Luca De Pascali

Sebastian Erhart

Luca Zaccarian

Sandra Hirche
email: hirche@lsr.ei.tum.de

A Decoupling Scheme for Force Control in Cooperative Multi-Robot Manipulation Tasks

Keywords: Cooperative Manipulation, Internal Forces, Interaction Dynamics, Impedance Control, Input Allocation

The internal forces and torques arising in cooperative manipulators ensembles are the grasp forces/torques and it is obviously desirable to control them. We present a novel approach that describes the internal loading as the interaction forces/torques arising in a multi-body system formed by multiple manipulators that behave like a formation of robots. We show that these quantities belong to the null space of the grasp matrix, thus they do not affect the dynamics of the object. The main contribution of this paper is a decoupling control scheme for tracking the internal and the motion-inducing forces and torques in a physically consistent way. The scheme is based on a physically and mathematically consistent model of the dynamics of the constrained interaction.

I. INTRODUCTION AND BACKGROUND MATERIAL

D EXTERITY and payload capacity of cooperative manip- ulator ensembles have moved the interest of the scientific community from single-arm robots configurations towards multi-arm robot technologies. In fact multi-robot cooperation is necessary in applications that require high load capacity, or when flexible objects or objects with extra degrees of freedom have to be handled, or even in the assembling of multiple parts. They can be employed in manufacturing, construction, forestry, medical applications and other domains. The challenge of the increased dexterity lies in the higher complexity of the robot configuration, which is mainly due to the redundancy of the system and to the coordination of the manipulators.

Force control in a cooperative manipulator ensemble is a quite considered topic in the literature and was first addressed in [START_REF] Alberts | Force control of a multi-arm robot system[END_REF]. In many application it is desirable to control the squeezing (also named internal) forces and torques acting on the object. Undesired and uncontrolled internal forces can damage the manipulated object as well as the robotic arm itself; on the other hand, in many operations we would like to impose a desired value of the internal loading, for example in those situation where the contact between the object and the tips of the manipulators is guaranteed by the friction in the contact points. Thus the design of a decoupled control scheme for tracking the desired object trajectory and the internal wrenches plays a vital role in the cooperative manipulation scenario. The relevance of a consistent internal force and torque model is mentioned in [START_REF] Heck | Internal and external force-based impedance control for cooperative manipulation[END_REF] and is consistently addressed in [START_REF] Erhart | Load distribution in cooperative manipulation tasks[END_REF]. An interesting approach for the computation of internal forces is the virtual linkage model proposed in [START_REF] Williams | The virtual linkage: A model for internal forces in multi-grasp manipulation[END_REF]. In a cooperative task each manipulator applies individually a force and a torque (or equivalently a wrench) to the object to obtain, in cooperation with the other manipulators, the desired object motion. The allocation of the manipulators wrenches is called load distribution problem [START_REF] Siciliano | Springer handbook of robotics[END_REF]. The cooperative system is over-actuated, thus there exist infinitely many solutions to the load distribution problem, varying to each other for the different internal components that do not induce the motion of the object. Assuming the existence of a non-squeezing [START_REF] Walker | Analysis of motion and internal loading of objects grasped by multiple cooperating manipulators[END_REF] load distribution strategy (in [START_REF] Erhart | Load distribution in cooperative manipulation tasks[END_REF] it is shown that there exist infinitely many non-squeezing solutions), we can define the setpoints for the wrenches exerted by the manipulators on the object, that are free of internal components. The control scheme presented in [START_REF] Erhart | Model and analysis of the interaction dynamics in cooperative manipulation tasks[END_REF] aims at tracking the desired object motion and the coordination of the manipulators by means of an impedance control scheme, but the control of the internal wrenches is not addressed. In this paper we show that the onset of the squeezing wrenches coupled with the motion-inducing ones does not affect the motion of the object, thus the tracking of the desired trajectory can be achieved neglecting the internal components. The main contribution of this paper is the design of a decoupling control scheme able to track the desired trajectory and the internal wrenches. A promising approach for the study of cooperative manipulators dynamics is proposed in [START_REF] Bruyninckx | Gauss' principle and the dynamics of redundant and constrained manipulators[END_REF] and [START_REF] Sapio | Operational space control of multibody systems with explicit holonomic constraints[END_REF], where the cooperative ensemble is considered as a constrained multi-body system. In [START_REF] Erhart | Model and analysis of the interaction dynamics in cooperative manipulation tasks[END_REF] this approach is employed to express analytically the forces and torques arising in the interaction dynamics; the same authors compute the internal wrenches as the formation-violating forces and torques arising in a system formed by the manipulators that moves together as a formation of robots 1 . In Section II we present a number of fundamental results derived from the definition of internal wrenches. These results are useful for the design of the decoupling scheme, performed in Section III; finally, in Section IV we illustrate the outcomes of a numerical simulation.

A. Preliminaries

We assume that N manipulators grasp an object and the grasp is assumed to be rigid. Each manipulator can exert both forces and torques on the object. As a consequence of the interaction with the cooperative setup, a force f i and a torque t i act on the i-th manipulator, for i = {1, . . . , N}. We define the i-th wrench as the stacked vector

h i = [f T i , t T i ] T .
In Fig. 1, the kinematic quantities and the wrenches exchanged in the system are depicted. The vectors are expressed in the world reference frame {w} if not indicated otherwise with a leading superscript. The object-fixed reference frame {o} has the origin in the object center of mass. Moreover we define the i-th end effector reference frame {i}, for i = {1, . . . , N}. The pose of the i-th end effector x i is composed by a translation p i 2 R3 and a rotation denoted by the unit quaternion q i 2 Spin(3)2 , namely

p i p o {!} {o} {i} r i q o q i h i h i h o
x i = [p T i , q T i ] T .
Similarly the pose of the object center of mass is

x o = [p T o , q T o ] T .
The distance between the object center of mass and the i-th manipulator is r i = p i p o 2 R 3 , for each i. The end effectors wrenches h = [h T 1 , . . . , h T N ] T are mapped in the wrenches acting at the object center of mass h o through the so called grasp matrix G(r) 2 R 6⇥6N as

h o =  I 3 0 3 . . . I 3 0 3 S(r 1 ) I 3 . . . S(r N ) I 3 h = G(r)h (1)
where S(•) indicates the skew symmetric matrix 3 and I m and 0 m denote respectively the m ⇥ m identity matrix and the m ⇥ m zero matrix. The vector r = [r T 1 , . . . , r T N ] T contains the relative distances between the object center of mass and the manipulators, in fact r i = w R o (q o ) o r i , for all i where o r i 2 R 3 is the (constant) position of the i-th end effector in the object reference frame {o} and w R o (q o ) 2 SO(3) is the rotation matrix from {o} to {w}. The definition of internal wrenches is linked to the geometry of the grasp, in fact, in [START_REF] Walker | Analysis of motion and internal loading of objects grasped by multiple cooperating manipulators[END_REF] the internal wrenches are defined as the wrenches lying in the null space of the grasp matrix. Recently, the authors of [START_REF] Erhart | Load distribution in cooperative manipulation tasks[END_REF] provide the following more general definition.

Definition 1. Internal forces/torques are end effectors forces and torques for which the total virtual work is zero for any virtual displacement of the end effectors, compliant with the kinematic constraints.

We will show in Section II that the two definitions are equivalent.

