Raphaël Taisant
email: raphael.taisant@eleves.enpc.fr

Mandar Datar
email: mandar.datar@inria.fr

Hélène Le Cadre
email: helene.le-cadre@inria.fr

Eitan Altman
email: eitan.altman@inria.fr

H Le

Learning Market Equilibria Using Performative Prediction: Balancing Efficiency and Privacy

We consider a peer-to-peer electricity market modeled as a network game, where End Users (EUs) minimize their cost by computing their demand and generation while satisfying a set of local and coupling constraints. Their nominal demand constitutes sensitive information, that they might want to keep private. We prove that the network game admits a unique Variational Equilibrium, which depends on the private information of all the EUs. A data aggregator is introduced, which aims to learn the EUs' private information. The EUs might have incentives to report biased and noisy readings to preserve their privacy, which creates shifts in their strategies. Relying on performative prediction, we define a decisiondependent game G stoch to couple the network game with a data market. Two variants of the Repeated Stochastic Gradient Method (RSGM) are proposed to compute the Performatively Stable Equilibrium solution of G stoch , that outperform RSGM with respect to efficiency gap minimization, privacy preservation, and convergence rates in numerical simulations.

I. INTRODUCTION

A. Problem Statement

In distributed control system literature, there is a usual trade-off between cost minimization and privacy preservation: to obtain a better outcome, each End User (EU) in the system relies on its own information and the information the others have reported. However, the EUs might be reluctant to release sensitive information because of privacy limitations or because, by acting strategically on their readingstherefore providing fake information, they might decrease their cost. The former situation might give rise to moral hazard, which could generate inefficiencies at the system level. To penalize free-rider behaviors and incentivize the EUs to minimize the variance of their readings, we aim to define contracts which remunerate the EUs as increasing functions of the data quality while discriminating between the EUs depending on their local privacy breach.

In energy systems, data -and the information it contains -is becoming an asset that the agents might want to acquire to learn market equilibria [START_REF] Le Cadre | Peer-to-peer electricity market analysis: From variational to generalized nash equilibrium[END_REF], [START_REF] Shilov | Privacy impact on generalized nash equilibrium in peer-to-peer electricity market[END_REF]. For example, in many demand-response programs, the aggregator uses the EUs' historical energy consumption to determine their nominal demand profiles, and design incentives to flatten peak times. The nominal demand profiles are then used to compute incentives depending on how much they curtail relative to their nominal profiles. However, EUs can act strategically

We consider a peer-to-peer electricity trading problem formulated as a network game, where strategic EUs want to minimize their cost by computing their demand (flexible around a nominal value), controllable generation, and trades with their peers, while satisfying a set of local constraints including the local balance of supply and demand, and trading reciprocity constraints that couple the EUs' optimization problems. Their nominal demand constitutes sensitive information, that is kept private to the EUs. Following [START_REF] Le Cadre | Peer-to-peer electricity market analysis: From variational to generalized nash equilibrium[END_REF], [START_REF] Shilov | Privacy impact on generalized nash equilibrium in peer-to-peer electricity market[END_REF] and the electricity market literature, the EUs' utility functions are assumed additive and quadratic. We prove that under mild assumptions, the network game admits a unique Variational Equilibrium (VE), which is efficient and stable. In addition, we prove that, at the VE, the EUs' best-responses depend on the private information of all the EUs. A Data Aggregator (DA) might then be introduced, which aims to learn the EUs' private information. Indeed the EUs might have incentives to report fake readings which might create shifts in the EUs' strategies. Relying on performative prediction, we define a decision-dependent game to couple the peer-to-peer trading problem with a data market. On the data market, the DA designs a privacy contract incentivizing the EUs to report their private information truthfully. The EUs' readings, demand and controllable generations are then determined as unique solutions of the decision-dependent game, relying on Nash (NE) and Performatively Stable (PSE) Equilibria. Finally, we propose three gradient method-based algorithms to compute NE and PSE, under different coordination schemes between the DA and the EUs, that we test against the Repeated Stochastic Gradient Method (RSGM) from [START_REF] Narang | Multiplayer performative prediction: Learning in decision-dependent games[END_REF]. Numerical results are presented.

C. Reviewed Literature

Over the last few years, there has been a growing body of literature on data markets. On the market design side, Agarwal et al. propose an innovative design involving robust real-time matching, to efficiently buy and sell data [START_REF] Agarwal | A marketplace for data: An algorithmic solution[END_REF], while Shilov et al. quantify the impact of privacy constraints on the market equilibrium [START_REF] Shilov | Privacy impact on generalized nash equilibrium in peer-to-peer electricity market[END_REF]. On the pricing side, Cai et al. in [START_REF] Cai | Optimum statistical estimation with strategic data sources[END_REF] study a data market with many strategic data sources and one DA. The DA aims to build an estimator of a function, thanks to the readings reported by the sources. The quality of the readings depends on the level of effort exerted by the sources, which determines the variance of the sample as a decreasing function of the effort exerted, and the aggregator wants to design incentives to obtain the best possible estimator. Cai et al. prove that there exists a unique NE, which is also a dominant strategy equilibrium. Westenbroek et al. extend this result in [START_REF] Westenbroek | Competitive statistical estimation with strategic data sources[END_REF] by taking into account the non-rivalry nature of data and considering multiple DAs among which the same data can be freely reproduced and shared. In the same vein, Gast et al. in [START_REF] Gast | Linear regression from strategic data sources[END_REF] propose a non-cooperative game model of strategic data sources, which optimize the variance of the noise level of their readings to minimize their cost, which can be split into a disclosure cost and an estimation cost based on linear regression. The impact of strategic information is quantified through explicit bounds on the price of stability. The above works differ from ours in that we explicitly take into account the dependence of the reading distributions in the decision variables of the EUs. Furthermore, our work differs significantly from the framework of ε-differential privacy, because in differential privacy, noise is added by a principal to the output of a computation which is subsequently publicly released [START_REF] Dwork | Calibrating noise to sensitivity in private data analysis[END_REF], [START_REF] Dwork | The algorithmic foundations of differential privacy[END_REF], and the principal performing the computation is trusted (i.e., it does not act strategically). In contrast, we do not assume that the EU is trusted, which motivates input perturbations that can be related to the more recent notion of local differential privacy, though ignoring the strategic aspect of the input perturbation [START_REF] Kairouz | Extremal mechanisms for local differential privacy[END_REF]. Finally, some papers study the trade-off between the accuracy of equilibrium computation and privacy or fairness guarantees [START_REF] Wang | Differentially-private distributed algorithms for aggregative games with guaranteed convergence[END_REF], [START_REF] Fioretto | Differential privacy and fairness in decisions and learning tasks: A survey[END_REF].

This paper is organized as follows. In Sec. II, we formulate the peer-to-peer electricity trading problem as a noncooperative network game. Relying on performative prediction, we define a decision-dependent game to couple the network game with a data market, in Sec. III. Algorithms for equilibrium computation are tested in Sec. IV. We conclude in Sec. V.

Notation: E X [f ] specifies that the expectation of the function f (.) is taken with respect to the random variable X. Vectors and matrices will be represented by bold letters. The stack of the N vectors x 1 , ..., x N is denoted x ≜ col(x 1 , ..., x N ). x -n is the stack of the vectors containing the decision variables of all the agents in N except n. For a closed set C ⊆ R m , the mapping proj C : R m → C denotes the projection onto C, i.e., proj C (x) ≜ argmin y∈C ∥y-x∥. SOL(P ) will denote the set of solutions of Problem P .

II. A NETWORK GAME MODEL

A. Description of the Agents 1) The End Users: We let N be a set of N EUs, forming a directed graph Γ ≜ (N , E), where E ⊆ N ×N is the set of links between the EUs. Let Γ n be the set of out-neighbors of EU n. Furthermore, we make the assumption that at node 0, i.e., the root node, Γ 0 = N \{0}.

