrees from the stripe axis. The shape of the signal can be well understood considering that the rectified voltage is proportional to sin(α)cos(α) with α the angle of the magnetization with respect to the stripe. Hence, peaks signs are reversed when the magnetization angle goes from slightly below 90 degrees to above it. Moreover, near zero field, the signal sign depends on the component of longitudinal magnetization, which is different when coming from positive or negative saturation fields. Nevertheless, apart from their signs, rather similar features are indeed observed in all these curves. Going from positive to negative fields, three different peaks can be seen:

(i) The notch vertex peak corresponding to the excitation of the precession in the area around the tip of the notch. It is the only peak we observe at both negative and positive fields (for example, at +25 and -25 mT when applying the magnetic field at 97 degrees with respect to the stripe axis).

(ii) The nucleation peak, when a DW starts to nucleate in the notch area.

(iii) The switching peak that happens just before the DW is depinned from the notch triggering magnetization reversal. This feature is actually not a real peak but should rather be viewed as a steep increase of the rectified signal completely suppressed as soon as the DW escapes the notch area.

The notch vertex excitation and the nucleation peaks are positive below 90 degrees and negative at angles above this value, and the opposite dependence is found for the switching peak, which is negative (positive) below (above) 90 degrees [see Fig. 4(a)], all these sign changes being consistent with the expected symmetry. As can be seen in Fig. 4(b), the switching field, in absolute value, is maximum at 90 degrees whereas the nucleation field is minimum at this angle.

The only peak that remains when measuring at angles not close to the transverse direction is the switching peak that can be observed at angles up (down) to 165 (15) degrees. This feature was previously reported by Grollier et al. [16], when the longitudinal magnetization reverses. In the longitudinal field geometry, the magnetization reversal does not take place by depinning of the pinned DW but rather by the nucleation of a reversed domain and subsequent DW propagation. In our measurements shown in Fig. 4(b), the nucleation field strongly increases when moving away from the transverse direction. A detailed study of its angular dependence would help us to gain a deeper insight into the precise mechanism of magnetization reversal, but this is not the aim of the present paper.

In order to understand in more detail the extra features in the rectified voltage, micromagnetic simulations were carried out using the MuMax2 code [17], which is a micromagnetic simulation tool running on graphical processing units. The simulations are performed in a 10 × 1 × 0.02 μm 3 volume with 20 × 8 × 5 nm 3 unit cells with the exact shape of our Py stripe with a notch. The Py parameters are set to: saturation magnetization M S = 800 kA m -1 , exchange constant A = 1.3 × 10 -11 J m -1 , and Gilbert damping parameter α = 0.01. We have followed a field scan procedure starting from positive transverse saturation and reducing the field. Each step uses as the initial state the magnetic configuration previously determined. The excitation frequency of the rf magnetic field has been chosen to be f = 3.0 GHz. The magnetic configurations obtained at different field values while sweeping the field, applied at an angle of 90 degrees, are shown on the left side of Fig. 5. The chosen field values correspond to the magnetic fields at which we observe each of the different peaks in our measurement [see Figs. On the right side of Fig. 5 we display in black the sample outline and in color the areas in which the magnetization is found to oscillate for an rf excitation of 3 GHz. Several interesting features can be pointed out from this figure:

(i) As discussed above, the peak we measure around +10 mT corresponds to the nucleation of a DW at the notch area and, as can be seen in Fig. 5(a.2), the only region that oscillates at +10 mT is the center of the DW. We measure a well-defined peak in the rectified voltage at this magnetic field value, in spite of the small oscillating area, because our detection technique is very sensitive to the magnetization precession around the notch (a significant voltage drop across the sample occurs in this region).

(ii) At zero field [Fig. 5(b.2)] the area in which the magnetization is found to oscillate at 3 GHz is only a tiny region, in good agreement with the absence of a distinguishable feature in our measurements. It is interesting to note here that the vortex wall does not move at this frequency since vortex wall motion is usually observed well below 1 GHz.

(iii) Around -10 mT [Figs. 5(c) and 5(d)] we clearly observe the depinning of the DW when the force exerted by the magnetic field exceeds the pinning force of the notch. The region that is found to oscillate just before the switching field is the inhomogeneous part of the DW but also the domains on both sides. Nevertheless we think that the peak we observe in the rectified voltage comes mainly from the DW part. This is due to the increased sensitivity of our detection in the notch area where current lines are denser and the fact that our gold contacts select a reduced region as represented by the red lines in Fig. 5. This is also supported by the fact that the two peaks we measure-the nucleation peak, due to the DW oscillation, and the switching peak-have comparable heights. Indeed the latter peak would be much higher if the domain contribution were dominant.

(iv) Finally, around -33 mT, the magnetic configuration is not fully saturated but the field is high enough to remove the closure domains at both ends of the stripe and the region that oscillates is the area around the notch vertex.

It is possible to use these simulations more quantitatively (along with the simulations of the current lines) in order to estimate the AMR changes expected. The AMR contribution due to magnetization precession at 3 GHz when sweeping the static field at angle θ of 90 degrees is shown in Fig. 6. We find that there is a good quantitative agreement with the data including in the peak positions, which proves the accuracy of these simulations.