B. Kinematic Constraints

The manipulated object is assumed to be rigid and the end effectors are assumed to be rigidly connected to it. Therefore we can study the ensemble of the end effectors and the object as a system of bodies constrained to each other by rigidity constraints. The position o r i and the relative orientation o q i of the i-th end effector in {o} remain constant for all i, thus we can write the velocity and the acceleration in {w} as detailed in [START_REF] Erhart | Model and analysis of the interaction dynamics in cooperative manipulation tasks[END_REF] differentiating the constrained positions, namely

ẋi =  ṗi ! i =  ṗo + S(r i ) T ! o ! o (2a) ẍi =  pi !i =  po + S(r i ) T !o + S(! o ) 2 r i !o . ( 2b
)
Introducing the stacked velocity 4

ẋ = [ ẋT 1 , . . . , ẋT N ] T 2 R 6N and acceleration ẍ = [ẍ T 1 , . . . , ẍT N ] T 2 R 6N leads to the following compact form ẋ = G(r) T ẋo (3a) ẍ = G(r) T ẍo + b, (3b)
where the constraint acceleration condition in (3b) can be reformulated as

A  ẍo ẍ = b. (4)
The constraint matrix A 2 R 6N ⇥6(N +1) and the vector b 2 R 6N of the centripetal terms have the form

A = ⇥ G(r) T I 6N ⇤ , and b = 2 6 6 4 S(!o) 2 r1 03⇥1 . . . S(!o) 2 r N 03⇥1 3 7 7 5 . (5)

II. CHARACTERIZATION OF THE INTERACTION DYNAMICS

Recalling [START_REF] Erhart | Model and analysis of the interaction dynamics in cooperative manipulation tasks[END_REF], the equations describing the motion of the manipulators are

M (x)ẍ = h ⌃ + h, (6)
where the inertia matrix is defined as

M (x) = blkdiag(m 1 I 3 , J 1 , . . . , m N I 3 , J N ) 2 R 6N ⇥6N
. Notice that the wrenches acting on the manipulators are split into interaction wrenches h and non-interaction wrenches h ⌃ . The wrenches h ⌃ are all the wrenches acting on the manipulators except those arising as a consequence of the constrained interaction, which are named h. The equations of motion for the object are obtained through Lagrangian mechanics yielding

M o (x o )ẍ o + C o (x o , ẋo ) ẋo + h g = ho + h o , (7)
where

M o (x o ), C o (x o , ẋo ) 2 R 6⇥6 and h g 2 R 6
denote respectively the inertial, centrifugal and gravitational terms.

The disturbances ho are due to external unmeasured wrenches or model uncertainties. The wrench h o comes from the interaction with the manipulators and is related to the end effectors interaction wrenches h through [START_REF] Alberts | Force control of a multi-arm robot system[END_REF]. Similarly to (6), we split the interaction h and non-interaction h ⌃ wrenches, as follows

M (x o )ẍ o = h ⌃ o + h o , (8)
wherein h ⌃ o = ho C(x o , ẋo ) ẋo h g . A.

Interaction Dynamics

We aim at finding an explicit expression of the interaction wrenches h and h o . In [START_REF] Udwadia | A new perspective on constrained motion[END_REF] the Gauss' principle of least constraints is solved on a system with acceleration constraints of the form (4) and leads to the result

 h o h = M 1 2 (A M 1 2 ) † ✓ b A M 1  h ⌃ o h ⌃ ◆ , (9)
where the superscript † denotes the Moore-Penrose pseudoinverse and matrix M = blkdiag(M o , M). It is possible to show that matrix A exhibits some properties that guarantee the existence of the pseudo-inverse, thus the following result holds.

Theorem 1. The interaction wrenches h o 2 R 6 and h 2 R 6N (resulting from the interaction between the object and the manipulators) are proportional to the extent to which the accelerations imposed by the non-interaction wrenches h ⌃ o 2 R 6 and h ⌃ 2 R 6N acting on the system tend to violate the acceleration constraints (3b). In particular, the following equation holds

 h o h = A T (A M 1 A T ) 1 ✓ b A M 1  h ⌃ o h ⌃ ◆ . ( 10
)
Proof. Matrix A in ( 5) is full row-rank equal to 6N by construction. We know from [11, p. 88] that, since M 1 2 has full rank equal to 6N , then also A M 1 2 is full row-rank. Therefore we can write its right inverse as

(A M 1 2 ) † = ( M 1 2 ) T A T (A M 1 2 ( M 1 2 ) T A T ) 1 .
The inertia matrix M is symmetric, thus also M 1 2 is symmetric and we can write

(A M 1 2 ) † = M 1 2 A T (A M 1 A T ) 1 . ( 11
)
Substituting [START_REF] Gentle | Matrix algebra: theory, computations, and applications in statistics[END_REF] in ( 9) yields [START_REF] Udwadia | A new perspective on constrained motion[END_REF].

With this result we can write the overall wrenches acting on the cooperative ensemble as a function of the non-interaction wrenches. Now we further inspect the dynamics of the object expressed in [START_REF] Erhart | Model and analysis of the interaction dynamics in cooperative manipulation tasks[END_REF] in order to understand how the wrenches h ⌃ are involved in the motion of the object. In a cooperative manipulators scenario these wrenches are the commanded wrenches provided at the end effectors. The new formulation in [START_REF] Udwadia | A new perspective on constrained motion[END_REF] allows to state the following Lemma, associated to Theorem 1.

Lemma 1. The actual object acceleration is described by

equation ẍo = M 1 (h ⌃ o + Gh ⌃ b), (12)
where b = GM b takes into account the centripetal terms, and the equivalent inertia matrix

M = (M o + GM G T ) 2 R 6⇥6
represents the actual inertia of the system comprising the object and the manipulators.

Proof. First we disclose the term A M 1 A T in (10) and, having in mind that A can be expressed as in [START_REF] Siciliano | Springer handbook of robotics[END_REF],we obtain

(A M 1 A T ) 1 = ⇥ G T I6N ⇤  M 1 o 06⇥6N 06N⇥6 M 1 " G I6N #! 1 = (M 1 + G T M 1 o G) 1 = Q. ( 13
)
We focus now on the term A M 1 , namely

A M 1 = ⇥ G T M 1 o M 1 ⇤ (14)
Substituting ( 13) and ( 14) in [START_REF] Udwadia | A new perspective on constrained motion[END_REF] yields

 h o h = " G I 6N # Q ✓ b ⇥ G T M 1 o M 1 ⇤  h ⌃ o h ⌃ ◆ . (15)
In particular the interaction wrench acting on the object becomes

h o = GQG T M 1 o h ⌃ o + GQM 1 h ⌃ GQb. (16)
We replace h o as in (16) into the object equations of motion [START_REF] Bruyninckx | Gauss' principle and the dynamics of redundant and constrained manipulators[END_REF] obtaining

M o ẍo = (I 6 GQG T M 1 o )h ⌃ o + GQM 1 h ⌃ GQb. ( 17
)
Using the Woodbury matrix identity to expand Q in (13) yields

Q = M MG T (Mo + ,⇥ z }| { GM G T ) 1 GM , (18)
We thus substitute (18) in (17) and inspect each term of the expression; first consider the term multiplying h ⌃ o :

I6 GQG T M 1 o = I6 ⇥M 1 o ⇥(Mo + ⇥) 1 ⇥M 1 o , = I6 ⇥(Mo + ⇥) 1 (Mo + ⇥ ⇥)M 1 o = I6 ⇥(Mo + ⇥) 1 = Mo(Mo + ⇥) 1 . ( 19
)
Now we focus on the term multiplying h ⌃ , namely

GQM 1 = G ⇥(M o + ⇥) 1 G = M o (M o + ⇥) 1 G (20)
Similarly we inspect the term multiplying b and get

GQ = M o (M o + ⇥) 1 GM . ( 21
)
We define the matrix M = M o + GM G T which is the actual inertia of the overall system as stated in [7, th. 2]. Finally we substitute (19), ( 20) and ( 21) in ( 17) and obtain [START_REF] Caccavale | Six-dof impedance control of dual-arm cooperative manipulators[END_REF].

B. Internal Wrenches

Many authors identify the internal wrenches as those ones that do not induce any wrench h o at the object center of mass; recalling [START_REF] Alberts | Force control of a multi-arm robot system[END_REF], this means that internal wrenches have to be in the null space of the grasp matrix G. The Definition 1 proposed in [START_REF] Erhart | Load distribution in cooperative manipulation tasks[END_REF] introduces a new approach. The following theorem shows that the two formulations are equivalent.

Theorem 2. Given a constant set of end effectors wrenches acting on the object and collected in the stacked vector h 2 R 6N , the following are equivalent:

• the wrenches h are internal (as specified in Definition 1);

• the wrenches h belong to the null space of the grasp matrix G.