Each EU n updates the vector of decision variables

xn ≜ (Dn, G c n , (qmn)m∈Γ n ) ∈ Xn,
which contains its demand, controllable generation, and bilateral trades with its neighbors, where X n ⊆ R mn , m n = |Γ n |+2, is the strategy set of EU n that captures only local constraints. In addition, we impose the following convention on EU n's bilateral trades: q mn ≥ 0 means that n buys q mn from m; while q mn < 0 means that n sells -q mn to m. Coupling constraints are added to capture bilateral trading reciprocity between EU n and its neighbors such that EU n's strategy set takes the form:

Cn(x-n) ≜ xn = (Dn, G c n , (qmn)m∈Γ n ) ∈ Xn | qmn = -qnm, ∀m ∈ Γn , with Xn ≜ xn = (Dn, G c n , (qmn)m∈Γ n ) ∈ R mn | D n ≤ Dn ≤ Dn, G c n ≤ G c n ≤ G c n , Dn ≤ G c n +G nc n +Qn
where D n and D n are lower and upper-bounds on n's demand, G nc n is the uncontrollable generation of EU n (solar PV panels generation), and Q n ≜ m∈Γn q mn is the net import at node n. The constraint D n ≤ G c n +G nc n +Q n captures the fact that demand at node n should not exceed the sum of local generation and net import.

Typically, the EUs' utility functions J n (.) depend on some parameters that are sensitive, and that they might not want to disclose [START_REF] Shilov | Privacy impact on generalized nash equilibrium in peer-to-peer electricity market[END_REF], [START_REF] Le Cadre | Peer-to-peer electricity market analysis: From variational to generalized nash equilibrium[END_REF]. We define θ n ∈ Θ n ⊆ R as the private information of EU n. Let J n : R mn ×Θ n → R be the utility function of EU n. Following [START_REF] Le Cadre | Peer-to-peer electricity market analysis: From variational to generalized nash equilibrium[END_REF], we define EU n's utility (in the cost minimization sense) as the difference between the sum of the total cost incurred from the bilateral trades with its neighbours and the controllable generation cost, and its usage benefit U n (.) (-U n (.) is the consumption cost of EU n):

Jn(xn, θn) = m∈Γn cnmqmn -Un(Dn, θn)+C G n (G c n ), (1)
where c n > 0 is a known vector of parameters capturing EU n's preferences (willingness-to-pay or taxes), and -U n (.) is its consumption cost. The controllable generation cost is assumed to be quadratic in

G c n , i.e., C G n (G c n ) = an 2 G c 2 n + b n G c
n +c n where a n , b n , c n are positive input coefficients. We assume U n (.) to be a quadratic function in EU n's demand, i.e., we can write:

Un(Dn, θn) = -ãn Dn -θn 2 + bn, (2)
where ãn , bn are positive input coefficients, and θ n is EU n's nominal demand. EU n's usage benefit is a strictly concave function in its demand D n . It leads to the following interpretation: by being flexible around its nominal demand, e.g., agreeing to slightly reduce its demand around its target value, EU n will decrease its usage benefit but it might also decrease its bilateral trading or controllable generation costs.

2) The Market Operator: The economic dispatch is classically obtained through a centralized clearing, i.e., the market operator aims to determine the vector x that minimizes the social cost defined as the sum of the EUs' costs, i.e., SC(x, θ) ≜ n J n (x n , θ n ) under network coupling constraints, i.e., it solves the following optimization problem:

min x SC(x, θ), (3a)
s.t. x ∈ C ≜ n Cn(x-n). (3b)
We prove in [START_REF] Le Cadre | Peer-to-peer electricity market analysis: From variational to generalized nash equilibrium[END_REF] that the solution of (3) can be reached through a distributed clearing, where each EU n ∈ N solves the parametrized optimization problem:

min xn∈Cn(x -n ) Jn(xn, θn). (4)
The latter setting gives rise to a non-cooperative game G between the EUs.

3) The Data Aggregator: The goal of the DA is to collect readings from the EUs to build an estimate x a of the EUs' decisions x at equilibrium and use it to measure the quality of the readings. To this end, it builds contracts with the EUs, and pays p a n (.) to each EU n, which is an increasing function in the accuracy of x a n with respect to x n .

B. Existence, Uniqueness and Stability of the Variational Equilibrium

To generalize the notion of NE in the presence of coupling constraints, we use the Generalized Nash Equilibrium (GNE). It is defined as a strategy profile vector x such that x n solves the system KKT n for each n (see Appendix A). The coupling constraint q mn = -q nm written in the problem of n (respectively m, ∀m ∈ Γ n ) is associated to the duals ζ nm (respectively ζ mn ). In this work, we focus on the special case of a GNE known as a Variational Equilibrium (VE), i.e., a solution x = (x n ) n such that x n solves the system KKT n for each n and, in addition, such that the duals of the coupling constraints are equal, i.e.,

ζ nm = ζ mn , ∀n ∈ N , ∀m ∈ Γ n . Set F x (x) ≜ ∇ xn J n (x n , θ n ) n .
Definition 1 (Variational Equilibrium [START_REF] Scutari | Convex optimization, game theory, and variational inequality theory[END_REF]): The Variational Inequality problem VI(F x , C) consists in finding a vector

x * ∈ C such that (y-x * ) T F x(x * ) ≥ 0, ∀y ∈ C. A solution of VI(F x , C) is called a VE.
Proposition 1: If no energy surplus is available at node n, then EU n's best-response strategy x * n at equilibrium depends linearly on the nominal demands of all the EUs, i.e., Proof: See Appendix B. Equilibrium stability is a sufficient condition for distributed learning methods to converge in [START_REF] Mertikopoulos | Distributed stochastic optimization via matrix exponential learning[END_REF]. Definition 2 (Stability [START_REF] Mertikopoulos | Distributed stochastic optimization via matrix exponential learning[END_REF]): An equilibrium point x * ∈ C is said to be variationally stable (or simply stable) if there exists a neighborhood U of x * such that (y-x * ) T F x (y) ≥ 0 for all y ∈ U. In particular, if this property holds for all y ∈ C, we say that x * is globally stable.

x * n = An,nθn +An,-nθ-n + Bn,nG nc n +Bn,-n n ′ ̸ =n G nc n ′ +Cn, with A n,n , B n,n , B n,-
Proposition 2: The VE solution of G is globally stable. Proof: In Thm. 1 proof, we check that G belongs to the class of generalized potential games (GPG) [START_REF] Facchinei | Decomposition algorithms for generalized potential games[END_REF]Sec. 2] for which the EUs' utility functions are aligned along the game's potential whose minimizers are VEs. From [START_REF] Mertikopoulos | Distributed stochastic optimization via matrix exponential learning[END_REF]Rk. 1], VEs solutions of convex potential games are automatically stable in the sense of Definition 2.

The impact on (c nm ) n,m on G equilibria has been analyzed in [START_REF] Le Cadre | Peer-to-peer electricity market analysis: From variational to generalized nash equilibrium[END_REF]. In case (c nm ) n,m capture taxes, it seems natural to take them homogeneous among EUs belonging to same community.

Assumption 3: We assume that the EUs' preferences for trading are uniform and normalized, i.e., c nm = 1, ∀n, m.

We notice that under Asms. 1 and 3, controlling demand and generation allows to control the net import. Therefore Q will be treated as a dependent decision variable in the rest of the work. Furthermore, replacing the (q mn ) m by Q n allows to use the linear dependence of x n , ∀n in θ (see Appendix A).