It is interesting to go further with the simulations and explore the response at higher frequencies for which we do not measure any particular feature in the rectified voltages. We show in Fig. 7 the areas in which the magnetization is found to oscillate at different frequencies below and above 6 GHz. As can be seen in this figure, at any frequency below 6 GHz, some region around the DW can resonate, but nothing happens above. This again reinforces our conclusion that the largely inhomogeneous internal field of the DW provides regions matching the resonance condition for any frequency below a cutoff found to be 6 GHz in our system. Now we turn our attention to the influence of a dc current on these DW resonances. For this aim we inject different dc currents into the sample while measuring the rectified voltage in the same conditions as described above. The field dependence of the rectified voltage measured at 3 GHz while applying different dc current values up to 10 6 A cm -2 for a magnetization at an angle of 85 degrees from the stripe axis is displayed in Fig. 8. The voltage signals are noisier but still clear and significant. As the current density increases above 8 × 10 5 A cm -2 , the nucleation peak broadens demonstrating that spin torque starts to play a role on the DW FMR profiles at current densities two orders of magnitude lower than the depinning ones. A quantitative description of this effect is beyond the scope of the present paper. However, the sensitivity of the rectified voltage measurement allows us to conclude that the domain wall behavior is modified at rather low current densities. This is consistent with models predicting that the internal structure of domain walls should start to distort largely before depinning [18][19][20].

IV. CONCLUSIONS

In summary, we have demonstrated here that internal DW resonances can be detected by an electrical technique. The measurement is based on the voltage signal generated by rectification of the induced current coupled to the AMR oscillation. This rectification technique is extremely sensitive as it allows us to measure signals from very localized precessing regions within a single pinned DW. Micromagnetic simulations, performed using the freely available code MuMax2, have enabled a full understanding of the measured voltage signal and the proper identification of the different resonances we observe.

Our measurements, along with the micromagnetic simulations, demonstrate that DWs provide a sufficiently varying internal field so that some resonance can always occur below a cutoff frequency of 6 GHz. Lastly, we showed that the DW resonances are affected by a DC current two orders of magnitude below the depinning threshold.

FIG. 1 .

 1 FIG. 1. (Color online) (a) Sketch of the samples designed for rectified voltage measurements. (b) (Artificially colored) optical image of a sample containing two Py stripes.

  (b), the rectified voltage is much richer with the presence of several peaks in the field region where a DW is pinned at the notch. It is instructive to compare the rectified voltage to the field derivative of the

FIG. 2 .

 2 FIG. 2. (Color online) (a) Sketch of the measurement geometry. (b) Field dependence of the measured rectified voltage at 9 GHz and 0 dBm for different angles between the static magnetic field and the longitudinal axis of the stripe.

FIG. 3 .

 3 FIG. 3. (Color online) (a) Field dependence of the measured rectified voltage at an angle of 90 degrees between the static magnetic field and the longitudinal axis of the stripe for different frequencies up to 4 GHz (the measurement at 3 GHz has been shifted downwards by -3 μV for the sake of clarity). (b) Measurement of the total stripe resistance (black or red squares, left axis) and the rectified voltage (blue or red dots, right axis) at 3 GHz as a function of the static magnetic field applied at an angle of 90 degrees with respect to the axis of the stripe. (c) Comparison of the rectified signal with the DC susceptibility dR/dH extracted from the R(H) curve.

FIG. 4 .

 4 FIG. 4. (Color online) (a) Field dependence of the measured rectified voltage at 3 GHz for different angles around the transverse direction. (b) Angular dependence of the nucleation and the switching fields (the switching field, negative, is represented in absolute value for the sake of clarity) at 3 GHz. Blue or black (red) symbols mean positive (negative) peak.

  3(a) or 4(a)]: the nucleation peak at around 10 mT [Fig. 5(a)], the switching peak at -10.4 mT [Figs. 5(c) and 5(d)], and the notch vertex excitation peak at -32.5 mT [Fig. 5(e)]. We have also included the magnetization configuration at zero field [Fig. 5(b.1)] in order to show the vortex wall pinned at the notch.

FIG. 5 .

 5 FIG. 5. (Color online) (x.1) Static magnetization configuration as found during the simulation of the experimental field scan (when sweeping the field downwards from positive transverse saturation) using MuMax2. (x.2) In color are the areas in which the magnetization oscillates when a 3 GHz AC field is superimposed at five different DC magnetic field values: 10 mT (a), 0 mT (b), -10 mT (c), -10.5 mT (d), and -32.5 mT (e). The color code used for the in-plane magnetization orientation is shown in the inset . Bold red lines on the right indicate the position of the voltage contacts on the stripe.

FIG. 6 .

 6 FIG. 6. (Color online) AMR contribution obtained from the micromagnetic simulations due to magnetization precession at 3 GHz when sweeping the static field at an angle θ of 90 degrees.

FIG. 7 .

 7 FIG. 7. (Color online) In color are the areas in which the magnetization oscillates at 100 Oe at different frequencies: 1.71 GHz (a), 2.40 GHz (b), 4.00 GHz (c), 4.86 GHz (d), and 7.40 GHz (e). The color code used for the in-plane magnetization orientation is shown in the inset of Fig. 5.

FIG. 8 .

 8 FIG. 8. (Color online) Field dependence of the measured rectified voltage when applying different dc current densities to the Py stripe at 3 GHz when the angle between the static magnetic field and the stripe axis is 85 degrees.

ACKNOWLEDGMENTS

We would like to acknowledge financial support from the European Commission through the Marie Curie Intra European Fellowship Project No. 301656: AtomicFMR, funded by the 7th Framework Programme. We also would like to thank J. V. Kim for his kind help with MuMax2 simulations and S. Rohart for assistance with Py deposition.