Proof. According to Definition 1, the virtual work of a set of internal wrenches h along the virtual displacements x compliant with the kinematic constraints is zero, namely

h T x = 0 6N ⇥1 .
Dividing all members by the time variation t we get the constrained virtual velocities ẋ, namely h T x t = h T ẋ = 0 6N ⇥1 . Notice that the ratio between the infinitesimal values x and t gives the finite velocity ẋ. The constrained velocities of the end effectors are described by (3a), thus we can write

h T ẋ = h T G T ẋo = (Gh) T ẋo = 0 6N ⇥1 , ( 22
)
where ẋo is any virtual velocity of the object, therefore (22) holds if and only if Gh = 0 6N ⇥1 . This means that the wrenches h are internal according to Definition 1 if and only if h 2 Ker(G).

In order to obtain a closed form equation for the internal wrenches we will adopt the approach proposed in [START_REF] Erhart | Model and analysis of the interaction dynamics in cooperative manipulation tasks[END_REF] that characterizes the internal wrenches as the formation-maintaining wrenches arising in a subsystem built by the manipulators as depicted in Fig. 2. The end effectors can be thought of as a rigid formation of robots that exchange wrenches to maintain the formation, these wrenches are exchanged through the object and thus can be considered internal. This novel approach presented in [START_REF] Erhart | Model and analysis of the interaction dynamics in cooperative manipulation tasks[END_REF] will be employed in this paper to design a decoupling control scheme for motion-inducing and internal wrenches. With the following theorem we show that the interaction wrenches arising in the subsystem in Fig. 2 are actually internal wrenches.

{!} {i} {j} {1} q j q i q 1 r 1j r 1i r 1 r i
Fig. 2. Reduced end effectors system consisting in the manipulators.

Theorem 3. Let r = [r T 12 , . . . , r T 1N ] T (with r 1i = r 1 r i the distance between the 1 st and the i-th end effector for i 2 {2, . . . , N}) and the formation maintaining wrenches h int

h int = ĀT ( ĀM 1 ĀT ) 1 b ĀM 1 h⌃ 2 R 6N (23a) Ā = ⇥ G( r) T I 6(N 1 ) ⇤ 2 R 6(N 1)⇥6N (23b) b = [ (S(!o) 2 r21) T 01⇥3 ... (S(!o) 2 r N 1 ) T 01⇥3 ] T 2 R 6(N 1) , (23c)
which arise in the end effectors system subject to the noninteraction wrenches h⌃ . Then h int are internal in the sense of Definition 1 and Ker(G) ⌘ Im( ĀT ).

Proof. First we prove the equivalence Ker(G) ⌘ Im( ĀT ). It is sufficient to show that (i) the dimensions of Ker(G) and Im( ĀT ) coincide and (ii) the following equation holds

G ĀT = 0 6⇥6(N 1) . ( 24
)
The first condition is trivial to verify, in fact from the fundamental theorem of algebra we have that dim(Im( ĀT )) = 6(N 1) and dim(Ker(G)) = 6N dim(Im(G)) = 6(N 1).

In order to prove the second condition we expand (24) obtaining i and recalling that S(r 1i ) = S(r 1 ) S(r i ), 8 i = 2, . . . , N yields (24). As stated in [START_REF] Erhart | Model and analysis of the interaction dynamics in cooperative manipulation tasks[END_REF], equation (23a) results by applying Gauss' principle of least constraints to the end effectors system in Fig. 2 subject to the constraints Āẍ = b, with Ā and b as in (23). To prove that the wrenches in (23a) are internal in the sense of Definition 1, it is sufficient to show, according to Theorem 2, that h int 2 Ker(G). This comes trivially by multiplying G and h int keeping in mind the result in (24), i.e.

Gh int = G ĀT | {z } 0 6⇥6(N 1) ( ĀM 1 ĀT ) 1 b ĀM 1 h⌃ , ( 25
)
therefore h int 2 Ker(G).

Lemma 2. If the non-interaction wrenches h⌃ make the manipulators move with an acceleration compatible with the constraints (3b), then they do not induce any internal wrench.

Proof. We write the non-interaction wrenches h⌃ = M ẍ = M (G T ẍo + b), since by assumption they make the manipulators move with accelerations ẍ compatible with the constraints (3b). We substitute them in (23a), yielding

h int = ĀT ( ĀM 1 ĀT ) 1 b Ā(G T ẍo + b) . ( 26
)
We know from (24) that G ĀT = ( ĀG T ) T = 0 6⇥6(N 1) , therefore ĀG T = 0 6(N 1)⇥6 . Moreover, it is easy to see that Āb = b. Substituting these results in (26) yields h int = 0 6N ⇥1 .

Notice that the overall interaction wrenches h are the sum of the external (motion-inducing) wrenches h ext and the internal ones, namely h = h ext + h int .

III. DECOUPLING CONTROL

The control scheme proposed in [START_REF] Erhart | Model and analysis of the interaction dynamics in cooperative manipulation tasks[END_REF] is based on an impedance control law to obtain the compliant behavior of the end effectors. We add to this scheme an allocator that makes the internal wrenches h int coincide with desired ones h int,d , so that the impedance control still achieves the tracking of x d o , ẋd o , ẍd o , in a fully decoupled way. In Fig. 3 the overall proposed control scheme is depicted.

A. Motion-Inducing Wrenches Control

Following [START_REF] Erhart | Model and analysis of the interaction dynamics in cooperative manipulation tasks[END_REF], each of the N manipulators is individually controlled by an impedance control law, which renders the end effectors compliant according to

Mi(xi)[ẍi ẍd i ]+Di(xi, ẋi)[ ẋi ẋd i ]+h K i (xi, x d i ) = hi +h d i . ( 27
)
The impedance parameters matrices M i (x), D i (x i , ẋi ) 2 R 6⇥6 represent the apparent mass and damping of the i-th manipulator. They are uniformly positive definite matrices and exhibit a block diagonal structure to decouple the translational and the rotational effects, namely M i = blkdiag(m i I 3 , J i ) and

D i = blkdiag(d i I 3 , i I 3 ). The term h K i (x i , x d i )
represents the geometrically consistent translational and rotational stiffness [START_REF] Caccavale | Six-dof impedance control of dual-arm cooperative manipulators[END_REF], id est

h K i (x i , x d i ) =  (k i I 3 ) p i (2 ⌘ i  i I 3 ) ✏ i ,
where the relative position is p i = p i p d i and the relative orientation is q i = [ ⌘ i , ✏ T i ] T . The terms k i ,  i 2 R + denote the impedance translational and rotational stiffnesses. Inspired by [START_REF] Erhart | Model and analysis of the interaction dynamics in cooperative manipulation tasks[END_REF], we rewrite (27) as

M (x)ẍ = ,h ⌃ z }| { h x + h d +h (28)
where

h x = M (x)ẍ d D(x, ẋ)[ ẋ ẋd ] h K (x, x d
) represents the wrenches involved in the local end effectors dynamics. The matrices M (x) and D(x, ẋ) are block diagonal matrices and have respectively on their diagonal the terms M i (x) and D i (x, ẋ). In the nominal case h x = M (x)ẍ d denotes the wrenches needed to obtain the desired acceleration ẍd of the manipulators according to (3b). The desired end effectors wrenches h d can be obtained by distributing the desired object wrench h d o among the manipulators with a load distribution strategy. The term h denote the interaction wrenches. Notice that we have defined the non-interaction wrenches h ⌃ = h x + h d , consistent with the formulation in [START_REF] Walker | Analysis of motion and internal loading of objects grasped by multiple cooperating manipulators[END_REF]. The stability of this system (without the allocator) has been already proven in [START_REF] Erhart | Model and analysis of the interaction dynamics in cooperative manipulation tasks[END_REF] and is based on the fact that the system of the object and the manipulators is strictly output passive [7, Lemma 1].

B. Internal Wrenches Control

The choice of the time-varying reference value h int,d for the internal wrenches has a pivotal importance in the formulation of the control law.

Proposition 1. The reference internal wrenches h int,d must satisfy h int,d 2 Ker(G), or equivalently h int,d 2 Im( ĀT ) (see Theorem 3). Thus we can write h int,d as

h int,d = ĀT z (29)
for a suitable vector z 2 R 6(N 1) . Note also that the grasp matrix G is constant in the object reference frame {o}, thus we choose first o h int,d in the object reference frame where Ker(G) is constant and then we transform each of its N components in the world reference frame {w}, namely

h int,d i = blkdiag( w R o (q o ), w R o (q o )) o h int,d
i , for all i. The control of the internal wrenches is achieved by means of the following decoupling control law.