III. COUPLING NETWORK GAME AND DATA MARKET AS A DECISION-DEPENDENT GAME

Decomposition methods are another way to compute equilibrium with minimum information exchange [START_REF] Marques | Grid impact aware tso-dso market models for flexibility procurement: Coordination, pricing efficiency, and information sharing[END_REF], [START_REF] Le Cadre | Parametrized inexact-admm based coordination games: A normalized nash equilibrium approach[END_REF]. Nevertheless, works in this category do not distinguish between local and global objectives. In this paper, we focus on the performative prediction setting with decisiondependent distributions which has been introduced in [START_REF] Narang | Multiplayer performative prediction: Learning in decision-dependent games[END_REF] as a game-theoretic model for decision-dependent learning in the presence of competition. It leads to a weaker solution concept than Nash Equilibrium, but allows to endogenously capture the drift (performative effect) caused by the strategic information contained in the EUs' readings on the market equilibrium, and leads to faster converging algorithms.

A. Balancing Cost and Privacy

We introduce randomization in the reading sent by EU n to the DA, which takes the form θ n = θ n +ε n . The first part of the reading captures the ability of EU n to add a deterministic bias to its reported information. The second part is a noise following a randomized mechanism M (.), added in order to preserve privacy. We assume that the noise in the readings is independent across the EUs [START_REF] Westenbroek | Competitive statistical estimation with strategic data sources[END_REF].

Assumption 4: For each EU n ∈ N , ε n follows a centered Gaussian distribution with variance V n . Furthermore, the random variables (ε n ) n∈N are independent.

To incentivize the EUs to report truthful readings that can later be monetized through targeted advertisements, the DA comes into play. The DA defines a payoff contract to each EU; the payment is decreasing in the distance between the EUs' strategy x n and the estimate x a n ≜ (D a n , G c a n ) built by the DA, defined as follows:

x a n = A a n θ+B a n G nc +C a n , (5)
where A a n , B a n , C a n are estimates of A n , B n , C n introduced in Prop. 1. The form of the DA's estimator is known by all the EUs. It comes from Prop. 1, assuming that Asm. 1 holds. The contract between the EU n and the DA takes the form:

p a n (xn, x a n , θ) = γn -βn (Dn -D a n ) 2 +(G c n -G c a n ) 2 , ( 6
)
where γ n , β n are non-negative parameters which can be optimized by the DA to check specific market properties [START_REF] Westenbroek | Competitive statistical estimation with strategic data sources[END_REF].

Each EU n chooses the neighboring input [START_REF] Chatzikokolakis | Broadening the scope of differential privacy using metrics[END_REF]. Let L M,θn, θn be the privacy loss of M (.), which follows the distribution N ( (θn-θn) 2 2Vn , 2 (θn-θn) 2 2Vn

θ n such that d( θ n , θ n ) ≜ ( θ n -θ n ) 2 ≤ α n
) [START_REF] Balle | Improving the gaussian mechanism for differential privacy: Analytical calibration and optimal denoising[END_REF]. How much information can be extracted from the reading θ n is measured by the expected privacy loss of the mechanism M (.):

I n (θ n , θ n ) ≜ -E εn L M,
θn, θn , which coincides with the Kullback-Leibler divergence (relative entropy) between M (.)'s output distributions on θ n and θ n . Under Asm. 4, the Kullback-Leibler divergence takes the closed form:

I n (θ n , θ n , V n ) = ( θn-θn) 2 2Vn .

B. Formulation of the Stochastic Game

In the data market, EU n's extended utility is split in three terms. The first term captures the cost it bears in G; the second term measures the cost of information release, while the third term is the contract payment it receives from the DA:

Jn(xn, θ, V, ε) = Qn + an(Dn -θn) 2 -bn + an 2 G c 2 n + +bnG c n +cn +c I n ( θn -θn) 2 2Vn -γn +βn((Dn -D a n ) 2 +(G c n -G c a n ) 2 )+ξ( N m=1 Qm) 2 , (7)
with c I n , ξ non-negative parameters. From (5), since D a n , G c a n depend on θ, ε, and I n (.) on θ, V, J n (.) also depends on θ, V, ε. Under Asms. 1 and 3, the bilateral trading reciprocity constraint at each node can be equivalently reformulated as N n=1 Q n = 0. To avoid coupling between the feasibility sets of the EUs, we put the constraint as a penalty in their utility function, with a weight ξ ≥ 0 that is the same for all the EUs. The penalty ξ calibration is discussed in Lemma 1, later. The first term in Q in [START_REF] Westenbroek | Competitive statistical estimation with strategic data sources[END_REF] refers to the benefit/cost due to the trades, while the second one encourages the EUs to ensure trading reciprocity among the community. We introduce V ≜ diag(V). Each EU n solves:

min xn, θn,Vn E ε∼N (0,V) Jn(xn, θ, V, ε) , (8a)
s.t. D n ≤ Dn ≤ Dn, (8b)
G c n ≤ G c n ≤ G c n , (8c) Dn = G c n +G nc n +Qn, (8d)
d( θn, θn) ≤ αn, (8e)

V n ≤ Vn ≤ V n. (8f)
This gives rise to the stochastic game G stoch . Each EU wants to ensure a minimum privacy level, so it imposes a lower bound V n on V n . To limit the instability of the market, the DA imposes an upper bound V n on V n . This upper bound does not need to be shared by the EUs, instead it might be encoded by the DA in the communication interface deployed in the EUs' nodes when communicating with the DA.

The game between the EUs, which involves the DA as a coordinator of the data market, takes place as follows: a) Each EU n reports a reading θ n = θ n +ε n , with ε n ∼ N (0, V n ), to the DA by solving [START_REF] Gast | Linear regression from strategic data sources[END_REF] with respect to θ n , V n . The readings θ are then broadcasted to all the EUs by the DA. b) Each EU n computes its x n by solving [START_REF] Gast | Linear regression from strategic data sources[END_REF]. Note that depending on the coordination level between the DA and EUs, steps a) and b) can take place simultaneously or sequentially.

We define Y n as the feasible set of Problem [START_REF] Gast | Linear regression from strategic data sources[END_REF]. We denote

y n ≜ (x n , θ n , V n ) ∈ Y n the decision variable of EU n in G stoch . C. Main Results Definition 3: A Nash Equilibrium (NE) is a strategy pro- file y * = (y * n ) n ∈ Y such that for all n ∈ N : y * n ∈ argmin y n ∈Yn E ε∼N (0,diag(Vn,V * -n )) J n (y n , y * -n , ε
) . There is another solution concept which is more amenable to numerical methods [START_REF] Narang | Multiplayer performative prediction: Learning in decision-dependent games[END_REF], and reduces to Performatively Stable points (PSEs). The idea of the PSE is to decouple the effect of a decision y on the integrand J n (y n , y -n , ε) and on the distribution N (0, V). Any vector V s induces a static game wherein the distribution for EU n is fixed at N (0, V s ), that is: Definition 4: [START_REF] Narang | Multiplayer performative prediction: Learning in decision-dependent games[END_REF] A Performatively Stable Equilibrium (PSE) is a strategy profile y s = (y s n ) n ∈ Y such that for all n ∈ N : y s n ∈ argmin

y n ∈Yn E ε∼N (0,V s ) J n (y n , y s -n , ε) . ( 9
)
Proposition 3: The utility function E ε J n (y n , y -n , ε) is strongly convex in y n , ∀n.