Theorem 4. The control signal

h x w = ĀT ( ĀM 1 ĀT ) 1 ( ĀM 1 h int,d + v) (30)
where v = b ĀM 1 h x makes the internal wrenches h int coincide with the desired reference h int,d . Moreover h x w does not affect the interaction dynamics [START_REF] Caccavale | Six-dof impedance control of dual-arm cooperative manipulators[END_REF] and the stability properties of the closed-loop system still holds.

Proof. First inspect the internal wrenches in (23a) replacing h⌃ = h x + h x w as depicted in Fig. 3 and obtain

h int = ĀT ( ĀM 1 ĀT ) 1 ⇣ v z }| { b ĀM 1 h x + ĀM 1 ĀT ( ĀM 1 ĀT ) 1 ( ĀM 1 h int,d + v) ⌘ = ĀT ( ĀM 1 ĀT ) 1 (v v + ĀM 1 h int,d ) = ĀT ( ĀM 1 ĀT ) 1 ĀM 1 h int,d
and knowing that, as stated in Proposition 1, the reference h int,d can be written as h int,d = ĀT z for a suitable function z, we finally get: prove that the interaction dynamics is not affected by h x w we substitute h ⌃ = h x + h x w + h d in [START_REF] Caccavale | Six-dof impedance control of dual-arm cooperative manipulators[END_REF] and get

h int = ĀT ( ĀM 1 ĀT ) 1 ĀM 1 ĀT z = ĀT z = h int,
ẍo = M 1 (h ⌃ o + G(h x + h x w + h d ) b).
Recalling the expression of h x w in (30) we know, from Theorem 3, that Gh x w = 0 6⇥1 , therefore h x w does not alter the acceleration of the manipulated object ẍo . Thanks to this property, we can infer that the control signal h x w does not affect the stability properties of the system addressed in [START_REF] Erhart | Model and analysis of the interaction dynamics in cooperative manipulation tasks[END_REF].

IV. EXAMPLE

In this section we illustrate the results obtained by means of an example. We analyze a rigid bar grasped by two cooperative manipulators independently controlled, accordingly to the control scheme in impedance inertial parameters for both manipulators are m i = 10 Kg, J i = 0.5 Kgm 2 , the impedance damping parameters are d i = 180 Ns m and i = 10 Nms rad and finally the impedance translational and rotational stiffnesses are k i = 300 N m and  i = 50 Nm rad . We want the object to follow the trajectory depicted in Fig. 5. The bar is lifted up along the z-axis and rotated about the same axis in the world reference frame {w}. A disturbance wrench ho = 10[1, 1, 0, 0, 0, 0] T N in the world reference frame {w} acts on the object. At the same time a load cycle of traction and compression is applied by the manipulators to the bar as indicated in Fig. 6. As expected, the static allocator makes be seen from Fig. 7 that the object follows very well the desired trajectory and the actual velocities and accelerations correspond to the reference ones, regardless of the internal wrenches applied to it. Thus we have shown that our control approach allows to independently control the motion-inducing wrenches (necessary to obtain the desired motion of the object) and the internal wrenches exerted by the manipulators in a fully decoupled manner. This simple academic simulation illustrates properly the effectiveness of our control approach. We have shown that the motion of the object is not affected by the application of internal wrenches and, vice versa, internal wrenches follow the desired setpoint independently of the motion-inducing wrenches.

V. CONCLUSIONS AND FUTURE WORK

In this paper we design a control scheme able to achieve the tracking of the motion-inducing and the internal wrenches involved in a cooperative manipulators setup, in a fully decoupled way. This result is based on a physically and mathematically consistent characterization of the interaction dynamics and of the internal wrenches. An impedance control low is used to achieve the compliant behavior of the end effectors, while a static allocator make the internal wrenches coincide with the desired reference, without affecting the motion of the object. Future developments of this paper include an experimental evaluation of the proposed technique.

Fig. 1 .

 1 Fig. 1. Kinematic and dynamic quantities of interest in the cooperative manipulators setup.

  03 I3 03 ... I3 03 S(r1) I3 S(r2) I3 ... S(r N ) I3 i )+S(r12)+S(r2) I3+I3 ... S(r1)+S(r 1N )+S(r N ) I3+I3

Fig. 3 .

 3 Fig.3. Overall control scheme: impedance control for motion-inducing wrenches and static allocator (in red) for internal wrenches.

Fig 3 .Fig. 4 .

 34 Fig. 4. Coordinate systems involved in the simulation.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Trajectory of the object center of mass and of the manipulators.

Fig. 7 .

 7 Fig. 7. Translational and rotational position, velocity and accelerations of the object center of mass along and about the world z-axes.

  d , therefore we have shown that the control law in (30) makes the internal wrenches h int coincide with the reference h int,d . To

					h int,d
	x d o , ẋd o , ẍd o	Kin. Coord. Eq. (3)	x d , ẋd , ẍd x, ẋ	Imp. Control Eq. (18)	h x Allocator Eq. (21)	+	+ + h x w	h ⌃ h⌃	Interact. dyn. Internal Dyn. Eq. (14)	ẍo h int
		Inv. Dynamics	h d o	Load Distrib. [3]	h d				ho Eq. (12)
		x o		x o

Internal wrenches can be seen equivalently as formation-violating (from the object point of view) or as formation-maintaining wrenches (from the manipulators point of view).

The

3D rotation group Spin(3) is a double cover of SO[START_REF] Erhart | Load distribution in cooperative manipulation tasks[END_REF], that is the group of all rotations about the origin of the three-dimensional Euclidean space R 3 .

The skew symmetric matrix function a 2 R 3 ! S(a) 2 R 3⇥3 implements the cross product, id est S(a)b = a ⇥ b.

We consider the twist velocities and accelerations. For this reason ẋ and ẍ have dimensions 6N .

ACKNOWLEDGMENT T

This work was supported in parts by the German Research Foundation (DFG) within the Joint Sino-German research project "Control and Optimization for Event-triggered Networked Autonomous Multi-agent Systems (COVEMAS)" and ANR grant "HANDY" (number ANR-18-CE40-0010).


Joshua Asirvatham $

Minh Anh Luong

Kiran Baraik

Tapas Ganguli

Alain Claverie

Aloke Kanjilal
email: aloke.kanjilal@snu.edu.in

Revealing the Impact of Pre-structural Ordering in GaSb Thin Films

Keywords: Phase Change Memory, Conductive Atomic Force Microscopy, GaSb, Crystallization Dynamics, Surface Modification, Synchrotron Radiation Photoelectron Spectroscopy

For phase change materials (PCMs) to become a viable universal memory candidate and obsolete von-Neumann architecture, materials with very low crystallization speed are needed. Moreover, introducing pre-structural ordering inside the material is also being touted as one of the promising techniques to reach the speed of SRAM or further down. In this aspect, GaSb alloys are showing much promise for not only having very low crystallization times by themselves and also showing an enormous drop in the programming time while crystallizing the re-amorphized material. Here we demonstrate how the threshold switching behaviour changes for the fully amorphous film in contrast with the disordered film with nucleation sites using conductive-Atomic Force Microscopy. It is found that the required power and programming current for memory switching with nominally stoichiometric GaSb (44:56) are 23 nW and 6.2 nA, respectively. As expected for a nucleation oriented PCM, the deposited thin film has two voltage thresholds during the local programming process, one for conduction and another for memory switching. However, when the nucleation sites are introduced inside the disordered film, the conduction and memory switching become simultaneous. This is also found to be reducing threshold power and SET current by 93%

(1.5 nW) and 91% (550 pA), respectively. Furthermore, we also reveal the origin behind this behavioural change observed between these two thin films by using high-resolution transmission electron microscopy and synchrotron radiation photoelectron spectroscopy.

Introduction

The ever-increasing hunger for high-density data storage and to remove the von-Neumann bottleneck led to the objective of inventing a non-volatile, high speed and self-computing memory.

Out of all those currently available memory candidates, Phase Change Material (PCM) based memory devices are very promising due to the maturity reached in the current understanding of those materials and their behaviour. [START_REF] Wuttig | Towards a universal memory?[END_REF] However, they suffer from a serious problem in terms of material aspect as the multi-element PCM alloys tend to segregate when it is crystallized and reamorphized multiple times. [START_REF] Krusin-Elbaum | Evidence for segregation of Te in Ge 2 Sb 2 Te 5 films: Effect on the "phase-change" stress[END_REF] We should note here that the PCMs are the materials that can switch their structure between amorphous and crystalline, and vice-versa, via small electrical power. This structural difference is exploited to store the data bits as 0 and 1. These two different structural properties associated with covalent bonding in amorphous and meta-valent bonding at crystalline states give rise to the contrast in optical reflectivity and electrical resistance. They are detected by the peripheral instrumentation and circuits, called 'reading' of those stored bits. [START_REF] Wełnic | Reversible switching in phase-change materials[END_REF][START_REF] Wuttig | Incipient metals: functional materials with a unique bonding mechanism[END_REF] In PCM based memories, one of the limiting factors which are solely based on the chosen material is the programming time. Up to now, there has been a great deal of research which is mainly focused on finding novel PCM alloys [START_REF] Zhang | Investigation of environmental friendly Te-free SiSb material for applications of phase-change memory[END_REF][START_REF] Lu | Investigation of HfO2 doping on GeTe for phase change memory[END_REF] and modifying the prototype material like Ge2Sb2Te5 (GST) by doping [START_REF] Zhu | Ni-doped GST materials for high speed phase change memory applications[END_REF][START_REF] Wu | High-speed and large-window C-doped Sb-rich GeSbTe alloy for phase-change memory applications[END_REF][START_REF] Wang | Scandium doped Ge2Sb2Te5 for high-speed and low-power-consumption phase change memory[END_REF] for reducing the programming speed. In recent years, Antimony (Sb)-rich materials are also emerging as the primary PCMs for universal memory devices since their crystallization speed is significantly less and fulfils the requirement of lower reamorphization energy than Te-based PCMs. [START_REF] Chang | Phase stability, bonding and electrical conduction of amorphous carbon-added Sb films[END_REF]11 In that particular subset of alloys, GaSb alloys are getting special attention due to the fact that they possess high transition temperature, and the phase segregation can be stopped by having novel device architectures. [START_REF] Chang | Crystallization behaviors of an ultra-thin Ga-Sb film[END_REF][START_REF] Putero | Phase transition in stoichiometric GaSb thin films: anomalous density change and phase segregation[END_REF] One of the simple, repeatable and robust ways to reduce the crystallization speed is to exploit the loopholes in the incubation time. Loke et al. [START_REF] Loke | Breaking the speed limits of phase-change memory[END_REF] demonstrated this by constantly supplying a small electric field to induce the nucleation called 'voltage priming' and had a breakthrough speed in the order of picoseconds for the GST itself. The origin behind this considerable reduction in speed is attributed to the carry-over of the pre-structurally arranged atoms at the nucleation sites to the crystallization state. Whereas, the atomic arrangement is found to be random, without any preferred nucleation sites in the fully amorphous films and thereby losing the initial atomic arrangement during crystallization. The process of nucleation and crystal growth in the fully amorphous films follow the random generation and annihilation of the bonds during the crystallization process. However, in the pre-structurally ordered films, the crystal growth happens from the already introduced nucleation sites. [START_REF] Loke | Breaking the speed limits of phase-change memory[END_REF] One of the materials which shows this kind of behaviour is GaSb based alloys. Even without voltage priming, there is a pronounced decrease in the crystallization speed from 57 ns to 19 ns, where a remnant short-range crystalline order is introduced by the previous re-amorphization process. [START_REF] Cheng | The crystallization behavior of stoichiometric and offstoichiometric Ga-Sb-Te materials for phase-change memory[END_REF] One of the hindrances to study the material aspects of this pre-structural ordering is that the difficulty to introduce the nucleation sites in a controlled way inside the film. Recently, Wu et al. [START_REF] Wu | Bipolar Resistive Switching in the Ag/Sb2Te3/Pt Heterojunction[END_REF] have indicated that, when the 150 W sputtering power is utilized, a good polycrystalline film is achieved for Sb2Te3 without any external annealing. It is a very interesting result as the sputtering power can also be considered as one of the tools to crystallize the PCMs other than the conventional methods like annealing and joule heating. [START_REF] Wu | Bipolar Resistive Switching in the Ag/Sb2Te3/Pt Heterojunction[END_REF] In this communication, we have utilized this idea in such a way that a small and relatively higher sputtering power were chosen to introduce nucleation sites in the films. The local atomic arrangement of these nucleation sites was thoroughly studied by cross-sectional transmission electron microscopy (XTEM). We have also demonstrated how the threshold switching behaviour of this thin film with nucleation sites differs from that of a fully amorphous film by conductive atomic force microscopy (c-AFM). In addition, we have investigated the origin of the change in threshold switching by studying those films' valence band structure and the chemical bonding nature using synchrotron radiation photoelectron spectroscopy.

Experimental Methods

Commercially purchased SiO2(200 nm) /Si and Pt(100 nm)/ Ti(20 nm)/ SiO2(200 nm)/Si substrates were used for structural and electrical study, respectively. Before GaSb film deposition, all the substrates were chemically cleaned in a systematic way. For that, the substrates were ultrasonicated in acetone, isopropyl alcohol, and de-ionized water for 5 min in each step and dried in air. Thin GaSb films were prepared by the RF magnetron sputtering technique, where the substrate-to-target distance was kept at 12 cm for all depositions. A 2" target (4N purity) with Ga: Sb of 43:57 was commercially bought from MTI. The substrate temperature was maintained at 15 o C during thin film deposition by water cooling. All the samples were deposited over cleaned Pt/Ti/SiO2/Si substrates. Chamber base pressure of ~2×10 -7 mbar has been reached every time before deposition, where the working pressure was maintained at ~3 ×10 -3 mbar by purging argon (Ar) gas at 20 sccm. For the growth of amorphous GaSb thin films, RF power of 20W (S1) was used, whereas 30W (S2) was employed for engineering the nucleation sites. The deposited film thickness was measured by the stylus surface profilometer (Bruker, DektakXT). An optimized film thickness of ~80 nm was considered for the present study. Note that an ultra-thin native oxide layer was formed on the surface of S1 due to the exposure to the ambient atmosphere, while the surface of S2 was passivated by the nitridation process. The surface oxidation and nitridation were studied by diffuse reflectance mode using Shimadzu Solidspec-3700i spectrophotometer and the details are provided in the supporting information. The nitridation process was carried out with an RF power of 20W for 1 min by injecting a 3:1 ratio of Ar: N2 gases, giving a working pressure of 4.3 × 10 -3 mbar. [START_REF] Lu | GaSb-oxide removal and surface passivation using an electron cyclotron resonance hydrogen source[END_REF][START_REF] Gotow | Surface cleaning and pure nitridation of GaSb by in-situ plasma processing[END_REF] The crystallization dynamics was studied in a custom-made gas analyzer set-up attached with a Source/Measure Unit (Keysight B2902A) in Ar atmosphere by annealing up to 430 o C with a ramp rate of ~27 o C/min. The resulting annealed samples of 20W (S3) and 30W (S4) were later characterized structurally by Grazing Incidence X-ray Diffraction (GIXRD) in a Bruker D8 Discover system at a grazing angle of 0.75 o .

The morphology of the deposited films was characterized by Field-Emission Scanning Electron Microscopy, FE-SEM (MIRA II LMH from TESCAN). The elemental property was further examined by Energy-Dispersive X-ray spectroscopy (EDX) with the attached INCA PentaFET3 detector from Oxford. To reduce the volatility of antimony, moderate electron energy of 7 kV was used for SEM measurements. For in-depth structural analysis, High-Resolution Transmission Electron Microscopy (HR-TEM) investigation was performed in cross-sectional geometry with an image corrected FEI-Tecnai F20 (200kV) and the I 2 TEM from Hitachi (300 kV) at CEMES, Toulouse -France. A 4 nm thick SiO2 capping layer was deposited for handling the sample in air, where a Pt layer was deposited further to avoid damage during the XTEM sample preparation by Focussed Ion Beam. Similarly, the SiO2 coated samples were transported to the Xray Photoelectron Spectroscopy (XPS) station of the Angle Resolved Photoelectron Spectroscopy, ARPES beamline (BL-10) of Indus-2 synchrotron radiation facility with a storage ring operated at 2.5 GeV at Raja Ramanna Centre for Advanced Technology Indore, India. Here, the SiO2 capping was removed by Ar ion sputtering at 500 eV for 30 min. After removing the capping layer, the photoelectron spectroscopy was carried out with synchrotron radiation for in-depth understanding.