Proof: We explicit the expectation of EU n's utility function:

E ε∼N (0,V) Jn(xn, θ, V, ε) = (Dn -G c n -G nc n )+ an(Dn -θn) 2 -bn + an 2 G c 2 n +bnG c n +cn +c I n ( θn-θn) 2 2Vn -γn + β (Dn -D a n ( θ)) 2 +(G c n -G a n ( θ)) 2 + N m=1 (A a n ) 2 (1,m) Vm + N m=1 (A a n ) 2 (2,m) Vm +ξ( N m=1 Dm -G c m -G nc m ) 2 .
Using the former expression, we compute the partial derivative F n ≜ F y n and the partial Hessian H n ≜ H y n of E ε J n (.) with respect to y n . We get:

Hn =      c I n Vn +2βn((A a n ) 2 1,n +(A a n ) 2 1,n ) -c I n ( θn-θ) V 2 n -2βn(A a n )1,n -2βn(A a n )2,n -c I n ( θn-θ) 2 V 2 n c I n ( θn-θ) V 3 n 0 0 -2βn(A a n )1,n 0 2 an +2βn +2ξ -2ξ -2βn(A a n )2,n 0 -2ξ an +2βn +2ξ      .
We note that z T H n z > 0, ∀z ∈ Y n , i.e., E ε J n (.) is strictly convex in y n , which will be used later in the proof. In a second time, we prove the strong monotonicity of F n , i.e., that

⟨F n(y n )-F n(y ′ n ), y n -y ′ n ⟩ ≥ α∥y n -y ′ n ∥ 2 , ∀y n , y ′ n ∈
Yn. Using the multidimensional Mean Value Theorem on the convex set Y n , we obtain the existence of a z ∈ [y n , y ′ n ], i.e., component-wise

z i ∈ [(y n ) i , (y ′ n ) i ] ∀i, such that ⟨F n(y n )-F n(y ′ n ), y n -y ′ n ⟩ = (y n -y ′ n ) T •∇nF n(z)•(y n - y ′ n ).
We then use the Spectral Theorem on ∇ n F n = H n to obtain the existence of D, a diagonal matrix such that H n (z) = P DP -1 , and P the orthogonal transition matrix. By replacing H n by its expression, identifying the variable in the new base, and using the property that the product of a vector by an orthogonal matrix preserves the Euclidean norm of the vector, we obtain:

(y n -y ′ n ) T •P DP -1 •(y n -y ′ n ) ≥ mini{λi(Hn(z))}∥(y n -y ′ n )∥ 2 where the (λ i (H n (z))) i∈ 1,4
are the eigenvalues of H n (z). We obtain the α-strong monotonicity by taking the minimum over all the feasible set of the eigenvalues of H n (which are positive due to the strict convexity), with α = minz∈Y n min i∈ 1,4 λi(Hn(z)). Since all the conditions listed in [START_REF] Scutari | Convex optimization, game theory, and variational inequality theory[END_REF] are met, we conclude that E ε J n (.) is strongly convex.

Let ∇ n h(y) be the partial derivative of h(.) with respect to y n .

Lemma 1: If the penalty coefficient ξ in J n (.) is bigger than the Lagrangian multiplier of the constraint ( N m=1 Q m ) 2 = 0, then the penalized game has the same solutions as G stoch .

Proof:

We adapt the proof of [21, Thm. 2.2]. We call J ♯ n (.) the function J n (.) without the penalty:

J ♯ n (xn, θ, V, ε) = Jn(xn, θn)+c I n In(θn, θn, Vn)- p a n (xn, x a n , θ). We set h(y) ≜ ( N m=1 Q m ) 2
and ϕ is the Lagrangian multiplier related to the constraint h(y) = 0. We denote y the point that minimizes J ♯ n (.) over Y and satisfies the constraint h( y) = 0. For any y ∈ Y, we have:

J ♯ n (y)+ξh(y) ≥ J ♯ n (y)+ϕh(y) (10a)
≥ J ♯ n (y)+ϕ(h( y)+∇nh( y) T (y-y)), (10b) = J ♯ n (y)+ϕ∇nh( y) T (y-y),

= J ♯ n (y)-∇nJ ♯ n ( y) T (y-y), (10c)
≥ J ♯ n ( y) = J ♯ n ( y)+ξh( y). (10d)
To go from (10a) to (10b), we use the convexity of h(.). To go from (10c) to (10d), we use the stationarity condition. To go from (10d) to (10e), we use the convexity of J ♯ n (.), which can be proved by invoking similar arguments as in the proof of the strict convexity of E ε [ Jn (.)] in Proposition 3. Hence, y also minimizes the penalized problem.

Since for all n, E ε J n (.) is strongly convex and the feasibility set Y n is closed and convex, we derive the following result.

Corollary 1: There exists a unique NE solution of G stoch . Theorem 2: If the DA's estimate x a is perfect for the game G, i.e., x a (θ, G nc ) = x G with x G ∈ SOL(G), then the NE solution of G stoch is efficient.

Proof: We denote x G the VE solution of the game G. We have, ∀ θ, V ,

x n : Jn(x G ) ≤ Jn(xn)+c I n ( θn-θn) 2 2Vn + βn((Dn -D a n ( θ)) 2 +(Gn -G a n ( θ) 2 )+ξ( N m=1 Qm) 2 =⇒ Πn(x G )+ N m=1 KmV m ≤ Jn(xn)+c I n ( θn-θn) 2 2Vn +βn((Dn - D a n ( θ)) 2 +(G c n -G a n ( θ)) 2 )+ξ( N m=1 Qm) 2 + N m=1 KmVm
where we set

K m ≜ ((A a n ) 2 (1,m) +(A a n ) 2 (2,m)
). Note that the term -γ n is constant and does not change the equilibrium, thus we omit it. This lower bound is reached in θ = θ,

V n = V n ∀n and x n = x G n .
This means that we have proved that assuming x a is perfect for G, x G , θ, V is the NE solution of G stoch . If we sum the former inequalities over n, we observe that θ = θ, V n = V n ∀n and x = x G also minimizes the EUs' expected social cost. Thus, assuming x a is perfect for G, the NE solution of G stoch is efficient.

Next, we look at the properties of the PSE. Proposition 4: There exists a unique PSE solution of G stoch .

Proof: To prove this result, we rewrite E ε J n (.) as follows, to use [4, Thm. 1]:

E ε∼N (0,V) Jn(xn, θ, V , ε) = E εn∼N (0,Vn) Jn(xn, θ, V , εn) , (11)
where Jn(xn, θ, V , εn)

= (Dn -G c n -G nc n )+ an(Dn - θn) 2 -bn + an 2 G c 2 n +bnG c n +cn +c I n ( θn-θn) 2 2Vn
-γn +β (Dn -

D a n ( θ)) 2 +(G c n -G a n ( θ)) 2 + N m=1,m̸ =n (A a n ) 2 (1,m) Vm + N m=1,m̸ =n (A a n ) 2 (2,m) Vm +(A a n ) 2 (1,n) ε 2 n +(A a n ) 2 (2,n) ε 2 n + ξ( N m=1 Dm -G c m -G nc m ) 2 .
To prove the strong convexity of J n (.), we proceed the same way as to prove the strong convexity of J n (.), because the partial Hessians of both objective functions are the same. We also check the C 1 continuity of J n (.), the Lipschitz-continuity of the application ε → J n (y, ε), and that the distribution of the ε varies in a Lipschitz way according to [START_REF] Narang | Multiplayer performative prediction: Learning in decision-dependent games[END_REF]Asm. 2]. From [4, Thm. 1], we obtain the existence and uniqueness of the PSE.

Assuming G stoch takes place sequentially, θ n , V n and x n are computed by solving two distinct parametrized optimization problems, by each EU n ∈ N . This leads us to introduce the game G ♯ ( θ, V ), where each EU n ∈ N solves:

min xn E ε∼N (0,V) J ♯ n (xn, θ, V , ε) , (12a)
s.t. D n ≤ Dn ≤ Dn, (12b)
G c n ≤ G c n ≤ G c n , (12c) Dn = G c n +G nc n +Qn, (12d)
Qm = 0. (12e)
Proposition 5: G ♯ ( θ, V ) is a generalized potential game (GPG), ∀ θ, V .

Proof: Relying on the GPG definition introduced in [16, Sec. 2], this results follows from the fact that the KKTs of G ♯ ( θ, V ) coincide with the KKTs of the minimization problem with n E ε J ♯ n (x n , θ, V , ε) in the objective, under the concatenation of the constraints (12b)-(12d), ∀n, and (12e).

IV. SIMULATIONS Numerical results are presented on a three-node network, with θ = [0.23, 0.32, 0.55] and G nc = [0.1, 0.2, 0.4]. G stoch parameters are available on GitHub [START_REF] Taisant | Balancing efficiency and privacy[END_REF].