We should note here that the Fermi level was calibrated to zero by using a gold sample. The deconvolution and component analysis of the XPS data were done after the Shirley background subtraction using the CASAXPS software. The parameters which are necessary for XPS data fitting such as area ratio, intensity ratio, line shape and doublet splitting were taken from the NIST database for both Ga and Sb atoms. [START_REF] Naumkin | NIST X-ray Photoelectron Spectroscopy Database[END_REF] Park XE7 AFM system coupled with VECA amplifier was utilized in characterizing the samples in c-AFM mode. For the top electrode, a conductive Au coated cantilever (NSC18-Cr-Au) was used in the present study. The physical contact area, Ac, was calculated to be ~ 41 nm 2 for the above-mentioned cantilever from the Hertz contact theory. [START_REF] Benstetter | Fundamentals of CAFM Operation Modes[END_REF] All the current-voltage (I-V) measurements were carried out by a triangular sweeping pulse of 0-5V with a 1s period which results in a 1.95 ms holding time for each data point. The time period was set in a higher range so that the amplifier could follow through with the sample's current response properly. [START_REF] Bhaskaran | Nanoscale phase transformation in Ge2Sb2Te5 using encapsulated scanning probes and retraction force microscopy[END_REF] All the prepared samples were characterized on the same day to avoid the uncertainty induced by the atmospheric conditions. [START_REF] Bosse | Nanosecond switching in GeSe phase change memory films by atomic force microscopy[END_REF] Though the morphology has a granular nature, the maximum peak height (Sp) and maximum valley depth (Sv) are found to be < 2 nm. This makes the morphology images very difficult to view, and thus, we have utilized the enhanced colour option to display. The typical plan view SEM image of S1 deposited on Pt/Ti/SiO2/Si substrate is displayed in Fig. 1(a), showing the formation of granular features in grey on dark background. Because of the weak contrast of the features, the size distribution has been extracted by using a correlation function g(r)

Results and Discussion

with radius, as shown in Fig. 1(b). As discerned, the feature size distribution peaking at ~100 nm seems to be dominant. The corresponding EDX spectrum is displayed in Fig. 1(c To understand the transition temperature effects between S1 and S2, we have studied the crystallization dynamics using a ~27 o C/min ramping rate [see Fig. 2(a)]. The transition temperature of S1 is found to be 341 o C, whereas the transition temperature S2 is reduced by ~13 o C.

After annealing, the resistance of S4 is found to be slightly lower than that of S3. To understand whether this lower resistance is intrinsic or due to any other effects, we performed the GIXRD on these samples. It can be clearly seen from Fig. 2(b) that there is a moderate increase in intensity of the Sb-(012) reflection of S4 compared to S3. This indicates that after annealing to the same temperature, the antimony segregation is relatively more in S4 than S3. The segregated Sb inside the GaSb matrix has a unique characteristic in crystallization where the resistance of the thin film, in conjecture with the antimony segregation, reduces linearly with respect to the thermal budget provided to it without any sudden drop. [START_REF] Raoux | Phase transitions in Ga-Sb phase change alloys[END_REF] In regards to the nucleation in S2, which is deposited at relatively higher RF sputtering power, the GIXRD pattern shows reflections from the multiple planes of the stoichiometric GaSb grains, revealing the formation of randomly oriented nanocrystals. Close inspection of the GIXRD pattern of S2 film (within the violet box) in the inset after the background subtraction shows a superimposition of a sharp reflection at 25.2 o from the (111) planes of the stoichiometric GaSb on a prominent 'hump' from the (012) planes of the elemental Sb, mainly associated with sub-1 nm crystals in rhombohedral structure. 24 This is also confirmed by XTEM investigation, discussed below. In S1, no distinct crystalline peaks are observed, indicating that nominally stoichiometric GaSb is in the amorphous state. To understand further about the location of the nucleation sites, S2 film was investigated by XTEM. One of the best ways to find out the formation of crystallites in the deposited film is the application of dark-field mode, as shown in Fig. 3(a). Here, the bright spots represent the crystalline region. Besides the dark-field XTEM image, the diffused ring in the corresponding SAED pattern [see, Fig. 3(b)] elucidates the formation of small crystallites of elemental Sb with an interplanar distance of ~0.301 nm. Moreover, the HR-TEM image in Fig. 3(c) further confirms the formation of GaSb nano-crystallites of about 1-2 nm along the bottom interface only where the interplanar distances of some of the nuclei (indicated by the red boxes) are revealed by the FFT analysis. They are mainly found to be 0.2076 nm and 0.348 nm, which belong to the (220) and (111) planes of the stoichiometric GaSb, respectively. From this study, it can be confirmed that the GaSb prefers heterogeneous and interfacial nucleation, whereas elemental Sb favours homogeneous nucleation all over the film. This is understandable as the interfacial activation energy is lower for the nucleation oriented PCM, where it behaves like an under-cooled liquid at the interface when capped with dielectric materials. [START_REF] Wełnic | Reversible switching in phase-change materials[END_REF][START_REF] Kalb | Kinetics of crystal nucleation in undercooled droplets of Sb-and Te-based alloys used for phase change recording[END_REF] On the other hand, the Sb and Sb-rich compositions are the growth-oriented PCMs, and they tend to have homogeneous nucleation before the crystallization growth. As it has been previously illustrated by Kalb et al., growthoriented homogenous nucleation centres tend to be minuscule in volume compared to the nucleation oriented interfacial nucleation sites. [START_REF] Wełnic | Reversible switching in phase-change materials[END_REF][START_REF] Kalb | Kinetics of crystal nucleation in undercooled droplets of Sb-and Te-based alloys used for phase change recording[END_REF] This is observed in nominally stoichiometric GaSb too, [see, Figs. 3(a) and(c)], where the homogenous Sb nucleation centres are in the subnanometer (~0.6 nm) region. But, the nucleation oriented stoichiometric GaSb nucleation centres have a higher dimension of 1 -2 nm.

To understand the electronic switching behaviour of S1 and S2, we have performed c-AFM study as it is known to be a powerful tool for local analysis. For that, we have considered films, which have been grown on the Pt/Ti/SiO2/Si substrates with the same deposition parameters. We should note here that the surface of S1 is kept bare for this study and nitridated for S2. [START_REF] Yang | Nanoscale multilevel switching in Ge2Sb2Te5 thin film with conductive atomic force microscopy[END_REF] The first one belongs to the conduction threshold of S1 at 2.6V, where the charge carrier concentration increases gradually, albeit non-linear, with respect to the applied voltage until reaching the second threshold at 3.7V. Above this threshold limit, the conduction becomes abrupt and increases exponentially. This is attributed to memory switching, which by conjecture signifies the crystallization process due to Joule heating. Here, the programming current required for this memory switching is 6.2 nA, which is in the same range of current required for other prototype materials, like Ge2Sb2Te5 26 and GeTe6 27 in the c-AFM configuration. We have kept a lower compliance limit of 10 nA for (uncapped) S1 to avoid the formation of oxide layers with increasing local temperature. Since it is an 80 nm film, the programming voltage and current correspond to the electric field and switching power of 46.5 MV/m and 23 nW, respectively. Once the programming cycle is done, we have repeated the DC sweep on the same point to detect the read voltage as well as to study the characteristics of the non-volatile filament. Although the switching power is extremely low, the sample's current response from the read DC sweep cycle indicates the requirement of a higher threshold voltage to read the bit stored in a particular region. The appearance of a higher conduction threshold voltage (1.22 V) indicates that surface oxidation affects the electrical behaviour of the non-volatile filament. Mizokawa et al. [START_REF] Mizokawa | Long-time air oxidation and oxide-substrate reactions on GaSb, GaAs and GaP at room temperature studied by X-ray photoelectron spectroscopy[END_REF] have established the requirement of only a few seconds to form the native oxide layer at the surface of GaSb by exposing it to the ambient atmosphere. Compared to other semiconductors, the formation of segregated native oxides like Ga2O3 (ΔG = -238.6 Kcal/mol) and Sb2O3 (ΔG = -151.5 Kcal/mol) are far more energetically favourable and thermodynamically stable compared to GaSb (ΔG = -9.3 Kcal/mol). [START_REF] Schwartz | Oxide-substrate and oxideoxide chemical reactions in thermally annealed anodic films on GaSb, GaAs, and GaP[END_REF] This makes the GaSb oxidize more rapidly at room temperature, and native oxide formation on S1 has also been confirmed (Fig. S1 in Supporting Information).