To compute the NE, we formulate (8) as a deterministic optimization problem. This is possible because the (ε n ) n are independent and ε n ∼ N (0, V n ), ∀n. To compute the NE and the social optimum (SO), we apply a classic Gradient Descent Method (GDM). To compute the PSE, we use the Repeated Stochastic Gradient Method (RSGM) [START_REF] Narang | Multiplayer performative prediction: Learning in decision-dependent games[END_REF]. In RSGM, having initialized y 0 , at each step t, each EU samples its ε n , and updates its variable y t n using the formula:

y t+1 n = proj Yn (y t n -η∇nJn(y t , ε t n )), (13)
where J n (.) is defined in Prop. 4 proof. We used a backtracking line search for the choice of the step size η. However, RSGM requires that the EUs share their decision variables with the other EUs at each iteration. Also, depending on the structure of the game, it might not be possible to reformulate the objective function E ε∼D(y) J n (x n , θ, V, ε) , in the form E εn∼Dn(y) J n (x n , θ, V, ε n ) , ∀n -for example, in case the (ε n ) n are correlated. For this reason, we propose an alternative algorithm, Coupled Privacy Repeated Stochastic Gradient Method (CP-RSGM), in which at each time step t, each EU n samples its ε t n and shares its θ t n with the DA, which then broadcasts it to the other EUs. In fact, in our game J n (y t , ε t ) ≡ J n (x t n , θ t n , V t n , θ t

). Each EU updates its decision variables following the rule:

y t+1 n = proj Yn (y t n -η∇n Jn(y t , ε t )). (14)
In CP-RSGM, the update of readings parameters, demand and generation variables is done simultaneously. In case where the readings parameters on the one side, demand and generation variables on the other side need to be updated sequentially, we introduce another algorithm, Privacy Split Repeated Stochastic Gradient Method (PS-RSGM). Algorithm Privacy Split Repeated Stochastic Gradient Method (PS-RSGM) n ) = 0. J ♯ n (.) is define above [START_REF] Dwork | The algorithmic foundations of differential privacy[END_REF]. In Fig. 1, we plot the final value of θ-θ reached by GDM, RSGM, CP and PS-RSGM, considering 50 different initializations of y 0 . We proved that the NE and the PSE solutions of G stoch are unique. However, CP-RSGM and PS-RSGM only enable to approximate the PSE. This is due to the sampling on ε and to the fact that we impose an upper bound on the number of iterations. We depict the resulting neighborhoods as the convex hulls of the set of approximate PSEs that can be reached by CP and PS-RSGM. We observe that the NE is very close to the SO. This gap can be explained from Thm. 2 as resulting from the estimation bias of the DA estimator. We compare the speed of convergence of RSGM, CP and PS-RSGM in θ, in Fig. 2. To build this plot, we ran the algorithm for 50 random starting points, and plot a 95% confidence interval for each step of the algorithm. Due to the sampling on ε at each step, the algorithms only converge in expectation. We observe that PS-RSGM converges faster than the two other algorithms, and that CP-RSGM and PS-RSGM might provide upper and lower bounds on the approximate PSE computed by RSGM.

1: Find x G ∈ SOL(G). 2: Find θ ♯ n , V ♯ n = argmin θn,Vn E ε J n (x G n , θ, V ,
Let y so be the SO solution of the optimization problem with objective the sum over n of the expected utilities (12a), denoted SC(.), under the concatenation of the constraints (8b)-(8f) for all the EUs. In an equilibrium y, we evaluate the efficiency gap as the difference between the social cost in the equilibrium and the optimal social cost: SC(y)-SC(y so ). We plot the distribution of the efficiency gap for the GDM, RSGM, CP and PS-RSM in Fig. 3. The efficiency gap of the NE is close to 0, i.e., the NE is a desirable equilibrium to reach. The PS-RSGM and the CP-RSGM have a 5% lower efficiency gap than the RSGM. The total information released by the EUs on the market is defined as n I n (θ n , θ n , V n ). The smaller it is, the less information is released. The total information released per algorithm is illustrated in Fig. 4. We summarize the performance of each algorithm in Tab. I. GDM leads to the smallest efficiency gap, information release and bias. However, it requires full coordination between the DA and EUs, which is very hard to impose in practice. CP-RSGM generates a smaller efficiency gap than RSGM, that we interpret as benchmark, for almost the same information release but with smaller bias, i.e., higher indistinguishability for the EUs, and requires less coordination among the EUs. Allowing sequentiality in the DA and EUs' decisions through PS-RSGM leads to a smaller variance (more information release) and larger bias (less indistinguishability) than RSGM and CP-RSGM.

V. CONCLUSION

Relying on performative prediction, the coupling between network game and data market is formulated as a decisiondependent game. Nash and Performatively stable equilibria are characterized, and two variants (CP, PS-RSGM) of the Repeated Stochastic Gradient Method (RSGM) are proposed to compute equilibria. In our numerical experiments, CP-RSGM outperforms RSGM, leading to smaller efficiency gap and better privacy preservation for the end users. PS-RSGM achieves faster convergence rates than CP-RSGM, but its sequential implementation requires more information release and less indistinguisability for the end users.

In an extension of this paper, we will propose another variant of RSGM, relying on two-time-scale stochastic approximation [START_REF] Doan | Nonlinear two-time-scale stochastic approximation: Convergence and finite-time performance[END_REF]. The goal will be at each iteration t to apply a gradient-play update to compute θ t , V t , and then, a second time gradient-play to compute x t taking as input θ t , V t . Convergence proofs and convergence in terms of time of all the algorithms will be detailed. In addition, the algorithms will be tested on larger networks. Finally, we will provide a method for specifying a Pareto dominating solution that depends on the variance or hiding ability of the EUs.

  ε) under constraints (8e), (8f) ∀n. Set x = x G and compute a solution with CP-RSGM. 3: Find x ♯ n = argmin xn E ε∼N (0,V ♯ ) J ♯ n (x n , θ ♯ , V ♯ , ε) ∀n under constraints (8b), (8c) and N n=1 (D n -G c n -G nc

Fig. 1 .

 1 Fig. 1. Final value θ-θ reached by GDM, RSGM, CP, PS-RSGM for 50 different starting points.

Fig. 2 .

 2 Fig. 2. 95% confidence interval for θ while learning the PSE.

Fig. 3 .

 3 Fig. 3. Box plots of the efficiency gaps for each algorithm.

Fig. 4 .

 4 Fig. 4. Total information released by the EUs for each algorithm.

  n , C n vectors and A n,-n matrix of appropriate dimensions; otherwise x * n depends only on the EU n's own nominal demand, θ n . Proof: See Appendix A. Remark 2.1: The coefficients of the best-responses are not known by the EUs because they rely on coefficients (ã n , a n ) n that the other EUs might not want to share. Assumption 1: Demand and supply balance each other at each node, i.e., D n = G c n +G nc n +Q n , ∀n ∈ N . Assumption 2: We assume that there exists at least one EU n ∈ N such that at the VE, D *

n ̸ = 0 or G c * n ̸ = 0. Theorem 1: Under Asm. 2, the game G admits a unique VE, which is in addition efficient.

TABLE I COMPARISON

 I OF THE ALGORITHMS FOR EQUILIBRIUM LEARNING.

	Algo.	Coordination Eff. Gap Info. Release	n d( θn, θn)	Eq.
	GDM	simultaneous	0.26	0.025	0.077	NE
	RSGM	simultaneous	66.06	0.034	0.452	PSE
	CP-RSGM	simultaneous	61.61	0.034	0.378	PSE
	PS-RSGM	sequential	62.99	0.046	0.490	PSE

APPENDIX

A. Proof of Proposition 1

For any EU n ∈ N , Problem (4) can be formulated as follows:

min Dn,G c n ,(qmn)m

Jn(xn, θn), (15a)

To compute an equilibrium solution of G, we derive the KKT conditions of the Lagrangian function L n (.) with respect to D n , G c n , q mn , ∀m ∈ Γ n , m ̸ = n, the first order stationarity conditions write down as follows:

From (16c), we obtain:

Then, from [START_REF] Marques | Grid impact aware tso-dso market models for flexibility procurement: Coordination, pricing efficiency, and information sharing[END_REF], we infer that at a VE:

From (16a), we infer that at equilibrium:

From (16b), we infer that at equilibrium:

Complementarity constraints are:

, which leads us to distinguish between two cases depending on whether constraint (15e) is binding.

Case 1 [No surplus at node n] In case λ > 0 then

i.e., there is no surplus at node n. By substitution of ( 19), [START_REF] Balle | Improving the gaussian mechanism for differential privacy: Analytical calibration and optimal denoising[END_REF] in the net import at node n, Q n , and using the fact that n Q n = 0, we obtain:

But, from (18), we have the following relationship:

By substitution in [START_REF] Doan | Nonlinear two-time-scale stochastic approximation: Convergence and finite-time performance[END_REF], we obtain a closed form expression for λ 0 :

.

To simplify the notations, let

. Substituting λ 0 closed form (25) in ( 24), and then later in [START_REF] Chatzikokolakis | Broadening the scope of differential privacy using metrics[END_REF], [START_REF] Balle | Improving the gaussian mechanism for differential privacy: Analytical calibration and optimal denoising[END_REF], [START_REF] Taisant | Balancing efficiency and privacy[END_REF], we get:

an . We observe that D * n , G c * n , Q * n at equilibrium depend linearly on the target demands of all the EUs.

Case 2 [Possible surplus at node n] In case λ n = 0, then by substitution in [START_REF] Chatzikokolakis | Broadening the scope of differential privacy using metrics[END_REF], [START_REF] Balle | Improving the gaussian mechanism for differential privacy: Analytical calibration and optimal denoising[END_REF], [START_REF] Taisant | Balancing efficiency and privacy[END_REF] we obtain that

In both cases, Prop. 1 statement follows from the closed form expressions of D * n , G c * n , Q * n , ∀n ∈ N at equilibrium.

B. Proof of Theorem 1

In the non-cooperative game G, the strategy set of each EU is convex and compact; thus, the existence of a NE follows from [START_REF] Rosen | Existence and uniqueness of equilibrium points for concave n-person games[END_REF]Thm. 3]. Now let us consider that there exists two VEs x and x solutions of the game G. As both x and x points are equilibrium they must satisfy the KKT conditions, multiplying first order condition with [xn -xn] at point xn and xn for each n, we get:

Summing over all the n gives us:

where Λ and Λ represent matrices with Lagrangian multipliers from KKT at point x and x respectively. We observe that:

which contradicts the assumption that there exist two VEs.

In the above equation to prove that first term is positive, we show that game G is strongly monotone, i.e.:

under Asm. 2. Consider the pseudo Hessian matrix H of utilities of the EUs where each n th row and m th column block matrix is given as Hn,m = ∂F xn ∂xm , i.e., in details:

However, we observe that the matrix H is a block diagonal matrix with each diagonal matrix positive semi-definite, which implies that matrix H is positive semi-definite and in our case

∂Qn is constant for all n ∈ N , thus inequality (28) follows from [START_REF] Rosen | Existence and uniqueness of equilibrium points for concave n-person games[END_REF]Thm. 6] and the game G is strongly monotone. The efficiency of the VE follows from the generalized potential game structure of G, which results from the coincidence of the KKTs of (3) and G.


L Ghizzo

D Tremouilles´b

F Richardeau

S Vinnac

L Moreau

N Mauran

Preconditioning of p-GaN power HEMT for reproducible Vth measurements

Keywords: Schottky p-GaN transistors, preconditioning protocol, reliability, threshold voltage shift, trapping

In reliability studies, the instability of the threshold voltage (Vth) is problematic when Vth is used as an indicator as it totally blurs an eventual drift due to real device aging. This instability is observed during electric characterization measurements and is related to the "biasing history" of the transistor which can introduce carrier trapping/detrapping in different layers of the structure. New methods are therefore required to overcome this trapping related instability issues in order to accurately monitor device aging. To solve the repeatability issue of threshold voltage measurement, we investigated its instability on GaN transistors. A preconditioning step applied right before the actual Vth measurement was studied. The proposed preconditioning method is based on the application of a dedicated VGS (t) biasing on the gate terminal which leads to a stable and repeatable value of Vth. The mechanisms enabling the observed stability of Vth are identified through the analysis of the drain leakage measurement after the preconditioned Vth measurement. It demonstrates the role of hole injection into the structure. The preconditioned-Vth measurement method is proposed as a complementary measurement to correctly follow the ageing of pGaN HEMT in future reliability studies.

Introduction

Monitoring of electrical characteristics is often used to follow transistor degradation in reliability studies. However, in AlGaN/GaN High Electron Mobility Transistors (HEMTs), measurements are often not reproducible and could vary depending on the previous-operation history which blurs the shift related to the degradation of the transistor. In particular, the Vth instability has been observed [START_REF] Li | Observation of Dynamic VTH of p-GaN Gate HEMTs by Fast Sweeping Characterization[END_REF].

Nevertheless, reproducible characterization measurements are mandatory to study irreversible degradation and drift of electrical parameters of transistors.

We studied a preconditioning step before the Vth measurement to stabilize it and make it perfectly reproducible. Some preconditioning methods have been proposed for SiC devices [START_REF] Rescher | Preconditioned BTI on 4H-SiC: Proposal for a Nearly Delay Time-Independent Measurement Technique[END_REF][START_REF] Association | Guidelines for measuring the threshold voltage (VT) of SiC MOSFETs[END_REF][START_REF] Deb | On the Repeatability and Reliability of Threshold Voltage Measurements during Gate Bias Stresses in Wide Bandgap Power Devices[END_REF]. Preconditioning for GaN devices is very new and seems to be only reported in the recent work of A. Deb [START_REF] Deb | On the Repeatability and Reliability of Threshold Voltage Measurements during Gate Bias Stresses in Wide Bandgap Power Devices[END_REF]. Compared to the work proposed here, it adopts a more experimental view with limited physical explanations and presents preconditioning steps with different gate-voltage-pulse amplitudes and time durations than the one described in this paper.

To identify mechanisms that occurred during the proposed preconditioning step, additional measurements were done post preconditioned Vth measurements.

Section 2 illustrates the instability of the threshold voltage and presents the experimental setup. The gate preconditioning protocol is explained in details in section 3. Section 4 is about the observations on the drain leakage after the preconditioning and the physical mechanisms that ensure the stability of the preconditioned Vth value.

Experimental details

The main device under test (DUT) is a commercial, 100V, 90A, 7mΩ packaged p-GaN gate AlGaN/GaN HEMT. The same protocol was also applied on a 650V, 30A, 50mΩ device from the same manufacturer.

In our setup, the threshold voltage (Vth) is measured by ramping up gate and drain voltages simultaneously (The measurement thus behaves as if the drain and gate terminals were shorted together.). Vth value has been chosen as the gate voltage value when the drain current ID equals 400mA. This value is high enough to be on the linear part of the curves of Fig. 2 (a) and to measure the drift precisely. As the duration of the Vth measurement is very short (well below the second), the measured value is not significantly influenced by the selfheating effect.

Like in [START_REF] Li | Observation of Dynamic VTH of p-GaN Gate HEMTs by Fast Sweeping Characterization[END_REF] and [START_REF] Murukesan | On the Challenges of Reliable Threshold Voltage Measurement in Ohmic and Schottky Gate pGaN HEMTs[END_REF] we observed that the threshold voltage is unstable depending on the history of the device. For instance, Vth will be shifted positively, up to 40% after drain leakage measurement Idss = f(VDS) at VGS = 0V because of the high voltage and the high electric field Fig. 2 (b).