Compared to GST and other PCMs, there has been a lot of research carried out on the surface passivation effects of GaSb with various methods ranging from nitridation and hydrogen treatment, [START_REF] Lu | GaSb-oxide removal and surface passivation using an electron cyclotron resonance hydrogen source[END_REF] sulfidation, [START_REF] Perotin | Passivation of GaSb by sulfur treatment[END_REF] etc. Among various methods used so far for making the surface less susceptible to oxidation, passivating the surface in RF plasma with nitrogen gas holds special attention as it can be done immediately after the deposition without requiring any other materials, precursors, instrumentation, etc. It has already been confirmed that it possesses the lowest trap densities and, in turn, reduces the complexities involved in the charge transport. However, these surface passivation methods cannot act as a physical capping layer like DLC, TiN or ITO. Instead, these passivation methods form thermodynamically stable bonds separately with the constituent atoms (for III-V semiconductors) at the surface with the reactive nitrogen ions. [START_REF] Gotow | Surface cleaning and pure nitridation of GaSb by in-situ plasma processing[END_REF][START_REF] Preisler | Nitridation of epitaxially grown 6.1 Å semiconductors studied by X-ray photoelectron spectroscopy[END_REF] The ineffectiveness of nitrogen passivation of the 20W film surface is addressed in Fig. S2 of the Supporting Information. After deposition, the passivation of S2 was done by purging nitrogen and argon gases with the ratio of 1:3 in the RF plasma for 1 min. Figure 5(a) shows the typical morphology of this film, and the corresponding I-V characteristics are presented in Fig 5(b). Gallo et al. [START_REF] Gallo | Evidence for thermally assisted threshold switching behavior in nanoscale phase-change memory cells[END_REF] have already illustrated that a PCM can remember the thermal budget introduced to it, and the threshold voltage only depends on the remaining amount of amorphous material present locally to switch it to a crystalline state. Based on this argument, the observed reduction in threshold voltage and the memory switching can be mainly attributed to the crystallization of remaining amorphous material inside the S2. From Fig. 5(b), it appears that until the conduction threshold of 2.9V is reached, the conduction remains extremely low. But, once it reaches the conduction threshold, the conduction abruptly increases a lot, indicating a simultaneous memory switching effect. This I-V characteristic closely resembles the behaviour of the well-known growth dominated material GeTe6. [START_REF] Manivannan | Low power ovonic threshold switching characteristics of thin GeTe6 films using conductive atomic force microscopy[END_REF] Hence, it is reasonable to conclude that the self-focalization of the non-volatile filament formation in c-AFM configuration can categorize the PCM as nucleation based (with two voltage thresholds) and growth based (with one voltage threshold). Since S2 consists of nanocrystalline nucleated sites, the required electric field and power are also very less for memory switching, and the higher conduction threshold observed in this case can be attributed to the electrostatic screening arising from the amorphous-crystalline interfaces inside the film. [START_REF] Nardone | Electrical conduction in chalcogenide glasses of phase change memory[END_REF] Here, the memory switching requires the electric field of 36.4 MV/m and the power of 1.55 nW. As the memory switching and conduction threshold are happening simultaneously, it is possible to calculate the energy required for memory switching by taking the hold time of each data point as the time factor. [START_REF] Manivannan | Low power ovonic threshold switching characteristics of thin GeTe6 films using conductive atomic force microscopy[END_REF] In this way, we have calculated the required energy value for switching as 3 pJ. This value can be further reduced by having a lower pulse width to program the material. It can also be observed from Fig. 5(b) that the read voltage can be anywhere from the conduction threshold of the non-volatile filament (0.9V) to a voltage of slightly less than the memory switching threshold (2.9 V). This further confirms the efficacy of the nitridated surface in reducing the conducting threshold of Read DC cycles in comparison with the native oxide. As presented in Fig. 5(c), we programmed multiple clusters of points in close quarters with a 5V DC cycle. The cluster of programmed points shows excellent non-volatility, as represented in Fig. 5(d), by keeping the compliance current at 100 nA. This information further confirms that the nitridated surface can be an effective way to reduce the read voltage. To verify the uniformity of the nitridated surface and the effect on non-volatile filament formation, we have selected 16 separate points on a new sample area [see Fig. 6(a)] and programmed it as shown in Fig. 6(b). The average memory switching (indicated by red filled circles) threshold is found to be 2.91 V with a standard deviation of ± 0.15 V. The very low standard deviation of ~5% on the memory switching threshold indicates a uniform nitridation all over the sample surface, and thus it can serve as an effective surface passivation layer. 3.7 V 2.9 V Conduction Threshold Before Programming 2.6 V 2.9 V Conduction Threshold of the non-volatile filament 1.2 V 0.8 V To analyze the non-volatile filament formation and the filament itself, we have plotted the I-V characteristics in the log scale. As can be observed from Figs. 7(a) and (b), both the samples have a very low conductance until the conduction threshold is reached. In GST, the origin of this change in conductivity is generally assigned towards the reduction in valence band offset with respect to the Fermi level (EF) due to the delocalization at the p-orbitals of the Sb and Te atoms. [START_REF] Wełnic | Reversible switching in phase-change materials[END_REF] Due to this delocalization, the valence band offset is often tuned in PCM when it changes its phase from amorphous to crystalline. Prior hard XPS studies revealed that the valence band of the crystalline state falls onto the Fermi level and thereby explaining its metallicity, whereas an offset of ~0.4 eV in the amorphous state justifies its high resistance state. [START_REF] Kim | Electronic structure of amorphous and crystalline (Ge Te) 1-x (Sb 2 Te 3) x inves gated using hard x-ray photoemission spectroscopy[END_REF][START_REF] Klein | Changes in electronic structure and chemical bonding upon crystallization of the phase change material GeSb 2 Te 4[END_REF][START_REF] Richter | Hard x-ray photoelectron spectroscopy study of Ge2Sb2Te5; as-deposited amorphous, crystalline, and laser-reamorphized[END_REF] To get more information about the valence band structure and its changes, synchrotron radiation UPS was performed in all four samples. [START_REF] Sarkar | Origin of electronic localization in metal-insulator transition of phase change materials[END_REF] The amorphous sample has a valence band offset of ~ 0.1 eV, which justifies the high resistance state.

The UPS spectra of the crystalline and amorphous sates follow each other very closely, except two regions which are presented in the insets. There is a pronounced 'dip' near 4 eV [see insets ① and ③] for both the crystalline samples (S3 and S4) compared to their amorphous counterparts (S1 and S2) where the Sb 5s (~6 eV) overlaps with the p-orbitals. This has been observed in the prototype materials like GST and has theoretically predicted for GaSb too. [START_REF] Sarkar | Origin of electronic localization in metal-insulator transition of phase change materials[END_REF][START_REF] Dixon | Origin of the reverse optical-contrast change of Ga-Sb phase-change materials-An ab initio molecular-dynamics study[END_REF] This 'dip' is generally attributed to the sharpening of the DOS between the orbitals expected in the crystalline form. [START_REF] Dixon | Origin of the reverse optical-contrast change of Ga-Sb phase-change materials-An ab initio molecular-dynamics study[END_REF] As expected, the valence band maximum falls on EF in both the crystalline samples, indicating that the annealed samples are in the metallic state [see Fig. 8 insets ② and ④]. It is important here to note that the DFT simulation of stoichiometric GaSb (annealed 10% higher than the transition temperature) and the pulsed laser testing of nominally stoichiometric GaSb have shown a negative optical contrast indicating a semiconducting behaviour with a deeper valence band offset in the crystalline form. [START_REF] Dixon | Origin of the reverse optical-contrast change of Ga-Sb phase-change materials-An ab initio molecular-dynamics study[END_REF][START_REF] Raoux | Phase transitions in Ga-Sb phase change alloys[END_REF] However, in S3 and S4, which are annealed 26% well above the transition temperature and consists of Sb segregation, the valence band maximum is falling on the EF revealing the metallic nature of the crystalline samples similar to GST.