This positive shift (increase of Vth) is ascribed to two distinct mechanisms according to [START_REF] Chen | OFF-State Drain-Voltage-Stress-Induced VTH Instability in Schottky-Type p-GaN Gate HEMTs[END_REF]. One of them is an electron trapping mechanism at the AlGaN/GaN interface caused by the 100V applied on the drain at the OFF-state during the Idss measurement. The trapped electrons do not have the time to be released before the threshold voltage measurement that follows the Idss measurement (10 to 20 min are needed to retrieve the initial value of Vth). The other one appears when the leakage current of the Schottky diode DSch (represented on Fig. 1) is high enough to polarize and switch on the pin diode Dpin. A hole current, limited by DSch, can go through the gate stack and the barrier to the buffer. The displacement of the positive charges (holes) that were partially in the P-GaN makes the charges within the P-GaN layer dominated by electrons. The hole current leaves a net negative charge within the P-GaN layer under the gate. The two mechanisms increase the electrical potential necessary to activate the two-dimensional electron gas (2DEG) in the channel i.e. increase Vth. Even if the shift observed is fully recoverable, it makes the monitoring of the transistor degradation based on the monitoring of its electricalcharacteristics drift very challenging.

Preconditioning of the gate of the DUT

The preconditioning is applied right before the threshold voltage measurement using the direct control mode of the used power device analyzer and curve tracer (Keysight B1505A). Therefore, the time between the preconditioning step and the threshold voltage measurement is controlled precisely.

The preconditioning consists on applying a positive voltage to the gate during a few seconds. This time affects the stability of the measurement and could be minimized with further future studies but should be taken long enough to saturate the phenomena and to obtain a repeatable result on Vth value. Indeed, this duration must be sufficient compared to the time constant of the involved physical mechanism and in order to reach the saturation of the trapped charge density detailed later. We have not yet found other studies about this saturation.

Immediately after this preconditioning step the Vth measurement is carried out by increasing VGS a n d V DS simultaneously (Fig. 3).

In order to test the Vth measurement reproducibility and its independence from the biasing history of the device, a preconditioned-Vth measurement following an Idss measurement was done in the same way the non-preconditioned-Vth was measured in Fig. 2. It is striking that the preconditioned-Vth measurement is essentially reproducible and not affected by the history (Fig. 4). This was observed on all of the sixteen devices tested on this part number. To evaluate the robustness of the proposed preconditioning, this process was repeated successfully with other biasing conditions, besides Idss measurement (biasing on the gate and the drain without exceeding the manufacturer's limits given in the datasheet). The preconditioned-Vth measurement is recoverable and repeatable (Fig. 4 (a)). The two values obtained for the threshold voltage differ by less than 0.1% (≈ 1mV). Vth is then only affected by the preconditioning step, it allows to do the measurement of a steady value of Vth. This value varies depending on the preconditioning voltage VGSPC. It increases for VGSPC between 1 and 4V and decreases for VGSPC between 4 and 7V (Fig. 6). The reasons this evolution are explained in the last paragraph of section 3. Such influence of the gate forward bias on Vth shift has also been observed in [START_REF] Stockman | On the origin of the leakage current in p-gate AlGaN/GaN HEMTs[END_REF][START_REF] Stockman | Schottky Gate Induced Threshold Voltage Instabilities in p-GaN Gate AlGaN/GaN HEMTs[END_REF][START_REF] Tajalli | Impact of Sidewall Etching on the Dynamic Performance of GaN-on-Si E-Mode Transistors[END_REF] on transistors with similar structures.

It has to be underlined that only the highest values of VGS,PC (>5V) lead to a stable, repeatable and bias-history independent preconditioned-Vth as shown in Fig. 5. Therefore, it could be advised to select the highest VGS value allowed. For the tested device the value VGSPC = 7V allows obtaining a reproducible measurement of the threshold voltage. This value should not degrades the device nor affects its performances since it fits in the limits recommended by the manufacturer (VGS in the range [-10V;+7V] in DC mode) and no shift of the preconditioned value of the threshold voltage Vth,PC has been observed.

Drain leakage current after preconditioning.

Observations

The leakage current curve in between the two measurements of Vth on Fig. 4 (a) is visible on Fig. 7. In comparison with "normal" Idss measurement (Fig. 2 (b)), two changes can be observed due to the preconditioning: the average current ID is three orders of magnitude larger than before (comparison of drain current reached at 100V for VGSPC = 7V compared to VGSPC ≤ 3V), and a spike of 7 µA occurs in the first 20 sec that adds up to the existing drain leakage current. The excess current disappears after the Idss measurement is done, i.e. another following Idss measurement gives a "normal" curve (Fig. 2 (b)).

The drain currents for different values of VGSPC are represented on the Fig. 9 (a). Here the sweeping time is shorter than in the previous measurement (33V/s against 3V/s before) as shown on the Fig. 9 (b). Therefore, the "spike" of current is not entirely evacuated during the forward sweep (upper line of each color on the graph). The increase in the average current is visible starting at VGSPC = 5V, like the current spike, which corresponds to the hole injection threshold during the preconditioning step.

The bell shape of the spike is influenced by the structure of the HEMT part number especially its field plates layout. The field plates of the structure influence the evolution of the leakage current-path resistance R during the Idss measurement. At OFF state the spreading of the 2DEG in the region between gate and drain (D2DEGx) depends on the voltage VDS. As VDS increases the 2DEG will be depleted progressively between the gate and the drain which extends the blocking surface Sblocking, therefore the path resistance R increases with VDS. The increase of Sblocking also reduces the capacitance value CDS. The CDS evolution regarding VDS is often given by the manufacturer and visible on Fig. 10. The depletion of the 2DEG at OFF state can then be deducted. Indeed, when the 2DEG surface varies, the size of the electrode of the equivalent capacitance varies which affects the capacitance.

At OFF-state, the blocking region is under the p-GaN only for VDS = 0V. On Fig. 11 it is the uncolored region of the 2DEG with the associated path resistance of R0V.

As VDS increases between 0V and 18V the 2DEG is depleted progressively under the field plate: D2DEGx increases in the pink region on Fig. 11. The path resistance seen by Idss increases up to R0V + R0V<Vdss<18V at VDS = 18V. At this point, the maximum of the spike current is reached due to the strong increase of the leakage path resistance related to the polarized field plate that acts like a negatively polarized gate. As VDS continues to increase, the relative "negative polarization" of the field plate reaches a threshold value that depends on the thickness of the AlGaN barrier and the thickness of the isolating layer around the gate stack ("e" in Fig. 11), which continues to depletes the 2DEG. The following decreasing exponential form on Fig. 7 between 18 and 40V shows that there are still charges to evacuate. Above 18V the 2DEG continues to be depleted in the yellow region (slower than with a field plate) which increases the path resistance with the increase of RVdss>18V. The yellow part of the 2DEG on the Fig. 11 is fully depleted when COSS is constant (above 100V).

The following simplified formula as well as Fig. 11 describe this resistance evolution (∆R) as a function of Sblocking knowing 7 (33V/s against 3V/s), the additional current leakage spike does not have enough time to be evacuated only on the forward sweep phase. The additional current of the spike is evacuated more rapidly during the measurement with faster sweeping (≈ 12s against 20s in Fig. 7), since the charge to be evacuated is the same for the same VGSPC, the maximum current reached is higher at faster sweeping rates.

(a)

the evolution of the simplified equivalent capacitance CDS of the transistor given in Fig. 10 and illustrated in Fig. 11: (1)

Where e is its thickness (AlGaN barrier and isolating layer illustrated on Fig. 11), ϵ the equivalent permittivity of the equivalent capacitor, S the surface where the 2DEG can be formed and Sblocking is the blocking surface of the drain current at a given VDS. S -Sblocking can be modeled as follow:

(2

)
Where Dtrz is the length of the transistor on z-axis and Dtrx is the space between the drain and the source electrode at AlGaN/GaN interface. DtrzDtrxis therefore the surface where the 2DEG can be formed. D2DEGx is the spreading of the depletion of the 2DEG on x-axis that depends on VDS. For other part numbers of pGaN HEMT the spreading of the 2DEG will be affected in the same way by VDS . As 600V pGaN HEMT have a different field plate structure with additional field plates, the 2DEG will be depleted with more sections. Each plate border adds a curvature on the curve CDS = f(VDS ) and adds a virtual gate threshold that depletes the 2DEG beneath at a given VDS at OFF-state.

Physical explanation

The impact of VGSPC on Idss shows that hole injection in a sufficient amount allows to reach saturation and is responsible for both the success of the preconditioning and the consequences on the leakage current post preconditioned-Vth measurement. During the preconditioning step at VGSPC = 7V, the holes reach the buffer under the gate region, see Fig. 12 (a). This hole injection lowers Vth and affects the drain-source blocking capability at OFF-state of the transistor under voltage which results in a larger drain leakage current during the following Idss measurement (VGS = 0V and VDS = 100V) [START_REF] Wang | Investigation of Dynamic IOFF Under Switching Operation in Schottky-Type p-GaN Gate HEMTs[END_REF][START_REF] Li | Charge trapping related channel modulation instability in P-GaN gate HEMTs[END_REF]. Therefore, this phenomenon, that reduces the threshold voltage Vth, is also responsible for the increase of the average leakage current (the order of magnitude of Idss is µA compared to nA without this phenomenon).

The additional current that appears at the beginning of the measurement (Fig. 7) might be explained by the recombination of electron-hole pairs, this recombination would emit a photon that will ionize electrons deeply trapped in the buffer as illustrated in Fig. 12 (b). These electrons are in the GaN buffer independently from the preconditioning and are a consequence of the quality of the buffer layer [START_REF] Vigneshwara Raja | Deep-Level Traps in AlGaN/GaN-and AlInN/GaNBased HEMTs With Different Buffer Doping Technologies[END_REF]. Those will be released in the channel creating the observed current spike. This phenomenon is related to persistent photo-conductivity [START_REF] Tang | Mechanism of Threshold Voltage Shift in p -GaN Gate AlGaN/GaN Transistors[END_REF]. The same increase in the leakage current amplitude with similar duration until equilibrium (approximately 30 sec) was also observed in [START_REF] Tang | Effect of Hole-Injection on Leakage Degradation in a p-GaN Gate AlGaN/GaN Power Transistor[END_REF][START_REF] Li | Degradation of transient OFF-state leakage current in AlGaN/GaN HEMTs induced by ON-state gate overdrive[END_REF] for a transistor from the same manufacturer. The measurement of this spike is an indicator of the quality of the buffer layer of a transistor and could be studied in other future work.

Conclusion

By adding a preconditioning step right before the threshold voltage measurement, it is possible to obtain reproducible Vth measurements.

During this preconditioning step we demonstrate that the gate needs to be forward biased above Vth close to the highest VGS value admissible and long enough to saturate the negative shift of the threshold voltage related to hole injection into the buffer (Fig. 8).

The phenomena observed on the leakage current measurement (Idss) after the preconditioned threshold voltage measurement confirms that hole injection is the mechanism responsible for the preconditioned state that gives a stable value of Vth. The increase in the static leakage Idss current after the preconditioned Vth measurement in the OFF-state and the additional flow of electrons in the leakage current is a direct consequence of the presence of holes coming from the preconditioning sequence. The preconditioning method is currently tested to follow the ageing of devices and shows encouraging results.

Fig. 1 .

 1 Fig. 1. Simplified band diagram of the gate stack of the DUT, the mechanism (1) is electron trapping at the interface AlGaN/GaN, (2) is the hole depletion over the barrier to reach the buffer.

Fig. 2 .

 2 Fig. 2. (a) Observation of the instability of the threshold-voltage measurement after an Idss measurement with Vdss up to 100V (b) Measurement of the current leakage at VGS = 0V of the transistor during the Idss that causes the shift of Vth of (a) (forward and backward sweeps pointed by the arrows). All measurements were done at room temperature 20 • C. The sweep rate is 33V/s.

Fig. 3 .

 3 Fig. 3. Protocol used in measurements on the test system: first in red the gate preconditioning step to obtain reproducible values of Vth, then in orange the actual Vth measurement, in blue another measurement (e.g. Idss on the figure) that could influence the measurement of the threshold voltage.

Fig. 4 .

 4 Fig. 4. Preconditioned-Vth measurement (at VGS,PC = 7V) before and after an Idss measurement for a 100V and a 650V pGaN HEMT.

Fig. 5 .

 5 Fig. 5. Measurements of the threshold voltage before (solid lines with dots) and after (solid lines) an Idss measurement. The curve for the preconditioning of 6 and 7V are the only ones that give precisely the same value of Vth as the curves after and before Idss overlap for these two preconditioning values. For smaller preconditioning values, such as VGSPC= 4V, it can be seen that Idss measurement influences the value of Vth.

Fig. 6 .

 6 Fig. 6. Effect of the voltage applied during the preconditioning step on the measurement of the threshold voltage (all measurements are made after a leakage current Idss) measurement.

Fig. 7 .

 7 Fig. 7. (a) Drain leakage-current measurement (with slower and constant increase of VDS ≈ 3V/s) after a preconditioned-Vth measurement (VGS,PC = 7V), (log scale on (b)), to be compared with Fig.2 (b) in orange. The overall level of drain current has increased abnormally (three orders of magnitude) and a current spike appears at the beginning of the measurement and lasts approximately 20 sec.

Fig. 8 .

 8 Fig. 8. (a) Drain leakage-current measurement after a preconditioned-Vth measurement (VGS,PC = 7V) for the 650V part number. A current spike appears similarly as with the 100V part number the maximum is reached at VDS ≈ 100V. Another change near 150V (less pronounced) appears due to the different field plates layout of this part number. (b) Intrinsic capacitance at VGS = 0V of the transistor under VDS given by the transistor datasheet. The field plates layout is more elaborated for this part number which influences the capacitance variation with VDS .

Fig. 9 .

 9 Fig. 9. Drain leakage current for several values of preconditioning gate voltage applied before, (a) as function of the drain voltage applied, (b) evolution over time of the drain voltage and current. As the sweeping is faster than in Fig.7(33V/s against 3V/s), the additional current leakage spike does not have enough time to be evacuated only on the forward sweep phase. The additional current of the spike is evacuated more rapidly during the measurement with faster sweeping (≈ 12s against 20s in Fig.7), since the charge to be evacuated is the same for the same VGSPC, the maximum current reached is higher at faster sweeping rates.

Fig. 10 .

 10 Fig. 10. Intrinsic capacitances at VGS = 0V of the transistor under VDS given by the transistor datasheet.

Fig. 11 .

 11 Fig. 11. Simplified view of the transistor and its 2D electron gas location for a given VDS = Vdss. The equivalent model of the resistance path is illustrated below. The pink (R0V) and the yellow (R0V<Vdss<18V) resistance are varying, R0V is constant at OFF-state.

Fig. 12 .

 12 Fig. 12. (a) Illustration of the injected holes due to the application of a high VGSPC, (b) effect on the leakage current, the thick reddish arrow illustrates a high leakage flow of electron, the yellow clouds show recombination processes that emit electrons of energy hν that release trapped electron from the buffer and that participate in the increase of the leakage current.

L. Ghizzo et al., LAAS-CNRS, LAPLACE, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France -THALES SIX France SAS, 2023 https://hal.science/

Acknowledgment

This is part of a PhD study on power GaN reliability at Thales Toulouse, LAAS-CNRS and Laplace laboratory in Toulouse and the follow-up of Centre National d'Etudes Spatiales (CNES).

This work was supported by the LAAS-CNRS PROOF platform, partly financed by the Occitanie region.