Interestingly, when we compared the UPS spectra of the amorphous samples, [see Fig. 8 (c)], a steep increase in spectral weightage is revealed in S2 with respect to S1 near the valence band maximum region (0 -4 eV) consisting of the p-orbitals of both Sb and Ga atoms. Zhang et al. [START_REF] Zhang | Role of vacancies in metal-insulator transitions of crystalline phase-change materials[END_REF] has previously reported that this type of steep rise near the valence band maximum in PCM alloys corresponds to the increase in density of localized wavefunctions of the vacancy disorders. This observation reveals that once the nucleation sites are introduced in the GaSb film, the density of vacancies also arises along with it.

To reveal the origin of this increase in vacancies in S2 by chemical bonding, we performed hard-XPS using synchrotron radiation on S1 and S2, which are presented in Fig. 9. Here, the Sb 4d and Ga 3d core-orbitals are very important, as they are sensitive to the change in chemical environment resulted due to structural change. [START_REF] Richter | Hard x-ray photoelectron spectroscopy study of Ge2Sb2Te5; as-deposited amorphous, crystalline, and laser-reamorphized[END_REF] Since the change in the chemical nature is very small here, we have done the composition analysis. As discerned from the Fig. 9, the Sb 4d spectrum of the as-deposited sample (S1) shows well-defined spin-orbit splitting and have two different components. The primary peaks (63%) with a 2/3 area constraint correspond to the Sb-Ga bonds in the tetrahedral environment. The highly broadened (FWHM = 3 eV) smaller peaks are attributed to the Sb-Sb bonds in defective octahedral states. This observation of defective octahedral states in Te and Sb has been well reported in GST. [START_REF] Richter | Hard x-ray photoelectron spectroscopy study of Ge2Sb2Te5; as-deposited amorphous, crystalline, and laser-reamorphized[END_REF] Here, the same was observed in GaSb based PCM alloys. The broadening is generally attributed to Sb being in a disordered state.

The peak positions of 𝑆𝑏 / , 𝑆𝑏 / , 𝑆𝑏 / , 𝑆𝑏 / are found to be 32.07 eV, 32.51 eV, 33.28 eV and 33.97 eV, respectively. Ga 3d spectrum of S1 also consists of two components. The primary composition (~85%) is the GaSb in the tetrahedral environment, where the other 15% is in the homo-polar Ga-Ga environment. However, Kalikka et al. [START_REF] Kalikka | Density functional simulations of structure and polymorphism in Ga/Sb films[END_REF] described that this Ga-Ga bonds in nominally stoichiometric GaSb can arise from the three-fold coordinated Ga dimer being attached with defective octahedrally coordinated Sb. The peak positions of 𝐺𝑎 / , 𝐺𝑎 / , 𝐺𝑎 / , 𝐺𝑎 / are found to be 19.18 eV, 18.23 eV, 19.61 eV and 18.64 eV, respectively. From the XPS analysis of S1, even though we found only 6% compositionally Sb rich, there is a significant amount of defective octahedrally coordinated Sb in the amorphous state.

For Ga 3d spectrum of S2, there is a clear asymmetry in the lower binding energy region. This results in an increase of homopolar Ga bonds to 23%, where the remaining 74% Ga bonds are in the tetrahedrally coordinated Ga-Sb environment. Though there is a significant increase in the homopolar Ga bonds, the corresponding increase in Sb-Sb is not observed, instead 8% decreases, in the Sb 4d spectrum of S2. Moreover, the FWHM of the octahedral Sb bonds is decreased to ~ 2 eV, indicating the formation of crystalline structure. The Sb bond analysis for S2 confirms the work of Kalikka et al. on a melt quenched amorphous state, indicating that the Sb-Sb tends to decrease by forming the tetrahedrally coordinated heteropolar Ga-Sb bonds. [START_REF] Kalikka | Density functional simulations of structure and polymorphism in Ga/Sb films[END_REF] From the Fig. 3, we have already seen that, Sb nucleation sites are formed all over the film in S2. This in turn creates a short-range highly Sb-rich GaSb environment. Previous reports however indicate that the formation of homopolar Ga dimers is very difficult in a melt quenched amorphous state of Sb-rich GaSb (GaSb7), instead they form a 3-fold arrangement of Ga-Ga-Sb. [START_REF] Kalikka | Density functional simulations of structure and polymorphism in Ga/Sb films[END_REF] Therefore, the discrepancy found in the increase of Ga-Ga bonds can be attributed to the Ga dimers being attached to the Sb-rich local nucleation sites. However, the measured FWHM (2 eV) for the crystalline Sb-Sb sites is still higher than that of the generally observed FWHM (0.6 to 0.7 eV) of Sb components, [START_REF] Cotirlan | Aspects of native oxides etching on n-GaSb (1 0 0) surface[END_REF] and this indicates the randomness in atomic orientations by introducing vacancy disorders. [START_REF] Kim | Electronic structure of amorphous and crystalline (Ge Te) 1-x (Sb 2 Te 3) x inves gated using hard x-ray photoemission spectroscopy[END_REF] From the hard-XPS studies we can conclude that when the Sb nucleation sites is occurred inside the film as a pre-structural ordering, the disorder rich octahedrally bonded Sb sites make bonds with the 3-fold coordinated Ga dimers, and as a result, increases the total density of vacancy states near the EF [Fig. 8(c We believe that this study is a benchmark for understanding the electronic switching behaviour of high-speed crystallization materials like GaSb by utilizing pre-structural ordering.

Supporting information

Tauc plots confirming the formation of native oxides and the nitridated surface are shown in Fig. S1. The morphology, typical I-V characteristics of 20W GaSb film with nitridated surface and the PSD profiles are shown in Fig. S2.

Fig. 1 .

 1 Fig. 1. (a) Typical plan-view FE-SEM image of S1 showing granular features in grey on dark

  ), revealing the atomic concentration of Ga and Sb to be 44:56, which is well inside the composition range of the confirmed phase change materials. The oxygen and Pt signals are coming from the substrate, though we cannot neglect the surface oxidation by exposing to air.

Fig. 2 .

 2 Fig. 2. (a) exhibits the crystallization dynamics for the samples S1 and S2; (b) The GIXRD patterns

Fig. 3 .

 3 Fig. 3. (a)exhibits the dark field TEM image of S2 and (b) the corresponding SAED pattern. The

Fig. 4 (

 4 Fig. 4 (a) The AFM topography image of the GaSb film grown at 20W, with a scan area of 0.5 m

Figure 4 (

 4 Figure 4(a) represents the morphology of S1, showing a granular feature, where the colour

Fig. 5 (

 5 Fig. 5 (a) The morphology of S2 with an area of 1μm x 1μm, where (b) shows the typical I-V

Fig. 6 (

 6 Fig. 6 (a) This presents a typical morphology of a 1μm x 1μm scan and the position of the probed

Fig. 7 .

 7 Fig. 7. The log I versus V plots are shown for (a) S1 and (b) S2. The red circle in (b) indicates the

Fig. 8

 8 Fig. 8 The valence band structure of samples S1 and S3 (a), S2 and S4 (b) and S1 and S2 (c). The

  )]. This increase in disorder-localization near the Fermi level is possibly the origin of the change in the threshold switching behaviour in S2 compared to the fully amorphous S1. This also sheds light on the possible reasons behind the reduction in crystallization time in reamorphized GaSb-based PCM devices.

Fig 9 .

 9 Fig 9. Synchrotron radiation XPS study of samples S1 and S2 and their composition analysis of

Acknowledgement

AK and JA acknowledge the financial support from the Shiv Nadar University and UGC -DAE -CSR for the project No. CSR/Acctts/2016-17/1204. AK and JA thank Dr. Saif Khan from IUAC, Delhi, for his kind help during the SEM measurements. JA wants to acknowledge the help received from Dr. Dip Das, Mrs. Ananya Chattaraj, Dr. Debosmita Banerjee and Dr. Gourav Bhattacharya.

Note

The authors declare no competing financial interest.